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500 |
Alternating sign matrices and domino tilings
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math.CO
|
We introduce a family of planar regions, called Aztec diamonds, and study the
ways in which these regions can be tiled by dominoes. Our main result is a
generating function that not only gives the number of domino tilings of the
Aztec diamond of order $n$ but also provides information about the orientation
of the dominoes (vertical versus horizontal) and the accessibility of one
tiling from another by means of local modifications. Several proofs of the
formula are given. The problem turns out to have connections with the
alternating sign matrices of Mills, Robbins, and Rumsey, as well as the square
ice model studied by Lieb.
|
math
|
501 |
A Short Proof of Jacobi's Formula for the Number of Representations of an Integer as a Sum of Four Squares
|
math.CO
|
A short and elementary proof, and a finite-form generalization, are given of
Jacobi's formula for the number of ways of writing an integer as a sum of four
squares (that implies Lagrange's famous 1777 theorem.)
|
math
|
502 |
Theorems for a Price: Tomorrow's Semi-Rigorous Mathematical Culture
|
math.CO
|
The future of mathematics is described, by using the WZ algorithmic proof
theory as a parable.
|
math
|
503 |
A WZ proof of Ramanujan's Formula for Pi
|
math.CO
|
Ramanujan's series for Pi, that appeared in his famous letter to Hardy, is
given a one-line WZ proof.
|
math
|
504 |
Chu's 1303 Identity Implies Bombieri's 1990 Norm-Inequality [via an Identity of Beauzamy and Dégot]
|
math.CO
|
The Vandermonde-Chu Binomial Coefficients Identity is shown to imply
Bombieri's deep norm inequalities, via identities of Beauzamy-D\'egot, and
Reznick.
|
math
|
505 |
Combinatorial Proofs of Capelli's and Turnbull's Identities from Classical Invariant Theory
|
math.CO
|
Capelli's and Turnbull's classical identities are given elegant combinatorial
proofs.
|
math
|
506 |
The Dinitz problem solved for rectangles
|
math.CO
|
The Dinitz conjecture states that, for each $n$ and for every collection of
$n$-element sets $S_{ij}$, an $n\times n$ partial latin square can be found
with the $(i,j)$\<th entry taken from $S_{ij}$. The analogous statement for
$(n-1)\times n$ rectangles is proven here. The proof uses a recent result by
Alon and Tarsi and is given in terms of even and odd orientations of graphs.
|
math
|
507 |
The sandwich theorem
|
math.CO
|
This report contains expository notes about a function $\vartheta(G)$ that is
popularly known as the Lov\'asz number of a graph~$G$. There are many ways to
define $\vartheta(G)$, and the surprising variety of different
characterizations indicates in itself that $\vartheta(G)$ should be
interesting. But the most interesting property of $\vartheta(G)$ is probably
the fact that it can be computed efficiently, although it lies ``sandwiched''
between other classic graph numbers whose computation is NP-hard. I~have tried
to make these notes self-contained so that they might serve as an elementary
introduction to the growing literature on Lov\'asz's fascinating function.
|
math
|
508 |
How Joe Gillis Discovered Combinatorial Special Function Theory
|
math.CO
|
How Enumerative Combinatorics met Special Functions, thanks to Joe Gillis
|
math
|
509 |
The Graphical Major Index
|
math.CO
|
A generalization of the classical statistics ``maj'' and ``inv'' (the major
index and number of inversions) on words is introduced, parameterized by
arbitrary graphs on the underlying alphabet. The question of characterizing
those graphs that lead to equi-distributed "inv" and "maj" is posed and
answered.
|
math
|
510 |
Proof of the Alternating Sign Matrix Conjecture
|
math.CO
|
The number of $n \times n$ matrices whose entries are either -1, 0, or 1,
whose row- and column- sums are all 1, and such that in every row and every
column the non-zero entries alternate in sign, is proved to be $[1!4! >...
(3n-2)!]/[n!(n+1)! ... (2n-1)!]$, as conjectured by Mills, Robbins, and Rumsey.
|
math
|
511 |
A High-School Algebra and high-school (purely formal) calculus,. Wallet-Sized Proof, of the Bieberbach Conjecture [after L. Weinstein]
|
math.CO
|
L. Weinstein's brilliant short proof of de Branges's Theorem is made even
shorter by using computer algebra.
|
math
|
512 |
Counting pairs of lattice paths by intersections
|
math.CO
|
On an $r\times (n-r)$ lattice rectangle, we first consider walks that begin
at the SW corner, proceed with unit steps in either of the directions E or N,
and terminate at the NE corner of the rectangle. For each integer $k$ we ask
for $N_k^{n,r}$, the number of {\em ordered\/} pairs of these walks that
intersect in exactly $k$ points. The number of points in the intersection of
two such walks is defined as the cardinality of the intersection of their two
sets of vertices, excluding the initial and terminal vertices. We find two
explicit formulas for the numbers $N_k^{n,r}$. Next we note that $N_1^{n,r}= 2
N_0^{n,r}$, i.e., that {\em exactly twice as many pairs of walks have a single
intersection as have no intersection\/}. Such a relationship clearly merits a
bijective proof, and we supply one. We discuss a number of related results for
different assumptions on the two walks. We find the probability that two
independent walkers on a given lattice rectangle do not meet. In this
situation, the walkers start at the two points $(a,b+x+1)$ and (a+x+1,b)$ in
the first quadrant, and walk West or South at each step, except that when a
walker reaches the $x$-axis (resp. the $y$-axis) then all future steps are
constrained to be South (resp. West) until the origin is reached. We find that
if the probability $p(i,j)$ that a step from $(i,j)$ will go West depends only
on $i+j$, then the probabilty that the two walkers do not meet until they reach
the origin is the same as the probability that a single (unconstrained) walker
who starts at $(a, b+x+1)$ and and takes $a+b+x$ steps, finishes at one of the
points $(0,1), (-1,2), \ldots, (-x,1+x)$.
|
math
|
513 |
Inverting sets and the packing problem
|
math.CO
|
Given a set $V$, a subset $S$, and a permutation $\pi$ of $V$, we say that
$\pi$ permutes $S$ if $\pi (S) \cap S = \emptyset$. Given a collection $\cS =
\{V; S_1,\ldots , S_m\}$, where $S_i \subseteq V ~~(i=1,\ldots ,m)$, we say
that $\cS$ is invertible if there is a permutation $\pi$ of $V$ such that $\pi
(S_i) \subseteq V-S_i$. In this paper, we present necessary and sufficient
conditions for the invertibility of a collection and construct a polynomial
algorithm which determines whether a given collection is invertible. For an
arbitrary collection, we give a lower bound for the maximum number of sets that
can be inverted. Finally, we consider the problem of constructing a collection
of sets such that no sub-collection of size three is invertible. Our
constructions of such collections come from solutions to the packing problem
with unbounded block sizes. We prove several new lower and upper bounds for the
packing problem and present a new explicit construction of packing.
|
math
|
514 |
Invertible families of sets of bounded degree
|
math.CO
|
Let H = (H,V) be a hypergraph with edge set H and vertex set V. Then
hypergraph H is invertible iff there exists a permutation pi of V such that for
all E belongs to H(edges) intersection of(pi(E) and E)=0. H is invertibility
critical if H is not invertible but every hypergraph obtained by removing an
edge from H is invertible. The degree of H is d if |{E belongs to H(edges)|x
belongs to E}| =< d for each x belongs to V Let i(d) be the maximum number of
edges of an invertibility critical hypergraph of degree d. Theorem: i(d) =<
(d-1) {2d-1 choose d} + 1. The proof of this result leads to the following
covering problem on graphs: Let G be a graph. A family H is subset of (2^{V(G)}
is an edge cover of G iff for every edge e of G, there is an E belongs to
H(edge set) which includes e. H(edge set) is a minimal edge cover of G iff for
H' subset of H, H' is not an edge cover of G. Let b(d) (c(d)) be the maximum
cardinality of a minimal edge cover H(edge set) of a complete bipartite graph
(complete graph) where H(edge set) has degree d. Theorem: c(d)=< i(d)=<b(d)=<
c(d+1) and 3. 2^{d-1} - 2 =< b(d)=< (d-1) {2d-1choose d} +1. The proof of this
result uses Sperner theory. The bounds b(d) also arise as bounds on the maximum
number of elements in the union of minimal covers of families of sets.
