Showing papers in "Electronic Journal of Combinatorics in 2005"
TL;DR: It is proved that the sum of entries of the suitably normalized groundstate vector of the O(1) loop model with periodic boundary conditions on a periodic strip of size $2n$ is equal to the total number of alternating sign matrices.
Abstract: We prove that the sum of entries of the suitably normalized groundstate vector of the $O(1)$ loop model with periodic boundary conditions on a periodic strip of size $2n$ is equal to the total number of $n\times n$ alternating sign matrices. This is done by identifying the state sum of a multi-parameter inhomogeneous version of the $O(1)$ model with the partition function of the inhomogeneous six-vertex model on a $n\times n$ square grid with domain wall boundary conditions.
116 citations
TL;DR: Applications of the insertion encoding to the evaluation of generating functions for classes of permutations, construction of polynomial time algorithms for enumerating such classes, and the illustration of bijective equivalence between classes are demonstrated.
Abstract: We introduce the insertion encoding , an encoding of finite permutations. Classes of permutations whose insertion encodings form a regular language are characterized. Some necessary conditions are provided for a class of permutations to have insertion encodings that form a context free language. Applications of the insertion encoding to the evaluation of generating functions for classes of permutations, construction of polynomial time algorithms for enumerating such classes, and the illustration of bijective equivalence between classes are demonstrated.
80 citations
TL;DR: Two main new methods are introduced, one based on breadthrst search and another that uses the number theory and combinatorial structure inherent in the problem to speed up the Dijkstra approach.
Abstract: The Frobenius problem, also known as the postage-stamp problem or the moneychanging problem, is an integer programming problem that seeks nonnegative integer solutions to x1a1 + + xnan = M ,w hereai and M are positive integers. In particular, the Frobenius number f(A), where A =faig ,i s the largestM so that this equation fails to have a solution. A simple way to compute this number is to transform the problem to a shortest-path problem in a directed weighted graph; then Dijkstra’s algorithm can be used. We show how one can use the additional symmetry properties of the graph in question to design algorithms that are very fast. For example, on a standard desktop computer, our methods can handle cases where n =1 0 anda1 =1 0 7 . We have two main new methods, one based on breadthrst search and another that uses the number theory and combinatorial structure inherent in the problem to speed up the Dijkstra approach. For both methods we conjecture that the average-case complexity is O(a1 p n). The previous best method
75 citations
TL;DR: In this article, it was shown that the sequences formed by logarithms and by fractional powers of integers, as well as the sequence of prime numbers, are not holonomic.
Abstract: We establish that the sequences formed by logarithms and by \fractional" powers of integers, as well as the sequence of prime numbers, are non-holonomic, thereby answering three open problems of Gerhold [El. J. Comb. 11 (2004), R87]. Our proofs depend on basic complex analysis, namely a conjunction of the Structure Theorem for singularities of solutions to linear dierential equations and of an Abelian theorem. A brief discussion is oered regarding the scope of singularity-based methods and several naturally occurring sequences are proved to be non-holonomic.
64 citations
TL;DR: In this paper, the asymptotic value of B(m,s;n,t) was shown to hold regardless of the number of vertices in a bipartite graph.
Abstract: Let $s,\,t,\,m$ and $n$ be positive integers such that $sm=tn$. Let $B(m,s;n,t)$ be the number of $m\times n$ matrices over $\{0,1\}$ with each row summing to $s$ and each column summing to $t$. Equivalently, $B(m,s;n,t)$ is the number of semiregular bipartite graphs with $m$ vertices of degree $s$ and $n$ vertices of degree $t$. Define the density $\lambda=s/n=t/m$. The asymptotic value of $B(m,s;n,t)$ has been much studied but the results are incomplete. McKay and Wang (2003) solved the sparse case $\lambda(1-\lambda)=o\big((mn)^{-1/2}\big)$ using combinatorial methods. In this paper, we use analytic methods to solve the problem for two additional ranges. In one range the matrix is relatively square and the density is not too close to 0 or 1. In the other range, the matrix is far from square and the density is arbitrary. Interestingly, the asymptotic value of $B(m,s;n,t)$ can be expressed by the same formula in all cases where it is known. Based on computation of the exact values for all $m,n\le 30$, we conjecture that the same formula holds whenever $m+n\to\infty$ regardless of the density.
