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500 | Alternating sign matrices and domino tilings | 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 domin... | 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 ... | 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 interes... | 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 exa... | 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) \subse... | 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 in... | 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 ... | 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 pat... | 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 latti... | 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... | 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... | 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 bo... | 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 ... | 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 cen... | 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 exten... | 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... | 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 linea... | 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
$\{r,s\}$-leape... | 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... | 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 grap... | 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 o... | 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 positi... | 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 s... | 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 logar... | 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 ... | 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 coeffic... | 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 ... | 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... | 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 binom... | 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 t... | 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 sim... | 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 posit... | 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
obstruction... | 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 gi... | 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 strid... | 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 prop... | 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 originat... | 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 i... | 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
$(A,D)$ s... | 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... | 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 c... | 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 $Q_i$ c... | 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 i... | 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 fam... | 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 ... | 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 a... | 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 ... | 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 subgra... | 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 ... | 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 parallel... | 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 z... | 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 rank... | 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 indecomposab... | 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 e... | 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 su... | 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 th... | 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 tha... | 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 i... | 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... | 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 ... | 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 ... | 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 o... | 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... | 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... | 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 th... | 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 $... | 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... | 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 th... | 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 a... | 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 \S... | 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 motiva... | 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 thi... | 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... | math |
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