|
math
|
515 |
Graph generated union-closed families of sets
|
math.CO
|
Let G be a graph with vertices V and edges E. Let F be the union-closed
family of sets generated by E. Then F is the family of subsets of V without
isolated points. Theorem: There is an edge e belongs to E such that |{U belongs
to F | e belongs to U}| =< 1/2|F|. This is equivalent to the following
assertion: If H is a union-closed family generated by a family of sets of
maximum degree two, then there is an $x$ such that |{U belongs to H | x belongs
to U}| > 1/2|H|. This is a special case of the union-closed sets conjecture. To
put this result in perspective, a brief overview of research on the
union-closed sets conjecture is given. A proof of a strong version of the
theorem on graph-generated families of sets is presented. This proof depends on
an analysis of the local properties of F and an application of Kleitman's
lemma. Much of the proof applies to arbitrary union-closed families and can be
used to obtain bounds on |{U belongs to F | e belongs to U}|/|F|.
|
math
|
516 |
A geometric identity for Pappus' Theorem
|
math.CO
|
An expression in the exterior algebra of a Peano space yielding Pappus'
Theorem was originally given by Doubilet, Rota, and Stein. Motivated by an
identity of Rota, we give an identity in a Grassmann-Cayley algebra of step 3,
involving joins and meets alone, which expresses the Theorem of Pappus.
|
math
|
517 |
Restricted routing and wide diameter of the cycle prefix network
|
math.CO
|
The cycle prefix network is a Cayley coset digraph based on sequences over an
alphabet which has been proposed as a vertex symmetric communication network.
This network has been shown to have many remarkable communication properties
such as a large number of vertices for a given degree and diameter, simple
shortest path routing, Hamiltonicity, optimal connectivity, and others. These
considerations for designing symmetric and directed interconnection networks
are well justified in practice and have been widely recognized in the research
community. Among the important properties of a good network, efficient routing
is probably one of the most important. In this paper, we further study routing
schemes in the cycle prefix network. We confirm an observation first made from
computer experiments regarding the diameter change when certain links are
removed in the original network, and we completely determine the wide diameter
of the network. The wide diameter of a network is now perceived to be even more
important than the diameter. We show by construction that the wide diameter of
the cycle prefix network is very close to the ordinary diameter. This means
that routing in parallel in this network costs little extra time compared to
ordinary single path routing.
|
math
|
518 |
The lattice of closure relations of a poset
|
math.CO
|
In this paper we show that the set of closure relations on a finite poset P
forms a supersolvable lattice, as suggested by Rota. Furthermore this lattice
is dually isomorphic to the lattice of closed sets in a convex geometry (in the
sense of Edelman and Jamison). We also characterize the modular elements of
this lattice and compute its characteristic polynomial.
|
math
|
519 |
Spanning trees short or small
|
math.CO
|
We study the problem of finding small trees. Classical network design
problems are considered with the additional constraint that only a specified
number $k$ of nodes are required to be connected in the solution. A
prototypical example is the $k$MST problem in which we require a tree of
minimum weight spanning at least $k$ nodes in an edge-weighted graph. We show
that the $k$MST problem is NP-hard even for points in the Euclidean plane. We
provide approximation algorithms with performance ratio $2\sqrt{k}$ for the
general edge-weighted case and $O(k^{1/4})$ for the case of points in the
plane. Polynomial-time exact solutions are also presented for the class of
decomposable graphs which includes trees, series-parallel graphs, and bounded
bandwidth graphs, and for points on the boundary of a convex region in the
Euclidean plane. We also investigate the problem of finding short trees, and
more generally, that of finding networks with minimum diameter. A simple
technique is used to provide a polynomial-time solution for finding $k$-trees
of minimum diameter. We identify easy and hard problems arising in finding
short networks using a framework due to T. C. Hu.
|
math
|
520 |
Optimal pooling designs with error detection
|
math.CO
|
Consider a collection of objects, some of which may be `bad', and a test
which determines whether or not a given sub-collection contains no bad objects.
The non-adaptive pooling (or group testing) problem involves identifying the
bad objects using the least number of tests applied in parallel. The
`hypergeometric' case occurs when an upper bound on the number of bad objects
is known {\em a priori}. Here, practical considerations lead us to impose the
additional requirement of {\em a posteriori} confirmation that the bound is
satisfied. A generalization of the problem in which occasional errors in the
test outcomes can occur is also considered. Optimal solutions to the general
problem are shown to be equivalent to maximum-size collections of subsets of a
finite set satisfying a union condition which generalizes that considered by
Erd\"os \etal \cite{erd}. Lower bounds on the number of tests required are
derived when the number of bad objects is believed to be either 1 or 2. Steiner
systems are shown to be optimal solutions in some cases.
|
math
|
521 |
A simple linear-time algorithm for finding path-decompositions of small width
|
math.CO
|
We described a simple algorithm running in linear time for each fixed
constant $k$, that either establishes that the pathwidth of a graph $G$ is
greater than $k$, or finds a path-decomposition of $G$ of width at most
$O(2^{k})$. This provides a simple proof of the result by Bodlaender that many
families of graphs of bounded pathwidth can be recognized in linear time.
|
math
|
522 |
Symmetries of plane partitions and the permanent-determinant method
|
math.CO
|
In the paper [J. Combin. Theory Ser. A 43 (1986), 103--113], Stanley gives
formulas for the number of plane partitions in each of ten symmetry classes.
This paper together with results by Andrews [J. Combin. Theory Ser. A 66
(1994), 28-39] and Stembridge [Adv. Math 111 (1995), 227-243] completes the
project of proving all ten formulas.
We enumerate cyclically symmetric, self-complementary plane partitions. We
first convert plane partitions to tilings of a hexagon in the plane by
rhombuses, or equivalently to matchings in a certain planar graph. We can then
use the permanent-determinant method or a variant, the Hafnian-Pfaffian method,
to obtain the answer as the determinant or Pfaffian of a matrix in each of the
ten cases. We row-reduce the resulting matrix in the case under consideration
to prove the formula. A similar row-reduction process can be carried out in
many of the other cases, and we analyze three other symmetry classes of plane
partitions for comparison.
|
math
|
523 |
New large graphs with given degree and diameter
|
math.CO
|
In this paper we give graphs with the largest known order for a given degree
$\Delta$ and diameter $D$. The graphs are constructed from Moore bipartite
graphs by replacement of some vertices by adequate structures. The paper also
contains the latest version of the $(\Delta, D)$ table for graphs.
|
math
|
524 |
Generalized degrees and densities for families of sets
|
math.CO
|
Let F be a family of subsets of {1,2,...,n}. The width-degree of an element x
in at least one member of F is the width of the family {U in F | x in U}. If F
has maximum width-degree at most k, then F is locally k-wide. Bounds on the
size of locally k-wide families of sets are established. If F is locally k-wide
and centered (every U in F has an element which does not belong to any member
of F incomparable to U), then |F| <= (k+1)(n-k/2); this bound is best possible.