63 citations
TL;DR: It is proved that if $G$ is a connected, prime graph, then D(G^{^r}) = 2 whenever $r \geq 4$.
Abstract: Given a graph $G$, a labeling $c:V(G) \rightarrow \{1, 2, \ldots, d\}$ is said to be $d$-distinguishing if the only element in ${\rm Aut}(G)$ that preserves the labels is the identity. The distinguishing number of $G$, denoted by $D(G)$, is the minimum $d$ such that $G$ has a $d$-distinguishing labeling. If $G \square H$ denotes the Cartesian product of $G$ and $H$, let $G^{^2} = G \square G$ and $G^{^r} = G \square G^{^{r-1}}$. A graph $G$ is said to be prime with respect to the Cartesian product if whenever $G \cong G_1 \square G_2$, then either $G_1$ or $G_2$ is a singleton vertex. This paper proves that if $G$ is a connected, prime graph, then $D(G^{^r}) = 2$ whenever $r \geq 4$.
61 citations
TL;DR: This article describes conjectured combinatorial interpretations for the higher $q,t,r$-Catalan sequences introduced by Garsia and Haiman, which arise in the theory of symmetric functions and Macdonald polynomials.
Abstract: This article describes conjectured combinatorial interpretations for the higher $q,t$-Catalan sequences introduced by Garsia and Haiman, which arise in the theory of symmetric functions and Macdonald polynomials. We define new combinatorial statistics generalizing those proposed by Haglund and Haiman for the original $q,t$-Catalan sequence. We prove explicit summation formulas, bijections, and recursions involving the new statistics. We show that specializations of the combinatorial sequences obtained by setting $t=1$ or $q=1$ or $t=1/q$ agree with the corresponding specializations of the Garsia-Haiman sequences. A third statistic occurs naturally in the combinatorial setting, leading to the introduction of $q,t,r$-Catalan sequences. Similar combinatorial results are proved for these trivariate sequences.
58 citations
TL;DR: This paper addresses the minimum common string partition problem, a string comparison problem with tight connection to the problem of sorting by reversals with duplicates, a key problem in genome rearrangement, and shows that $2- MCSP (and therefore MCSP ) is NP-hard and, moreover, even APX-hard.
Abstract: String comparison is a fundamental problem in computer science, with applications in areas such as computational biology, text processing and compression. In this paper we address the minimum common string partition problem, a string comparison problem with tight connection to the problem of sorting by reversals with duplicates, a key problem in genome rearrangement. A partition of a string $A$ is a sequence ${\cal P} = (P_1,P_2,\dots,P_m)$ of strings, called the blocks , whose concatenation is equal to $A$. Given a partition ${\cal P}$ of a string $A$ and a partition ${\cal Q}$ of a string $B$, we say that the pair $\langle{{\cal P},{\cal Q}}\rangle$ is a common partition of $A$ and $B$ if ${\cal Q}$ is a permutation of ${\cal P}$. The minimum common string partition problem ( MCSP ) is to find a common partition of two strings $A$ and $B$ with the minimum number of blocks. The restricted version of MCSP where each letter occurs at most $k$ times in each input string, is denoted by $k$- MCSP . In this paper, we show that $2$- MCSP (and therefore MCSP ) is NP-hard and, moreover, even APX-hard. We describe a $1.1037$-approximation for $2$- MCSP and a linear time $4$-approximation algorithm for $3$- MCSP . We are not aware of any better approximations.
56 citations
TL;DR: Using residues modulo the primes dividing $n$, a representation of the vertices is introduced that reduces the problem to a purely combinatorial question of comparing strings of symbols and proves that the multiplicity of each prime dividing n has no effect on the length of the longest induced cycle in X_n.