Nearly exact bounds, linear in n and k, on the size of locally k-wide families
of arcs or segments are determined. If F is any locally k-wide family of sets,
then |F| is linearly bounded in n. The proof of this result involves an
analysis of the combinatorics of antichains. Let P be a poset and L a
semilattice (or an intersection-closed family of sets). The P-size of L is
|L^P|. For u in L, the P-density of u is the ratio |[u)^P|/|L^P|. The density
of u is given by the [1]-density of u. Let p be the number of filters of P. L
has the P-density property iff there is a join-irreducible a in L such that the
P-density of a is at most 1/p Which non-trivial semilattices have the P-density
property? For P=[1], it has been conjectured that the answer is: "all" (the
union-closed sets conjecture). Certain subdirect products of lower-semimodular
lattices and, for P=[n], of geometric lattices have the P-density property in a
strong sense. This generalizes some previously known results. A fixed lattice
has the [n]-density property if n is large enough. The density of a generator U
of a union-closed family of sets L containing the empty set is estimated. The
estimate depends only on the local properties of L at U. If L is generated by
sets of size at most two, then there is a generator U of L with estimated
density at most 1/2.
|
math
|
525 |
Notes on the connectivity of Cayley coset digraphs
|
math.CO
|
Hamidoune's connectivity results for hierarchical Cayley digraphs are
extended to Cayley coset digraphs and thus to arbitrary vertex transitive
digraphs. It is shown that if a Cayley coset digraph can be hierarchically
decomposed in a certain way, then it is optimally vertex connected. The results
are obtained by extending the methods used by Hamidoune. They are used to show
that cycle-prefix graphs are optimally vertex connected. This implies that
cycle-prefix graphs have good fault tolerance properties.
|
math
|
526 |
Algorithms for learning and teaching sets of vertices in graphs
|
math.CO
|
The learning complexity of special sets of vertices in graphs is studied in
the model(s) of exact learning by (extended) equivalence and membership
queries. Polynomial-time learning algorithms are described for vertex covers,
independent sets, and dominating sets. The complexity of learning vertex sets
of fixed size is also investigated, and it is shown that the k-element vertex
covers in a graph can be learned in a number of rounds of interaction that is
independent of the size of the graph. Apart from the elegance of these
algorithmic problems, the chief motivation is the surprising recently
established connection between the important unsolved problem of the learning
complexity of CNF (or DNF) formulas and the learning complexity of dominating
sets. The complexity of teaching sets of vertices in graphs is also considered.
|
math
|
527 |
Self-complementary plane partitions by Proctor's minuscule method
|
math.CO
|
A method of Proctor [European J. Combin. 5 (1984), no. 4, 331-350] realizes
the set of arbitrary plane partitions in a box and the set of symmetric plane
partitions as bases of linear representations of Lie groups. We extend this
method by realizing transposition and complementation of plane partitions as
natural linear transformations of the representations, thereby enumerating
symmetric plane partitions, self-complementary plane partitions, and
transpose-complement plane partitions in a new way.
|
math
|
528 |
Leaper graphs
|
math.CO
|
An $\{r,s\}$-leaper is a generalized knight that can jump from $(x,y)$ to
$(x\pm r,y\pm s)$ or $(x\pm s,y\pm r)$ on a rectangular grid. The graph of an
$\{r,s\}$-leaper on an $m\times n$ board is the set of $mn$~vertices $(x,y)$
for $0\leq x<m$ and $0\leq y<n$, with an edge between vertices that are one
$\{r,s\}$-leaper move apart. We call $x$ the {\it rank} and $y$ the {\it file}
of board position $(x,y)$. George~P. Jelliss raised several interesting
questions about these graphs, and established some of their fundamental
properties. The purpose of this paper is to characterize when the graphs are
connected, for arbitrary~$r$ and~$s$, and to determine the smallest boards with
Hamiltonian circuits when $s=r+1$ or $r=1$.
|
math
|
529 |
The degree-diameter problem for several varieties of Cayley graphs, I: the Abelian case
|
math.CO
|
We address the degree-diameter problem for Cayley graphs of Abelian groups
(Abelian graphs), both directed and undirected. The problem turns out to be
closely related to the problem of finding efficient lattice coverings of
Euclidean space by shapes such as octahedra and tetrahedra; we exploit this
relationship in both directions. In particular, we find the largest Abelian
graphs with 2 generators (dimensions) and a given diameter. (The results for 2
generators are not new; they are given in the literature of distributed loop
networks.) We find an undirected Abelian graph with 3 generators and a given
diameter which we conjecture to be as large as possible; for the directed case,
we obtain partial results, which lead to unusual lattice coverings of 3-space.
We discuss the asymptotic behavior of the problem for large numbers of
generators. The graphs obtained here are substantially better than traditional
toroidal meshes, but, in the simpler undirected cases, retain certain desirable
features such as good routing algorithms, easy constructibility, and the
ability to host mesh-connected numerical algorithms without any increase in
communication times.
|
math
|
530 |
A new series of dense graphs of high girth
|
math.CO
|
Let $k\ge 1$ be an odd integer, $t=\lfloor {{k+2}\over 4}\rfloor$, and $q$ be
a prime power. We construct a bipartite, $q$-regular, edge-transitive graph
$C\!D(k,q)$ of order $v \le 2q^{k-t+1}$ and girth $g \ge k+5$. If $e$ is the
the number of edges of $C\!D(k,q)$, then $e =\Omega(v^{1+ {1\over {k-t+1}}})$.
These graphs provide the best known asymptotic lower bound for the greatest
number of edges in graphs of order $v$ and girth at least $g$, $ g\ge 5$, $g
\not= 11,12$. For $g\ge 24$, this represents a slight improvement on bounds
established by Margulis and Lubotzky, Phillips, Sarnak; for $5\le g\le 23$,
$g\not= 11,12$, it improves on or ties existing bounds.
|
math
|
531 |
Aztec diamonds, checkerboard graphs, and spanning trees
|
math.CO
|
This note derives the characteristic polynomial of a graph that represents
nonjump moves in a generalized game of checkers. The number of spanning trees
is also determined.
|
math
|
532 |
Recent contributions to the calculus of finite differences: a survey
|
math.CO
|
We retrace the recent history of the Umbral Calculus. After studying the
classic results concerning polynomial sequences of binomial type, we generalize
to a certain type of logarithmic series. Finally, we demonstrate numerous
typical examples of our theory.
Nous passons en revue ici les resultats recents du calcul ombral. Nous nous
interessons tout d'abord aux resultats classique appliqu\'es aux suites de
polyn\^omes de type binomial, pius elargions le champ d'\'etude aux series
logarithmiques. Enfin nous donnons de nombreaux exemples types d'application de
cette th\'eorie.
|
math
|
533 |
DX-operator expansion
|
math.CO
|
We characterize those linear operators that can be expressed as a sum over k
of terms of the form f_k(D) x^k and give several examples.
|
math
|
534 |
Proof of a conjecture of Narayana on dominance refinements of the Smirnov two-sample test
|
math.CO
|
We prove the following conjecture of Narayana: there are no dominance
refinements of the Smirnov two-sample test if and only if the two sample sizes
are relatively prime.
|
math
|
535 |
Maple umbral calculus package
|
math.CO
|
We are developing a Maple package of functions related to Rota's Umbral
Calculus. A Mathematica version of this package is being developed in parallel.
|
math
|
536 |
Getting results with negative thinking
|
math.CO
|
Given a universe of discourse $U$, a {\em multiset} can be thought of as a
function $M$ from $U$ to the natural numbers ${\bf N}$. In this paper, we
define a {\em hybrid set} to be any function from the universe $U$ to the
integers ${\bf Z}$. These sets are called hybrid since they contain elements
with either a positive or negative multiplicity. Our goal is to use these
hybrid sets {\em as if} they were multisets in order to adequately generalize
certain combinatorial facts which are true classically only for nonnegative
integers.
|
math
|
537 |
A simpler characterization of Sheffer polynomial
|
math.CO
|
We characterize the Sheffer sequences by a single convolution identity $$
F^{(y)} p_{n}(x) = \sum _{k=0}^{n}\ p_{k}(x)\ p_{n-k}(y)$$ where $F^{(y)}$ is a
shift-invariant operator. We then study a generalization of the notion of
Sheffer sequences by removing the requirement that $F^{(y)}$ be
shift-invariant. All these solutions can then be interpreted as cocommutative
coalgebras. We also show the connection with generalized translation operators
as introduced by Delsarte. Finally, we apply the same convolution to symmetric
functions where we find that the ``Sheffer'' sequences differ from ordinary
full divided power sequences by only a constant factor.
|
math
|
538 |
Series with general exponents
|
math.CO
|
We define the Artinian and Noetherian algebra which consist of formal series
involving exponents which are not necessarily integers. All of the usual
operations are defined here and characterized. As an application, we compute
the algebra of symmetric functions with nonnegative real exponents. The
applications to logarithmic series and the Umbral calculus are deferred to
another paper.
On d\'efinit ici les alg\`ebres Artinienne et Noetherienne comme \'etant des
alg\`ebres constitu\'ees des s\'eries formelles \`a exposants pas
n\'ecessairement entiers. On definit sur ces alg\`ebres toutes les op\'erations
classiques et on les caracterise. Comme exemple d'exploitation de cette
th\'eorie, on s'interesse \`a alg\`ebre de fonctions sym\'etriques à
exponsants r\`eels en nonn\'egatifs. Une autre publication est consacr\'ee aux
applications aux series logarithmiques et au calcul ombral.
|
math
|
539 |
A generalization of Stirling numbers
|
math.CO
|
We generalize the Stirling numbers of the first kind $s(a,k)$ to the case
where $a$ may be an arbitrary real number. In particular, we study the case in
which $a$ is an integer. There, we discover new combinatorial properties held
by the classical Stirling numbers, and analogous properties held by the
Stirling numbers $s(n,k)$ with $n$ a negative integer.
On g\'{e}n\'{e}ralise ici les nombres de Stirling du premier ordre $s(a,k)$
au cas o\`u $a$ est un r\'eel quelconque. On s'interesse en particulier au cas
o\`u $a$ est entier. Ceci permet de mettre en evidence de nouvelles
propri\'et\'es combinatoires aux quelles obeissent les nombres de Stirling
usuels et des propri\'et\'es analougues auquelles obeissent les nombres de
Stirling $s(n,k)$ o\`u $n$ est un entier n\`egatif.
|
math
|
540 |
A generalization of the binomial coefficients
|
math.CO
|
We pose the question of what is the best generalization of the factorial and
the binomial coefficient. We give several examples, derive their combinatorial
properties, and demonstrate their interrelationships.
On cherche ici \`a d\'eterminer est la meilleure g\'en\'eralisation possible
des factorielles et des coefficients du bin\^oome. On s'interesse \`a plusieurs
exemples, \`a leurs propri\'et\'es combinatoires, et aux differentes relations
qu'ils mettent en jeu.
|
math
|
541 |
The iterated logarithmic algebra
|
math.CO
|
We generalize the Umbral Calculus of G-C. Rota by studying not only sequences
of polynomials and inverse power series, or even the logarithms studied in, but
instead we study sequences of formal expressions involving the iterated
logarithms and x to an arbitrary real power.
Using a theory of formal power series with real exponents, and a more general
definition of factorial, binomial coefficient, and Stirling numbers to all the
real numbers, we define the Iterated Logarithmic Algebra I. Its elements are
the formal representations of the asymptotic expansions of a large class of
real functions, and we define the harmonic logarithm basis of I which will be
interpreted as a generalization of the powers x^n since it behaves nicely with
respect to the derivative
We classify all operators over I which commute with the derivative
(classically these are known as shift-invariant operators), and formulate
several equivalent definitions of a sequence of binomial type. We then derive
many formulas useful towards the calculation of these sequences including the
Recurrence Formula, the Transfer Formula, and the Lagrange Inversion Formula.
Finally, we study Sheffer sequences, and give many examples.
|
math
|
542 |
The iterated logarithmic algebra II: Sheffer sequences
|
math.CO
|
An extension of the theory of the Iterated Logarithmic Algebra gives the
logarithmic analog of a Sheffer or Appell sequence of polynomials. This leads
to several examples including Stirling's formula and a logarithmic version of
the Euler-MacLaurin summation formula.
Gr\^ace \`a une g\'en\'eralisation de la th\'eorie de l'alg\`ebre des
logarithmes it\'er\'es, on definit un analogue logarithmique des suites de
polyn\^omes de Sheffer et d'Appell. Quelques exemples d'applications permettent
de d\'eduire la formule de Stirling ainsi qu'un version logarithmique de la
formule de sommation de Euler--MacLaurin.
|
math
|
543 |
Sequences of symmetric functions of binomial type
|
math.CO
|
We take advantage of the combinatorial interpretations of many sequences of
polynomials of binomial type to define a sequence of symmetric functions
corresponding to each sequence of polynomials of binomial type. We derive many
of the results of Umbral Calculus in this context including a Taylor's
expansion and a binomial identity for symmetric functions. Surprisingly, the
delta operators for all the sequences of binomial type correspond to the same
operator on symmetric functions.
On s'appuie ici sur les interpr\'etations combinatoires de nombreuses suites
de polyn\^omes de type binomial pour d\'efinir une suite de fonctions
sym\'etriques associ\'ee \`a chque suite de polyn\^omes de type binomial. On
retrouve dans ce cadre, de nombreaux r\'esultats du calcul ombral, en
particulier une version de la formule de Taylor et la formule d'identit\'e du
bin\^ome pour les fonctions sym\'etriques. On s'aper\oit que les op\'erateurs
differentiels de degr\'e un pour toutes les suite de polyn\^omes de type a
binomial correspondent \`a un op\'erateur unique sur les fonction
sym\'etriques.
|
math
|
544 |
Richman games
|
math.CO
|
A Richman game is a combinatorial game in which, rather than alternating
moves, the two players bid for the privilege of making the next move. We
consider both the case where the players pay each other and the case where the
players pay a neutral third party. We find optimal strategies considering both
the case where the players know how much money their opponent has and the case
where they do not.
|
math
|
545 |
A new proof of Monjardet's median theorem
|
math.CO
|
New proofs are given for Monjardet's theorem that all strong simple games
(i.e., ipsodual elements of the free distributive lattice) can be generated by
the median operation. Tighter limits are placed on the number of iterations
necessary. Comparison is drawn with the $\chi$ function which also generates
all strong simple games.
|
math
|
546 |
Symmetric chain decompositions of B_n and Pi_n
|
math.CO
|
We review the Green/Kleitman/Leeb interpretation of de Bruijn's symmetric
chain decomposition of ${\cal B}_{n}$, and explain how it can be used to find a
maximal collection of disjoint symmetric chains in the nonsymmetric lattice of
partitions of a set.
|
math
|
547 |
The combinatorics of Mancala-type games: Ayo, Tchoukaitlon, and 1/pi
|
math.CO
|
Certain endgame considerations in the two-player Nigerian Mancala-type game
Ayo can be identified with the problem of finding winning positions in the
solitaire game Tchoukaitlon. The periodicity of the pit occupancies in $s$
stone winning positions is determined. Given $n$ pits, the number of stones in
a winning position is found to be asymptotically bounded by $n^{2}/\pi$.
|
math
|
548 |
Obstructions to within a few vertices or edges of acyclic
|
math.CO
|
Finite obstruction sets for lower ideals in the minor order are guaranteed to
exist by the Graph Minor Theorem. It has been known for several years that, in
principle, obstruction sets can be mechanically computed for most natural lower
ideals. In this paper, we describe a general-purpose method for finding
obstructions by using a bounded treewidth (or pathwidth) search. We illustrate
this approach by characterizing certain families of cycle-cover graphs based on
the two well-known problems: $k$-{\sc Feedback Vertex Set} and $k$-{\sc
Feedback Edge Set}. Our search is based on a number of algorithmic strategies
by which large constants can be mitigated, including a randomized strategy for
obtaining proofs of minimality.
|
math
|
549 |
A finite partition theorem with double exponential bounds
|
math.CO
|
We prove that double exponentiation is an upper bound to Ramsey theorem for
colouring of pairs when we want to predetermine the order of the differences of
successive members of the homogeneous set.
|
math
|
550 |
The Knowlton-Graham partition problem
|
math.CO
|
A set partition technique that is useful for identifying wires in cables can
be recast in the language of 0--1 matrices, thereby resolving an open problem
stated by R.~L. Graham in Volume 1 of this journal. The proof involves a
construction of 0--1 matrices having row and column sums without gaps.
|
math
|
551 |
New constructions for covering designs
|
math.CO
|
A $(v,k,t)$ {\em covering design}, or {\em covering}, is a family of
$k$-subsets, called blocks, chosen from a $v$-set, such that each $t$-subset is
contained in at least one of the blocks. The number of blocks is the covering's
{\em size}, and the minimum size of such a covering is denoted by $C(v,k,t)$.