Abstract: In this paper we study the length of the longest induced cycle in the unit circulant graph $X_n = Cay({\Bbb Z}_n; {\Bbb Z}_n^*)$, where ${\Bbb Z}_n^*$ is the group of units in ${\Bbb Z}_n$. Using residues modulo the primes dividing $n$, we introduce a representation of the vertices that reduces the problem to a purely combinatorial question of comparing strings of symbols. This representation allows us to prove that the multiplicity of each prime dividing $n$, and even the value of each prime (if sufficiently large) has no effect on the length of the longest induced cycle in $X_n$. We also see that if $n$ has $r$ distinct prime divisors, $X_n$ always contains an induced cycle of length $2^r+2$, improving the $r \ln r$ lower bound of Berrezbeitia and Giudici. Moreover, we extend our results for $X_n$ to conjunctions of complete $k_i$-partite graphs, where $k_i$ need not be finite, and also to unit circulant graphs on any quotient of a Dedekind domain.
55 citations
TL;DR: It is shown that latin squares built by the ‘orthomorphism method’ have large automorphism groups and an invariant called the train of a latin square is introduced, which proves to be useful for distinguishing non-isomorphic examples.
Abstract: Atomic latin squares have indivisible structure which mimics that of the cyclic groups of prime order. They are related to perfect 1-factorisations of complete bipartite graphs. Only one example of an atomic latin square of a composite order (namely 27) was previously known. We show that this one example can be generated by an established method of constructing latin squares using cyclotomic orthomorphisms in finite fields. The same method is used in this paper to construct atomic latin squares of composite orders 25, 49, 121, 125, 289, 361, 625, 841, 1369, 1849, 2809, 4489, 24649 and 39601. It is also used to construct many new atomic latin squares of prime order and perfect 1-factorisations of the complete graph Kq+1 for many prime powers q. As a result, existence of such a factorisation is shown for the first time for q in {529,2809,4489,6889,11449,11881,15625,22201, 24389,24649,26569,29929,32041, 38809,44521,50653,51529,52441,63001,72361,76729,78125,79507,103823, 148877,161051,205379,226981,300763,357911,371293,493039,571787}. We show that latin squares built by the ‘orthomorphism method’ have large automorphism groups and we discuss conditions under which different orthomorphisms produce isomorphic latin squares. We also introduce an invariant called the train of a latin square, which proves to be useful for distinguishing non-isomorphic examples. ∗This work was undertaken at Christ Church, Oxford and at the Department of Computer Science, Australian National University. the electronic journal of combinatorics 12 (2005), #R22 1
51 citations
TL;DR: Based on a bijection between domino tilings of anAztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given by means of Hankel determinants of the large and small Schroder numbers.
Abstract: Based on a bijection between domino tilings of an Aztec diamond and non-intersecting lattice paths, a simple proof of the Aztec diamond theorem is given by means of Hankel determinants of the large and small Schroder numbers.
TL;DR: A higher order generalization of the Erdős-Ko-Rado theorem is proved for systems of pairwise $t-intersecting uniform $k$-partitions of an $n$-set and it is proved that for large enough, any such system contains at most 1-over-k partitions.
Abstract: Two set partitions of an $n$-set are said to $t$-intersect if they have $t$ classes in common A $k$-partition is a set partition with $k$ classes and a $k$-partition is said to be uniform if every class has the same cardinality $c=n/k$ In this paper, we prove a higher order generalization of the Erdős-Ko-Rado theorem for systems of pairwise $t$-intersecting uniform $k$-partitions of an $n$-set We prove that for $n$ large enough, any such system contains at most $${1\over(k-t)!} {n-tc \choose c} {n-(t+1)c \choose c} \cdots {n-(k-1)c \choose c}$$ partitions and this bound is only attained by a trivially $t$-intersecting system We also prove that for $t=1$, the result is valid for all $n$ We conclude with some conjectures on this and other types of intersecting partition systems
TL;DR: Every matroid of rank three or corank three satisfies a condition only slightly weaker than the conclusion of Stanley's theorem, and a nest of inequalities for weighted basis–generating polynomials that are related to these ideas are explored.