This paper gives three new methods for constructing good coverings: a greedy
algorithm similar to Conway and Sloane's algorithm for lexicographic
codes~\cite{lex}, and two methods that synthesize new coverings from
preexisting ones. Using these new methods, together with results in the
literature, we build tables of upper bounds on $C(v,k,t)$ for $v \leq 32$, $k
\leq 16$, and $t \leq 8$.%
|
math
|
552 |
Scenic trails ascending from sea-level Nim to alpine chess
|
math.CO
|
Aim: Present a systematic development of part of the theory of combinatorial
games from the ground up.
Approach: Computational complexity. Combinatorial games are completely
determined; the questions of interest are efficiencies of strategies.
Methodology: Divide and conquer. Ascend from Nim to chess in small strides at
a gradient that's not too steep.
Presentation: Informal; examples of games sampled from various strategic
viewing points along scenic mountain trails, which illustrate the theory.
|
math
|
553 |
Partitioned tensor products and their spectra
|
math.CO
|
A pleasant family of graphs defined by Godsil and McKay is shown to have
easily computed eigenvalues in many cases.
|
math
|
554 |
Overlapping Pfaffians
|
math.CO
|
A combinatorial construction proves an identity for the product of the
Pfaffian of a skew-symmetric matrix by the Pfaffian of one of its submatrices.
Several applications of this identity are followed by a brief history of
Pfaffians.
|
math
|
555 |
Error-correcting codes derived from combinatorial games
|
math.CO
|
The ``losing positions" of certain combinatorial games constitute linear
error detecting and correcting codes. We show that a large class of games that
can be cast in the form of *annihilation games*, provides a potentially
polynomial method for computing codes (*anncodes*). We also give a short proof
of the basic properties of the previously known *lexicodes*, which are defined
by means of an exponential algorithm, and are related to game theory. The set
of lexicodes is seen to constitute a subset of the set of anncodes. In the
final section we indicate, by means of an example, how the method of producing
lexicodes can be applied optimally to find anncodes. Some extensions are
indicated.
|
math
|
556 |
Algebraic constructions of efficient broadcast networks
|
math.CO
|
Cayley graph techniques are introduced for the problem of constructing
networks having the maximum possible number of nodes, among networks that
satisfy prescribed bounds on the parameters maximum node degree and broadcast
diameter. The broadcast diameter of a network is the maximum time required for
a message originating at a node of the network to be relayed to all other
nodes, under the restriction that in a single time step any node can
communicate with only one neighboring node. For many parameter values these
algebraic methods yield the largest known constructions, improving on previous
graph-theoretic approaches. It has previously been shown that hypercubes are
optimal for degree $k$ and broadcast diameter $k$. A construction employing
dihedral groups is shown to be optimal for degree $k$ and broadcast diameter
$k+1$.
|
math
|
557 |
New results for the degree/diameter problem
|
math.CO
|
The results of computer searches for large graphs with given (small) degree
and diameter are presented. The new graphs are Cayley graphs of semidirect
products of cyclic groups and related groups. One fundamental use of our
``dense graphs'' is in the design of efficient communication network
topologies.
|
math
|
558 |
Self Avoiding Walks, the Language of Science, and Fibonacci Numbers
|
math.CO
|
The Bordelaise philosophy, or rather a juvenile version of it, is used to
enumerate self avoiding walks in a $[0,1] \times (- \infty, \infty)$.
|
math
|
559 |
The Method of Undetermined Generalization and Specialization Illustrated with Fred Galvin's Amazing Proof of the Dinitz Conjecture
|
math.CO
|
Fred Galvin's amazing proof of the Dinitiz conjecture is used to illustrate
the method of undetermined generalization and specialization.
|
math
|
560 |
Four symmetry classes of plane partitions under one roof
|
math.CO
|
In previous paper, the author applied the permanent-determinant method of
Kasteleyn and its non-bipartite generalization, the Hafnian-Pfaffian method, to
obtain a determinant or a Pfaffian that enumerates each of the ten symmetry
classes of plane partitions. After a cosmetic generalization of the Kasteleyn
method, we identify the matrices in the four determinantal cases (plain plane
partitions, cyclically symmetric plane partitions, transpose-complement plane
partitions, and the intersection of the last two types) in the representation
theory of sl(2,C). The result is a unified proof of the four enumerations.
|
math
|
561 |
On non-even digraphs and symplectic pairs
|
math.CO
|
A digraph $D$ is called {\bf noneven} if it is possible to assign weights of
0,1 to its arcs so that $D$ contains no cycle of even weight. A noneven digraph
$D$ corresponds to one or more nonsingular sign patterns. Given an $n \times n$
sign pattern $H$, a {\bf symplectic pair} in $Q(H)$ is a pair of matrices
$(A,D)$ such that $A \in Q(H)$, $D \in Q(H)$, and $A^T D = I$. An unweighted
digraph $D$ allows a matrix property $P$ if at least one of the sign patterns
whose digraph is $D$ allows $P$. Thomassen characterized the noneven,
2-connected symmetric digraphs (i.e., digraphs for which the existence of arc
$(u,v)$ implies the existence of arc $(v ,u))$. In the first part of our paper,
we recall this characterization and use it to determine which strong symmetric
digraphs allow symplectic pairs. A digraph $D$ is called {\bf semi-complete}
if, for each pair of distinct vertices $(u,v)$, at least one of the arcs
digraph. In the second part of our paper, we fill a gap in these two
characterizations and present and prove correct versions of the main theorems
involved. We then pr oceed to determine which digraphs from these classes allow
symplectic pairs. $(u,v)$ and $(v,u)$ exists in $D$. Thomassen presented a
characterization of two classes of strong, noneven digraphs: the semi-complete
class and the class of digraphs for which each vertex has total degree which
exceeds or equals the size of the digraph. In the second part of our paper, we
fill a gap in these two characterizations and present and prove correct
versions of the main theorems involved. We then p oceed to determine which
digraphs from these classes allow symplectic pairs.
|
math
|
562 |
Reverend Charles to the aid of Major Percy and Fields-Medalist Enrico
|
math.CO
|
Dodgson's determinant condensation rule is shown to immediately imply the
evaluation of MacMahon's determinant expression that leads to the Box Theorem.
|
math
|
563 |
Finite canonization
|
math.CO
|
The canonization theorem says that for given m,n for some m^* (the first one
is called ER(n;m)) we have: for every function f with domain [{1, ...,m^*}]^n,
for some A in [{1, ...,m^*}]^m, the question of when the equality f({i_1,
...,i_n})=f({j_1, ...,j_n}) (where i_1< ... <i_n and j_1 < ... < j_n are from
A) holds has the simplest answer: for some v subseteq {1, ...,n} the equality
holds iff (for all l in v)(i_l = j_l).
In this paper we improve the bound on ER(n,m) so that fixing n the number of
exponentiation needed to calculate ER(n,m) is best possible.
|
math
|
564 |
Asymptotically optimal covering designs
|
math.CO
|
A (v,k,t) covering design, or covering, is a family of k-subsets, called
blocks, chosen from a v-set, such that each t-subset is contained in at least
one of the blocks. The number of blocks is the covering's size}, and the
minimum size of such a covering is denoted by C(v,k,t). It is easy to see that
a covering must contain at least (v choose t)/(k choose t) blocks, and in 1985
R\"odl [European J. Combin. 5 (1985), 69-78] proved a long-standing conjecture
of Erd\H{o}s and Hanani [Publ. Math. Debrecen 10 (1963), 10-13] that for fixed
k and t, coverings of size (v choose t)/(k choose t) (1+o(1)) exist (as v \to
\infty).
An earlier paper by the first three authors [J. Combin. Des. 3 (1995),
269-284] gave new methods for constructing good coverings, and gave tables of
upper bounds on C(v,k,t) for small v, k, and t. The present paper shows that
two of those constructions are asymptotically optimal: For fixed k and t, the
size of the coverings constructed matches R\"odl's bound. The paper also makes
the o(1) error bound explicit, and gives some evidence for a much stronger
bound.
|
math
|
565 |
An Explicit Formula for the Number of Solutions of X^2=0 in Triangular Matrices over a Finite Field
|
math.CO
|
We prove an explicit formula for the number of $n \times n$ upper triangular
matrices, over $GF(q)$, whose square is the zero matrix. This formula was
recently conjectured by Sasha Kirillov and Anna Melnikov[KM].
|
math
|
566 |
A Tverberg-type result on multicolored simplices
|
math.CO
|
Let $P_1, P_2,\ldots, P_{d+1}$ be pairwise disjoint $n$-element point sets in
general position in $d$-space. It is shown that there exist a point $O$ and
suitable subsets $Q_i\subseteq P_i \; (i=1, 2, \ldots, d+1)$ such that
$|Q_i|\geq c_d|P_i|$, and every $d$-dimensional simplex with exactly one vertex
in each $Q_i$ contains $O$ in its interior. Here $c_d$ is a positive constant
depending only on $d$.
|
math
|
567 |
Proof of the Refined Alternating Sign Matrix Conjecture
|
math.CO
|
Mills, Robbins, and Rumsey conjectured, and Zeilberger proved, that the
number of alternating sign matrices of order $n$ equals $A(n):={{1!4!7! ...
(3n-2)!} \over {n!(n+1)! ... (2n-1)!}}$. Mills, Robbins, and Rumsey also made
the stronger conjecture that the number of such matrices whose (unique) `1' of
the first row is at the $r^{th}$ column, equals $A(n) {{n+r-2} \choose
{n-1}}{{2n-1-r} \choose {n-1}}/ {{3n-2} \choose {n-1}}$. Standing on the
shoulders of A.G. Izergin, V. E. Korepin, and G. Kuperberg, and using in
addition orthogonal polynomials and $q$-calculus, this stronger conjecture is
proved.
|
math
|
568 |
Irredundant intervals
|
math.CO
|
This expository note presents simplifications of a theorem due to Gy\H{o}ri
and an algorithm due to Franzblau and Kleitman: Given a family $F$ of $m$
intervals on a linearly ordered set of $n$ elements, we can construct in
$O(m+n)^2$ steps an irredundant subfamily having maximum cardinality, as well
as a generating family having minimum cardinality. The algorithm is of special
interest because it solves a problem analogous to finding a maximum independent
set, but on a class of objects that is more general than a matroid. This note
is also a complete, runnable computer program, which can be used for
experiments in conjunction with the public-domain software of {\sl The Stanford
GraphBase}.
|
math
|
569 |
Combinatorics and topology of stratifications of the space of monic polynomials with real coefficients
|
math.CO
|
We study the stratification of the space of monic polynomials with real
coefficients according to the number and multiplicities of real zeros. In the
first part, for each of these strata we provide a purely combinatorial chain
complex calculating (co)homology of its one-point compactification and describe
the homotopy type by order complexes of a class of posets of compositions. In
the second part, we determine the homotopy type of the one-point
compactification of the space of monic polynomials of fixed degree which have
only real roots (i.e., hyperbolic polynomials) and at least one root is of
multiplicity $k$. More generally, we describe the homotopy type of the
one-point compactification of strata in the boundary of the set of hyperbolic
polynomials, that are defined via certain restrictions on root multiplicities,
by order complexes of posets of compositions. In general, the methods are
combinatorial and the topological problems are mostly reduced to the study of
partially ordered sets.
|
math
|
570 |
Proper and Unit Trapezoid Orders and Graphs
|
math.CO
|
We show that the class of trapezoid orders in which no trapezoid strictly
contains any other trapezoid strictly contains the class of trapezoid orders in
which every trapezoid can be drawn with unit area. This is different from the
case of interval orders, where the class of proper interval orders is exactly
the same as the class of unit interval orders.
|
math
|
571 |
On $k$-ordered Hamiltonian Graphs
|
math.CO
|
A Hamiltonian graph $G$ of order $n$ is $k$-ordered, $2\leq k \leq n$, if for
every sequence $v_1, v_2, \ldots ,v_k$ of $k$ distinct vertices of $G$, there
exists a Hamiltonian cycle that encounters $v_1, v_2, \ldots , v_k$ in this
order. In this paper, answering a question of Ng and Schultz, we give a sharp
bound for the minimum degree guaranteeing that a graph is a $k$-ordered
Hamiltonian graph under some mild restrictions. More precisely, we show that
there are $\varepsilon, n_0> 0$ such that if $G$ is a graph of order $n\geq
n_0$ with minimum degree at least $\lceil \frac{n}{2} \rceil + \lfloor
\frac{k}{2} \rfloor - 1$ and $2\leq k \leq \eps n$, then $G$ is a $k$-ordered
Hamiltonian graph. It is also shown that this bound is sharp for every $2\leq k
\leq \lfloor \frac{n}{2} \rfloor$.
|
math
|
572 |
An Algorithmic Version of the Blow-up Lemma
|
math.CO
|
Recently we have developed a new method in graph theory based on the
Regularity Lemma. The method is applied to find certain spanning subgraphs in
dense graphs. The other main general tool of the method, beside the Regularity
Lemma, is the so-called Blow-up Lemma. This lemma helps to find bounded degree
spanning subgraphs in $\varepsilon$-regular graphs. Our original proof of the
lemma is not algorithmic, it applies probabilistic methods. In this paper we
provide an algorithmic version of the Blow-up Lemma. The desired subgraph, for
an $n$-vertex graph, can be found in time $O(nM(n))$, where $M(n)=O(n^{2.376})$
is the time needed to multiply two $n$ by $n$ matrices with 0,1 entries over
the integers. We show that the algorithm can be parallelized and implemented in
$NC^5$.
|
math
|
573 |
The number of faces of a simple polytope
|
math.CO
|
Consider the question: Given integers $k<d<n$, does there exist a simple
$d$-polytope with $n$ faces of dimension $k$? We show that there exist numbers
$G(d,k)$ and $N(d,k)$ such that for $n> N(d,k)$ the answer is yes if and only
if $n\equiv 0\quad \pmod {G(d,k)}$. Furthermore, a formula for $G(d,k)$ is
given, showing that e.g. $G(d,k)=1$ if $k\ge
\left\lfloor\frac{d+1}{2}\right\rfloor$ or if both $d$ and $k$ are even, and
also in some other cases (meaning that all numbers beyond $N(d,k)$ occur as the
number of $k$-faces of some simple $d$-polytope).
This question has previously been studied only for the case of vertices
($k=0$), where Lee \cite{Le} proved the existence of $N(d,0)$ (with $G(d,0)=1$
or $2$ depending on whether $d$ is even or odd), and Prabhu \cite{P2} showed
that $N(d,0) \le cd\sqrt {d}$. We show here that asymptotically the true value
of Prabhu's constant is $c=\sqrt2$ if $d$ is even, and $c=1$ if $d$ is odd.
|
math
|
574 |
Trapezoid Order Classification
|
math.CO
|
We show the nonequivalence of combinations of several natural geometric
restrictions on trapezoid representations of trapezoid orders. Each of the
properties unit parallelogram, unit trapezoid and proper parallelogram, unit
trapezoid and parallelogram, unit trapezoid, proper parallelogram, proper
trapezoid and parallelogram, proper trapezoid, parallelogram, and trapezoid is
shown to be distinct from each of the others. Additionally, interval orders are
shown to be both unit trapezoid and proper parallelogram orders.
|
math
|
575 |
Piles of Cubes, Monotone Path Polytopes and Hyperplane Arrangements
|
math.CO
|
Monotone path polytopes arise as a special case of the construction of fiber
polytopes, introduced by Billera and Sturmfels. A simple example is provided by
the permutahedron, which is a monotone path polytope of the standard unit cube.