Abstract: In 1981, Stanley applied the Aleksandrov–Fenchel Inequalities to prove a logarithmic concavity theorem for regular matroids. Using ideas from electrical network theory we prove a generalization of this for the wider class of matroids with the "half–plane property". Then we explore a nest of inequalities for weighted basis–generating polynomials that are related to these ideas. As a first result from this investigation we find that every matroid of rank three or corank three satisfies a condition only slightly weaker than the conclusion of Stanley's theorem.
TL;DR: A partition structure regenerative as discussed by the authors is defined as a sequence of probability distributions for a random partition of the positive halfline of the Ewens partition structure, such that for each box of balls removed by uniform random deletion without replacement, the remaining partition of $n-x$ is distributed like the original partition.
Abstract: A partition structure is a sequence of probability distributions for $\pi_n$, a random partition of $n$, such that if $\pi_n$ is regarded as a random allocation of $n$ unlabeled balls into some random number of unlabeled boxes, and given $\pi_n$ some $x$ of the $n$ balls are removed by uniform random deletion without replacement, the remaining random partition of $n-x$ is distributed like $\pi_{n-x}$, for all $1 \le x \le n$. We call a partition structure regenerative if for each $n$ it is possible to delete a single box of balls from $\pi_n$ in such a way that for each $1 \le x \le n$, given the deleted box contains $x$ balls, the remaining partition of $n-x$ balls is distributed like $\pi_{n-x}$. Examples are provided by the Ewens partition structures, which Kingman characterised by regeneration with respect to deletion of the box containing a uniformly selected random ball. We associate each regenerative partition structure with a corresponding regenerative composition structure, which (as we showed in a previous paper) is associated in turn with a regenerative random subset of the positive halfline. Such a regenerative random set is the closure of the range of a subordinator (that is an increasing process with stationary independent increments). The probability distribution of a general regenerative partition structure is thus represented in terms of the Laplace exponent of an associated subordinator, for which exponent an integral representation is provided by the Levy-Khintchine formula. The extended Ewens family of partition structures, previously studied by Pitman and Yor, with two parameters $(\alpha,\theta)$, is characterised for $0 \le \alpha 0$ by regeneration with respect to deletion of each distinct part of size $x$ with probability proportional to $(n-x)\tau+x(1-\tau)$, where $\tau = \alpha/(\alpha+\theta)$.
TL;DR: The construction of graphs having a $(1,\le \ell)-identifying code of small cardinality is addressed from a connection between identifying codes and superimposed codes, which is described in this paper.
Abstract: In this paper the problem of constructing graphs having a $(1,\le \ell)$-identifying code of small cardinality is addressed. It is known that the cardinality of such a code is bounded by $\Omega\left({\ell^2\over\log \ell}\log n\right)$. Here we construct graphs on $n$ vertices having a $(1,\le \ell)$-identifying code of cardinality $O\left(\ell^4 \log n\right)$ for all $\ell \ge 2$. We derive our construction from a connection between identifying codes and superimposed codes, which we describe in this paper.
TL;DR: The multiple sum that is constructed is the generating function for the so-called $K$-restricted jagged partitions and the corresponding generalization of the Rogers-Ramunjan identities is displayed, together with a novel combinatorial interpretation.
Abstract: We present a natural extension of Andrews' multiple sums counting partitions of the form $(\lambda_1,\cdots,\lambda_m)$ with $\lambda_i\geq \lambda_{i+k-1}+2$. The multiple sum that we construct is the generating function for the so-called $K$-restricted jagged partitions. A jagged partition is a sequence of non-negative integers $(n_1,n_2,\cdots , n_m)$ with $n_m\geq 1$ subject to the weakly decreasing conditions $n_i\geq n_{i+1}-1$ and $n_i\geq n_{i+2}$. The $K$-restriction refers to the following additional conditions: $n_i \geq n_{i+K-1} +1$ or $n_i = n_{i+1}-1 = n_{i+K-2}+1= n_{i+K-1}$. The corresponding generalization of the Rogers-Ramunjan identities is displayed, together with a novel combinatorial interpretation.
TL;DR: It is shown that the number of fully packed loop congurations correspond- ing to a matching with m nested arches is polynomial in m if m is large enough, thus essentially proving two conjectures by Zuber.
Abstract: We show that the number of fully packed loop congurations correspond- ing to a matching with m nested arches is polynomial in m if m is large enough, thus essentially proving two conjectures by Zuber (Electronic J. Combin. 11(1) (2004), Arti- cle #R13).
TL;DR: The classification of Wilf-equivalence classes for pairs of permutations of length four is complete with bijections between the sets using the idea of a "block" and a generating function is found for S_n.
Abstract: $S_n(\pi_1,\pi_2,\dots, \pi_r)$ denotes the set of permutations of length $n$ that have no subsequence with the same order relations as any of the $\pi_i$. In this paper we show that $|S_n(1342,2143)|=|S_n(3142,2341)|$ and $|S_n(1342,3124)|=|S_n(1243,2134)|$. These two facts complete the classification of Wilf-equivalence classes for pairs of permutations of length four. In both instances we exhibit bijections between the sets using the idea of a "block", and in the former we find a generating function for $|S_n(1342,2143)|$.
TL;DR: This paper proposes a weakening of the definitions of uniform and perfect one-factorizations of the complete graph in such a way that the union of any two (cyclically) consecutive one-factors is always isomorphic to the same two-regular graph.
Abstract: In this paper, we consider a weakening of the definitions of uniform and perfect one-factorizations of the complete graph. Basically, we want to order the $2n-1$ one-factors of a one-factorization of the complete graph $K_{2n}$ in such a way that the union of any two (cyclically) consecutive one-factors is always isomorphic to the same two-regular graph. This property is termed sequentially uniform ; if this two-regular graph is a Hamiltonian cycle, then the property is termed sequentially perfect . We will discuss several methods for constructing sequentially uniform and sequentially perfect one-factorizations. In particular, we prove for any integer $n \geq 1$ that there is a sequentially perfect one-factorization of $K_{2n}$. As well, for any odd integer $m \geq 1$, we prove that there is a sequentially uniform one-factorization of $K_{2^t m}$ of type $(4,4,\dots,4)$ for all integers $t \geq 2 + \lceil \log_2 m \rceil$ (where type $(4,4,\dots,4)$ denotes a two-regular graph consisting of disjoint cycles of length four).
TL;DR: This work gives combinatorial proofs of two identities from the representation theory of the partition algebra of C A_k(n), n = 2k, where n is the number of standard tableaux of shape $\lambda, and m_k^\lambda is theNumber of "vacillating tableaux" of shape $2k and length 2k.
Abstract: We give combinatorial proofs of two identities from the representation theory of the partition algebra CAk(n) ;n 2k. The rst is n k = P f m k , where the sum is over partitions of n, f is the number of standard tableaux of shape ,a nd m k is the number of \vacillating tableaux" of shape and length 2k. Our proof uses a combination of Robinson-Schensted-Knuth insertion and jeu de taquin. The second identity is B(2k )= P (m k ) 2 ,w hereB(2k) is the number of set partitions of f1 ;:::; 2kg. We show that this insertion restricts to work for the diagram algebras which appear as subalgebras of the partition algebra: the Brauer, Temperley-Lieb, planar partition, rook monoid, planar rook monoid, and symmetric group algebras.
TL;DR: A general characterization of tightness for one-sided quotients is provided, all tight one- sided quotients of nite Coxeter groups are classified, and all tight double quotients are classified of ane Weyl groups.