The permutahedron is the zonotope polar to the braid arrangement. We show how
the zonotopes polar to the cones of certain deformations of the braid
arrangement can be realized as monotone path polytopes. The construction is an
extension of that of the permutahedron and yields interesting connections
between enumerative combinatorics of hyperplane arrangements and geometry of
monotone path polytopes.
|
math
|
576 |
The Tutte dichromate and Whitney homology of matroids
|
math.CO
|
We consider a specialization $Y_M(q,t)$ of the Tutte polynomial of a matroid
$M$ which is inspired by analogy with the Potts model from statistical
mechanics. The only information lost in this specialization is the number of
loops of $M$. We show that the coefficients of $Y_M(1-p,t)$ are very simply
related to the ranks of the Whitney homology groups of the opposite partial
orders of the independent set complexes of the duals of the truncations of $M$.
In particular, we obtain a new homological interpretation for the coefficients
of the characteristic polynomial of a matroid.
|
math
|
577 |
Exact enumeration of 1342-avoiding permutations: A close link with labeled trees and planar maps
|
math.CO
|
Solving the first nonmonotonic, longer-than-three instance of a classic
enumeration problem, we obtain the generating function $H(x)$ of all
1342-avoiding permutations of length $n$ as well as an {\em exact} formula for
their number $S_n(1342)$. While achieving this, we bijectively prove that the
number of indecomposable 1342-avoiding permutations of length $n$ equals that
of labeled plane trees of a certain type on $n$ vertices recently enumerated by
Cori, Jacquard and Schaeffer, which is in turn known to be equal to the number
of rooted bicubic maps enumerated by Tutte in 1963. Moreover, $H(x)$ turns out
to be algebraic, proving the first nonmonotonic, longer-than-three instance of
a conjecture of Zeilberger and Noonan. We also prove that $\sqrt[n]{S_n(1342)}$
converges to 8, so in particular, $lim_{n\rightarrow
\infty}(S_n(1342)/S_n(1234))=0$.
|
math
|
578 |
A Simple Bijection for the Regions of the Shi Arrangement of Hyperplanes
|
math.CO
|
The Shi arrangement ${\mathcal S}_n$ is the arrangement of affine hyperplanes
in ${\mathbb R}^n$ of the form $x_i - x_j = 0$ or $1$, for $1 \leq i < j \leq
n$. It dissects ${\mathbb R}^n$ into $(n+1)^{n-1}$ regions, as was first proved
by Shi. We give a simple bijective proof of this result. Our bijection
generalizes easily to any subarrangement of ${\mathcal S}_n$ containing the
hyperplanes $x_i - x_j = 0$ and to the extended Shi arrangements.
|
math
|
579 |
Automorphism groups with cyclic commutator subgroup and Hamilton cycles
|
math.CO
|
It has been shown that there is a Hamilton cycle in every connected Cayley
graph on each group G whose commutator subgroup is cyclic of prime-power order.
This paper considers connected, vertex-transitive graphs X of order at least 3
where the automorphism group of X contains a transitive subgroup G whose
commutator subgroup is cyclic of prime-power order. We show that of these
graphs, only the Petersen graph is not hamiltonian.
|
math
|
580 |
On non-Hamiltonian circulant digraphs of outdegree three
|
math.CO
|
We construct infinitely many connected, circulant digraphs of outdegree three
that have no hamiltonian circuit. All of our examples have an even number of
vertices, and our examples are of two types: either every vertex in the digraph
is adjacent to two diametrically opposite vertices, or every vertex is adjacent
to the vertex diametrically opposite to itself.
|
math
|
581 |
A Combinatorial proof of a result of Hetyei and Reiner on Foata-Strehl type permutation trees
|
math.CO
|
We give a combinatorial proof of the result of Hetyei and Reiner that there
are exactly $n!/3$ permutations of length $n$ in the minmax tree representation
of which the $i$th node is a leaf. We also prove the new result that the number
of $n$-permutations in which this node has one child is $n!/3$ as well,
implying that the same holds for those in which this node has two children.
|
math
|
582 |
A Suspension Lemma for Bounded Posets
|
math.CO
|
Let $P$ and $Q$ be bounded posets. In this note, a lemma is introduced that
provides a set of sufficient conditions for the proper part of $P$ being
homotopy equivalent to the suspension of the proper part of~$Q$. An application
of this lemma is a unified proof of the sphericity of the higher Bruhat orders
under both inclusion order (a known proved earlier by Ziegler) and single step
inclusion order (which was not previously known).
|
math
|
583 |
On Subdivision Posets of Cyclic Polytopes
|
math.CO
|
There are two related poset structures, the higher Stasheff-Tamari orders, on
the set of all triangulations of the cyclic $d$ polytope with $n$ vertices. In
this paper it is shown that both of them have the homotopy type of a sphere of
dimension $n-d-3$.
Moreover, we resolve positively a new special case of the \emph{Generalized
Baues Problem}: The Baues poset of all polytopal decompositions of a cyclic
polytope of dimension $d \leq 3$ has the homotopy type of a sphere of dimension
$n-d-2$.
|
math
|
584 |
Continued Fractions and Unique Additive Partitions
|
math.CO
|
A partition of the positive integers into sets $A$ and $B$ {\em avoids} a set
$S\subset\N$ if no two distinct elements in the same part have a sum in $S$. If
the partition is unique, $S$ is {\em uniquely avoidable.} For any irrational
$\alpha>1$, Chow and Long constructed a partition which avoids the numerators
of all convergents to $\alpha$, and conjectured that the set $S_\alpha$ which
this partition avoided was uniquely avoidable. We prove that the set of
numerators of convergents is uniquely avoidable if and only if the continued
fraction for $\alpha$ has infinitely many partial quotients equal to 1. We also
construct the set $S_\alpha$ and show that it is always uniquely avoidable.
|
math
|
585 |
Order complexes of noncomplemented lattices are nonevasive
|
math.CO
|
We reprove and generalize in a combinatorial way the result of A. Bj\"orner
[J.\ Comb.\ Th.\ A {\bf 30}, 1981, pp.~90--100, Theorem 3.3], that order
complexes of noncomplemented lattices are contractible, namely by showing that
these simplicial complexes are in fact nonevasive, in particular collapsible.
|
math
|
586 |
A Polynomial Time Algorithm for Vertex Enumeration and Optimization over Shaped Partition Polytopes
|
math.CO
|
We consider the {\em Shaped Partition Problem} of partitioning $n$ given
vectors in real $k$-space into $p$ parts so as to maximize an arbitrary
objective function which is convex on the sum of vectors in each part, subject
to arbitrary constraints on the number of elements in each part. In addressing
this problem, we study the {\em Shaped Partition Polytope} defined as the
convex hull of solutions. The Shaped Partition Problem captures ${\cal N}{\cal
P}$-hard problems such as the Max-Cut problem and the Traveling Salesperson
problem, and the Shaped Partition Polytope may have exponentially many vertices
and facets, even when $k$ or $p$ are fixed. In contrast, we show that when both
$k$ and $p$ are fixed, the number of vertices is polynomial in $n$, and all
vertices can be enumerated and the optimization problem solved in strongly
polynomial time. Explicitly, we show that any Shaped Partition Polytope has
$O(n^{k{p\choose 2}})$ vertices which can be enumerated in $O(n^{k^2p^3})$
arithmetic operations, and that any Shaped Partition Problem is solvable in
$O(n^{kp^2})$ arithmetic operations.
|
math
|
587 |
Diagrams of classifying spaces and $k$-fold Boolean algebras
|
math.CO
|
In this paper we study the problem of determining the homology groups of a
quotient of a topological space by an action of a group. The method is to
represent the original topological space as a homotopy limit of a diagram, and
then act with the group on that diagram. Once it is possible to understand what
the action of the group on every space in the diagram is, and what it does to
the morphisms, we can compute the homology groups of the homotopy limit of this
quotient diagram.
Our motivating example is the symmetric deleted join of a simplicial complex.