Abstract: It is a well-known theorem of Deodhar that the Bruhat ordering of a Coxeter group is the conjunction of its projections onto quotients by maximal parabolic subgroups. Similarly, the Bruhat order is also the conjunction of a larger number of simpler quotients obtained by projecting onto two-sided (i.e., \double") quotients by pairs of maximal parabolic subgroups. Each one-sided quotient may be represented as an orbit in the reflection representation, and each double quotient corresponds to the portion of an orbit on the positive side of certain hyperplanes. In some cases, these orbit representations are \tight" in the sense that the root system induces an ordering on the orbit that yields eective coordinates for the Bruhat order, and hence also provides upper bounds for the order dimension. In this paper, we (1) provide a general characterization of tightness for one-sided quotients, (2) classify all tight one-sided quotients of nite Coxeter groups, and (3) classify all tight double quotients of ane Weyl groups.
Journal Article•
TL;DR: This paper gives an algorithm for expanding the Kronecker product s(n p;p) s if 1 2 2p and obtains a formula for g (n p ;p);; in terms of the Littlewood-Richardson coecients which does not involve cancellations.
Abstract: The Kronecker product of two Schur functions s and s, denoted s s ,i s dened as the Frobenius characteristic of the tensor product of the irreducible representations of the symmetric group indexed by partitions of n, and , respectively. The coecient, g;; ,o fs in s s is equal to the multiplicity of the irreducible representation indexed by in the tensor product. In this paper we give an algorithm for expanding the Kronecker product s(n p;p) s if 1 2 2p .A s a consequence of this algorithm we obtain a formula for g (n p;p);; in terms of the Littlewood-Richardson coecients which does not involve cancellations. Another consequence of our algorithm is that if 1 2 2p then every Kronecker coecient in s(n p;p) s is independent of n, in other words, g(n p;p);; is stable for all
TL;DR: The probability that a random pair of elements from the alternating group A{n} generates all of A_{n} is shown to have an asymptotic expansion of the form $1-1/n-1 /n^{2}-4/n^{3}-23/ n^{4}-171/n 5}-... $.
Abstract: The probability that a random pair of elements from the alternating group $A_{n}$ generates all of $A_{n}$ is shown to have an asymptotic expansion of the form $1-1/n-1/n^{2}-4/n^{3}-23/n^{4}-171/n^{5}-... $. This same asymptotic expansion is valid for the probability that a random pair of elements from the symmetric group $S_{n}$ generates either $A_{n}$ or $S_{n}$. Similar results hold for the case of $r$ generators ($r>2$).
TL;DR: This paper gives a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolORED binary trees and presents generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.
Abstract: In this paper, we give a simple combinatorial explanation of a formula of A. Postnikov relating bicolored rooted trees to bicolored binary trees. We also present generalized formulas for the number of labeled k-ary trees, rooted labeled trees, and labeled plane trees.
TL;DR: In this paper, a shape datum is defined, which is equivalent to a commutation relation for certain operators acting on a $q$-deformed Fock space, obtained by Kashiwara, Miwa and Stern.