It can be represented as a diagram of symmetric deleted products. In the case
where the simplicial complex in question is a simplex, we perform the complete
computation of the homology groups with $\mathbb Z_p$ coefficients. For the
infinite simplex the spaces in the quotient diagram are classifying spaces of
various direct products of symmetric groups and diagram morphisms are induced
by group homomorphisms. Combining Nakaoka's description of the $\mathbb
Z_p$-homology of the symmetric group with a spectral sequence, we reduce the
computation to an essentially combinatorial problem, which we then solve using
the braid stratification of a sphere. Finally, we give another description of
the problem in terms of posets and complete the computation for the case of a
finite simplex.
|
math
|
588 |
Littlewood-Richardson semigroups
|
math.CO
|
This note is an extended abstract of my talk at the workshop on
Representation Theory and Symmetric Functions, MSRI, April 14, 1997. We discuss
the problem of finding an explicit description of the semigroup $LR_r$ of
triples of partitions of length $\leq r$ such that the corresponding
Littlewood-Richardson coefficient is non-zero. After discussing the history of
the problem and previously known results, we suggest a new approach based on
the ``polyhedral'' combinatorial expressions for the Littlewood-Richardson
coefficients.
|
math
|
589 |
Complexes of not $i$-connected graphs
|
math.CO
|
Complexes of (not) connected graphs, hypergraphs and their homology appear in
the construction of knot invariants given by V. Vassiliev. In this paper we
study the complexes of not $i$-connected $k$-hypergraphs on $n$ vertices. We
show that the complex of not $2$-connected graphs has the homotopy type of a
wedge of $(n-2)!$ spheres of dimension $2n-5$. This answers one of the
questions raised by Vassiliev in connection with knot invariants. For this case
the $S_n$-action on the homology of the complex is also determined. For
complexes of not $2$-connected $k$-hypergraphs we provide a formula for the
generating function of the Euler characteristic, and we introduce certain
lattices of graphs that encode their topology. We also present partial results
for some other cases. In particular, we show that the complex of not
$(n-2)$-connected graphs is Alexander dual to the complex of partial matchings
of the complete graph. For not $(n-3)$-connected graphs we provide a formula
for the generating function of the Euler characteristic.
|
math
|
590 |
2-stack sortable permutations with a given number of runs
|
math.CO
|
Using earlier results we prove a formula for the number $W_{(n,k)}$ of
2-stack sortable permutations of length $n$ with $k$ runs, or in other words,
$k-1$ descents. This formula will yield the suprising fact that there are as
many 2-stack sortable permutations with $k-1$ descents as with $k-1$ ascents.
We also prove that $W_{(n,k)}$ is unimodal in $k$, for any fixed $n$.
|
math
|
591 |
Extended Linial Hyperplane Arrangements for Root Systems and a Conjecture of Postnikov and Stanley
|
math.CO
|
A hyperplane arrangement is said to satisfy the ``Riemann hypothesis'' if all
roots of its characteristic polynomial have the same real part. This property
was conjectured by Postnikov and Stanley for certain families of arrangements
which are defined for any irreducible root system and was proved for the root
system $A_{n-1}$. The proof is based on an explicit formula for the
characteristic polynomial, which is of independent combinatorial significance.
Here our previous derivation of this formula is simplified and extended to
similar formulae for all but the exceptional root systems. The conjecture
follows in these cases.
|
math
|
592 |
$q$-Rook polynomials and matrices over finite fields
|
math.CO
|
Connections between $q$-rook polynomials and matrices over finite fields are
exploited to derive a new statistic for Garsia and Remmel's $q$-hit polynomial.
Both this new statistic $mat$ and another statistic for the $q$-hit polynomial
$\xi$ recently introduced by Dworkin are shown to induce different multiset
Mahonian permutation statistics for any Ferrers board. In addition, for the
triangular boards they are shown to generate different families of
Euler-Mahonian statistics. For these boards the $\xi$ family includes Denert's
statistic $den$, and gives a new proof of Foata and Zeilberger's Theorem that
$(exc,den)$ is jointly distributed with $(des,maj)$. The $mat$ family appears
to be new. A proof is also given that the $q$-hit polynomials are symmetric and
unimodal.
|
math
|
593 |
Linear inequalities for flags in graded posets
|
math.CO
|
The closure of the convex cone generated by all flag $f$-vectors of graded
posets is shown to be polyhedral. In particular, we give the facet inequalities
to the polar cone of all nonnegative chain-enumeration functionals on this
class of posets. These are in one-to-one correspondence with antichains of
intervals on the set of ranks and thus are counted by Catalan numbers.
Furthermore, we prove that the convolution operation introduced by Kalai
assigns extreme rays to pairs of extreme rays in most cases. We describe the
strongest possible inequalities for graded posets of rank at most 5.
|
math
|
594 |
Obstructions to Shellability
|
math.CO
|
We consider a simplicial complex generaliztion of a result of Billera and
Meyers that every nonshellable poset contains the smallest nonshellable poset
as an induced subposet. We prove that every nonshellable $2$-dimensional
simplicial complex contains a nonshellable induced subcomplex with less than
$8$ vertices. We also establish CL-shellability of interval orders and as a
consequence obtain a formula for the Betti numbers of any interval order.
|
math
|
595 |
Bases in Systems of Simplices and Chambers
|
math.CO
|
We consider a finite set $E$ of points in the $n$-dimensional affine space
and two sets of objects that are generated by the set $E$: the system $\Sigma$
of $n$-dimensional simplices with vertices in $E$ and the system $\Gamma$ of
chambers. The incidence matrix $A= \parallel a_{\sigma, \gamma}\parallel$,
$\sigma \in \Sigma$, $\gamma \in \Gamma$, induces the notion of linear
independence among simplices (and among chambers). We present an algorithm of
construction of bases of simplices (and bases of chambers). For the case $n=2$
such an algorithm was described in the author's paper {\em Combinatorial bases
in systems of simplices and chambers} (Discrete Mathematics 157 (1996) 15--37).
However, the case of $n$-dimensional space required a different technique. It
is also proved that the constructed bases of simplices are geometrical.
|
math
|
596 |
Chain Decomposition Theorems for Ordered Sets (and Other Musings)
|
math.CO
|
A brief introduction to the theory of ordered sets and lattice theory is
given. To illustrate proof techniques in the theory of ordered sets, a
generalization of a conjecture of Daykin and Daykin, concerning the structure
of posets that can be partitioned into chains in a ``strong'' way, is proved.
The result is motivated by a conjecture of Graham's concerning probability
correlation inequalities for linear extensions of finite posets.
|
math
|
597 |
Residue symbols and Jantzen-Seitz partitions
|
math.CO
|
Jantzen-Seitz partitions are those $p$-regular partitions of~$n$ which label
$p$-modular irreducible representations of the symmetric group $S_n$ which
remain irreducible when restricted to $S_{n-1}$; they have recently also been
found to be important for certain exactly solvable models in statistical
mechanics. In this article we study their combinatorial properties via a
detailed analysis of their residue symbols; in particular the $p$-cores of
Jantzen-Seitz partitions are determined.
|
math
|
598 |
Enumeration of rhombus tilings of a hexagon which contain a fixed rhombus on its symmetry axis
|
math.CO
|
We compute the number of rhombus tilings of a hexagon with sides
$N,M,N,N,M,N$, which contain a fixed rhombus on the symmetry axis. A special
case solves a problem posed by Jim Propp.
|
math
|
599 |
Determinant identities and a generalization of the number of totally symmetric self-complementary plane partitions
|
math.CO
|
We prove a constant term conjecture of Robbins and Zeilberger (J. Combin.
Theory Ser. A 66 (1994), 17-27), by translating the problem into a determinant
evaluation problem and evaluating the determinant. This determinant generalizes
the determinant that gives the number of all totally symmetric
self-complementary plane partitions contained in a $(2n)\times(2n)\times(2n)$
box and that was used by Andrews (J. Combin. Theory Ser. A 66 (1994), 28-39)
and Andrews and Burge (Pacific J. Math. 158 (1993), 1-14) to compute this
number explicitly. The evaluation of the generalized determinant is independent
of Andrews and Burge's computations, and therefore in particular constitutes a
new solution to this famous enumeration problem. We also evaluate a related
determinant, thus generalizing another determinant identity of Andrews and
Burge (loc. cit.). By translating some of our determinant identities into
constant term identities, we obtain several new constant term identities.
|
math
|
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