Abstract: The RSK correspondence generalises the Robinson-Schensted correspondence by replacing permutation matrices by matrices with entries in ${\bf N}$, and standard Young tableaux by semistandard ones. For $r\in{\bf N}_{>0}$, the Robinson-Schensted correspondence can be trivially extended, using the $r$-quotient map, to one between $r$-coloured permutations and pairs of standard $r$-ribbon tableaux built on a fixed $r$-core (the Stanton-White correspondence). Viewing $r$-coloured permutations as matrices with entries in ${\bf N}^r$ (the non-zero entries being unit vectors), this correspondence can also be generalised to arbitrary matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r$-core; the generalisation is derived from the RSK correspondence, again using the $r$-quotient map. Shimozono and White recently defined a more interesting generalisation of the Robinson-Schensted correspondence to $r$-coloured permutations and standard $r$-ribbon tableaux; unlike the Stanton-White correspondence, it respects the spin statistic on standard $r$-ribbon tableaux, relating it directly to the colours of the $r$-coloured permutation. We define a construction establishing a bijective correspondence between general matrices with entries in ${\bf N}^r$ and pairs of semistandard $r$-ribbon tableaux built on a fixed $r$-core, which respects the spin statistic on those tableaux in a similar manner, relating it directly to the matrix entries. We also define a similar generalisation of the asymmetric RSK correspondence, in which case the matrix entries are taken from $\{0,1\}^r$. More surprising than the existence of such a correspondence is the fact that these Knuth correspondences are not derived from Schensted correspondences by means of standardisation. That method does not work for general $r$-ribbon tableaux, since for $r\geq3$, no $r$-ribbon Schensted insertion can preserve standardisations of horizontal strips. Instead, we use the analysis of Knuth correspondences by Fomin to focus on the correspondence at the level of a single matrix entry and one pair of ribbon strips, which we call a shape datum. We define such a shape datum by a non-trivial generalisation of the idea underlying the Shimozono-White correspondence, which takes the form of an algorithm traversing the edge sequences of the shapes involved. As a result of the particular way in which this traversal has to be set up, our construction directly generalises neither the Shimozono-White correspondence nor the RSK correspondence: it specialises to the transpose of the former, and to the variation of the latter called the Burge correspondence. In terms of generating series, our shape datum proves a commutation relation between operators that add and remove horizontal $r$-ribbon strips; it is equivalent to a commutation relation for certain operators acting on a $q$-deformed Fock space, obtained by Kashiwara, Miwa and Stern. It implies the identity $$\sum_{\lambda\geq_r(0)}G^{(r)}_\lambda(q^{1\over2},X) G^{(r)}_\lambda(q^{1\over2},Y) =\prod_{i,j\in{\bf N}}\prod_{k=0}^{r-1}{1\over1-q^kX_iY_j}; $$ where $G^{(r)}_\lambda(q^{1\over2},X)\in{\bf Z}[q^{1\over2}][[X]]$ is the generating series by $q^{{\rm spin}(P)}X^{{\rm wt}(P)}$ of semistandard $r$-ribbon tableaux $P$ of shape $\lambda$; the identity is a $q$-analogue of an $r$-fold Cauchy identity, since the series factors into a product of $r$ Schur functions at $q^{1\over2}=1$. Our asymmetric correspondence similarly proves $$\sum_{\lambda\geq_r(0)}G^{(r)}_\lambda(q^{1\over2},X) \check G^{(r)}_\lambda(q^{1\over2},Y) =\prod_{i,j\in{\bf N}}\prod_{k=0}^{r-1}(1+q^kX_iY_j). $$ with $\check G^{(r)}_\lambda(q^{1\over2},X)$ the generating series by $q^{{\rm spin}^{\rm t}(P)}X^{{\rm wt}(P)}$ of transpose semistandard $r$-ribbon tableaux $P$, where ${\rm spin}^{\rm t}(P)$ denotes the spin as defined using the standardisation appropriate for such tableaux.
TL;DR: In this article, it was shown that there is a polynomial time algorithm to compute the minimum period of a rational polytope with integral vertices for a rational generating function.
Abstract: If $P\subset {\Bbb R}^d$ is a rational polytope, then $i_P(t):=\#(tP\cap {\Bbb Z}^d)$ is a quasi-polynomial in $t$, called the Ehrhart quasi-polynomial of $P$. A period of $i_P(t)$ is ${\cal D}(P)$, the smallest ${\cal D}\in {\Bbb Z}_+$ such that ${\cal D}\cdot P$ has integral vertices. Often, ${\cal D}(P)$ is the minimum period of $i_P(t)$, but, in several interesting examples, the minimum period is smaller. We prove that, for fixed $d$, there is a polynomial time algorithm which, given a rational polytope $P\subset{\Bbb R}^d$ and an integer $n$, decides whether $n$ is a period of $i_P(t)$. In particular, there is a polynomial time algorithm to decide whether $i_P(t)$ is a polynomial. We conjecture that, for fixed $d$, there is a polynomial time algorithm to compute the minimum period of $i_P(t)$. The tools we use are rational generating functions.
TL;DR: This paper gives a precise characterization of the structure of maximum sized $k$-sum-free sets in $\{1,\ldots,n\}$ for $k\ge 4$ and $n$ large.
Abstract: If $k$ is a positive integer, we say that a set $A$ of positive integers is $k$-sum-free if there do not exist $a,b,c$ in $A$ such that $a + b = kc$. In particular we give a precise characterization of the structure of maximum sized $k$-sum-free sets in $\{1,\ldots,n\}$ for $k\ge 4$ and $n$ large.
TL;DR: For any configuration of pebbles on the nodes of a graph, there exists a cover pebbling as discussed by the authors, which is a move sequence ending with no empty nodes, and it was conjectured that the maximum of these simple cover-pebbling numbers is indeed the general cover number of the graph.
Abstract: For any configuration of pebbles on the nodes of a graph, a pebbling move replaces two pebbles on one node by one pebble on an adjacent node. A cover pebbling is a move sequence ending with no empty nodes. The number of pebbles needed for a cover pebbling starting with all pebbles on one node is trivial to compute and it was conjectured that the maximum of these simple cover pebbling numbers is indeed the general cover pebbling number of the graph. That is, for any configuration of this size, there exists a cover pebbling. In this note, we prove a generalization of the conjecture. All previously published results about cover pebbling numbers for special graphs (trees, hypercubes et cetera) are direct consequences of this theorem. We also prove that the cover pebbling number of a product of two graphs equals the product of the cover pebbling numbers of the graphs.
TL;DR: The experience suggests us to conjecture that under the color degree condition G, an edge-colored graph, has a heterochromatic path of length at least $k-1$.
Abstract: Let $G$ be an edge-colored graph. A heterochromatic path of $G$ is such a path in which no two edges have the same color. $d^c(v)$ denotes the color degree of a vertex $v$ of $G$. In a previous paper, we showed that if $d^c(v)\geq k$ for every vertex $v$ of $G$, then $G$ has a heterochromatic path of length at least $\lceil{k+1\over 2}\rceil$. It is easy to see that if $k=1,2$, $G$ has a heterochromatic path of length at least $k$. Saito conjectured that under the color degree condition $G$ has a heterochromatic path of length at least $\lceil{2k+1\over 3}\rceil$. Even if this is true, no one knows if it is a best possible lower bound. Although we cannot prove Saito's conjecture, we can show in this paper that if $3\leq k\leq 7$, $G$ has a heterochromatic path of length at least $k-1,$ and if $k\geq 8$, $G$ has a heterochromatic path of length at least $\lceil{3k\over 5}\rceil+1$. Actually, we can show that for $1\leq k\leq 5$ any graph $G$ under the color degree condition has a heterochromatic path of length at least $k$, with only one exceptional graph $K_4$ for $k=3$, one exceptional graph for $k=4$ and three exceptional graphs for $k=5$, for which $G$ has a heterochromatic path of length at least $k-1$. Our experience suggests us to conjecture that under the color degree condition $G$ has a heterochromatic path of length at least $k-1$.
TL;DR: Two approaches are here systematically explored, using the flag-major index on the one hand, and theFlag-inversion number on the other hand, of calculating multivariable generating functions for this group by statistics involving record values and the length function.
Abstract: As for the symmetric group of ordinary permutations there is also a statistical study of the group of signed permutations, that consists of calculating multivariable generating functions for this group by statistics involving record values and the length function. Two approaches are here systematically explored, using the flag-major index on the one hand, and the flag-inversion number on the other hand. The MacMahon Verfahren appears as a powerful tool throughout.