Authors:Felix Breuer; Zafeirakis Zafeirakopoulos Abstract: Abstract Polyhedral Omega is a new algorithm for solving linear Diophantine systems (LDS), i.e., for computing a multivariate rational function representation of the set of all non-negative integer solutions to a system of linear equations and inequalities. Polyhedral Omegacombines methods from partition analysis with methods from polyhedral geometry. In particular, we combine MacMahon’s iterative approach based on the Omega operator and explicit formulas for its evaluation with geometric tools such as Brion decompositions and Barvinok’s short rational function representations. In this way, we connect two recent branches of research that have so far remained separate, unified by the concept of symbolic cones which we introduce. The resulting LDS solver Polyhedral Omegais significantly faster than previous solvers based on partition analysis and it is competitive with state-of-the-art LDS solvers based on geometric methods. Most importantly, this synthesis of ideas makes Polyhedral Omegathe simplest algorithm for solving linear Diophantine systems available to date. Moreover, we provide an illustrated geometric interpretation of partition analysis, with the aim of making ideas from both areas accessible to readers from a wide range of backgrounds. PubDate: 2017-04-26 DOI: 10.1007/s00026-017-0349-x

Authors:John Bamberg; Jan De Beule; Ferdinand Ihringer Abstract: Abstract We provide new proofs for the non-existence of ovoids in hyperbolic spaces of rank at least four in even characteristic, and for the Hermitian polar space H(5,4). We also improve the results of A. Klein on the non-existence of ovoids of Hermitian spaces and hyperbolic quadrics. PubDate: 2017-02-10 DOI: 10.1007/s00026-017-0346-0

Authors:Vojtěch Rödl; Andrzej Ruciński; Mathias Schacht; Endre Szemerédi Abstract: Abstract We show that every 3-uniform hypergraph with minimum vertex degree at least 0.8 \(\left(\begin{array}{c}n-1\\2\end{array}\right)\) contains a tight Hamiltonian cycle. PubDate: 2017-02-09 DOI: 10.1007/s00026-017-0345-1

Authors:Larry Rolen Abstract: Abstract We study generating functions which count the sizes of t-cores of partitions, and, more generally, the sizes of higher rows in t-core towers. We then use these results to derive an asymptotic results for the average size of the t-defect of partitions, as well as some curious congruences. PubDate: 2017-02-03 DOI: 10.1007/s00026-017-0343-3

Authors:Richard Arratia; Stephen DeSalvo Abstract: Abstract We provide completely effective error estimates for Stirling numbers of the first and second kinds, denoted by s(n, m) and S(n, m), respectively. These bounds are useful for values of \({m\geq n-O(\sqrt{n})}\) . An application of our Theorem 3.2 yields, for example, $$\begin{array}{ll}{s({10^{12}}, {10^{12}}-2 \times{10^6})/{10^{35664464}} \in [1.87669, 1.876982],}\\{S({10^{12}}, {10^{12}}-2 \times{10^6})/{10^{35664463}} \in [1.30121, 1.306975]}.\end{array}$$ The bounds are obtained via Chen-Stein Poisson approximation, using an interpretation of Stirling numbers as the number of ways of placing non-attacking rooks on a chess board. As a corollary to Theorem 3.2, summarized in Proposition 2.4, we obtain two simple and explicit asymptotic formulas, one for each of s(n, m) and S(n, m), for the parametrization \({m = n-t {n^a}, 0 \leq a \leq \frac{1}{2}}\) . These asymptotic formulas agree with the ones originally observed by Moser and Wyman in the range \({0 < a < \frac{1}{2}}\) , and they connect with a recent asymptotic expansion by Louchard for \({\frac{1}{2} < a < 1}\) , hence filling the gap at \({a = \frac{1}{2}}\) . We also provide a generalization applicable to rook and file numbers. PubDate: 2017-02-03 DOI: 10.1007/s00026-017-0339-z

Authors:I. P. Goulden; Mathieu Guay-Paquet; Jonathan Novak Abstract: Abstract Monotone Hurwitz numbers were introduced by the authors as a combinatorially natural desymmetrization of the Hurwitz numbers studied in enumerative algebraic geometry. Over the course of several papers, we developed the structural theory of monotone Hurwitz numbers and demonstrated that it is in many ways parallel to that of their classical counterparts. In this note, we identify an important difference between the monotone and classical worlds: fixed-genus generating functions for monotone double Hurwitz numbers are absolutely summable, whereas those for classical double Hurwitz numbers are not. This property is crucial for applications of monotone Hurwitz theory in analysis. We quantify the growth rate of monotone Hurwitz numbers in fixed genus by giving universal upper and lower bounds on the radii of convergence of their generating functions. PubDate: 2017-02-02 DOI: 10.1007/s00026-017-0341-5

Authors:Madeline Locus; Ian Wagner Abstract: Abstract Let \({p_{-t}}\) (n) denote the number of partitions of n into t colors. In analogy with Ramanujan’s work on the partition function, Lin recently proved that \({{p_{-3}}(11n+7) \equiv 0}\) (mod 11) for every integer n. Such congruences, those of the form \({{p_{-t}} (\ell n+a) \equiv 0}\) (mod \({\ell}\) ), were previously studied by Kiming and Olsson. If \({\ell \geq 5}\) is prime and \({-t {\epsilon} \{\ell-1, \ell-3\}}\) , then such congruences satisfy \({24a \equiv-t}\) (mod \({\ell}\) ). Inspired by Lin’s example, we obtain natural infinite families of such congruences. If \({\ell \equiv 2}\) (mod 3) ( \({\ell \equiv 3}\) (mod 4) and \({\ell \equiv 11}\) (mod 12), respectively) is prime and \({r \in \{4, 8, 14\}}\) ( \({r \in \{6, 10\}}\) and r = 26, respectively), then for \({t = \ell s-r}\) , where \({s \geq 0}\) , we have that $${p_{-t}}(\ell n + \frac{r(\ell^{2}-1)}{24}-\ell\lfloor\frac{r(\ell^{2}-1)}{24\ell}\rfloor)\equiv0 \,\,({\rm mod} \ell).$$ Moreover, we exhibit infinite families where such congruences cannot hold. PubDate: 2017-02-02 DOI: 10.1007/s00026-017-0342-4

Authors:Ting-Yuan Cheng; Sen-Peng Eu; Tung-Shan Fu; Yi-Lin Lee Abstract: Abstract In this paper, we establish a bijection between standard domino tableaux with at most three rows and partial Motzkin paths. Moreover, we establish a connection between skew standard domino tableaux with at most three rows and a variant of partial Motzkin paths within the nonnegative quadrant and enumerate such tableaux with n dominoes in terms of linear combinations of Motzkin numbers. PubDate: 2017-02-02 DOI: 10.1007/s00026-017-0340-6

Authors:Raimundas Vidunas Abstract: Abstract The maximal number of totally mixed Nash equilibria in games of several players equals the number of block derangements, as proved by McKelvey and McLennan. On the other hand, counting the derangements is a well-studied problem. The numbers are identified as linearization coefficients for Laguerre polynomials. MacMahon derived a generating function for them as an application of his master theorem. This article relates the algebraic, combinatorial, and game-theoretic problems that were not connected before. New recurrence relations, hypergeometric formulas, and asymptotics for the derangement counts are derived. An upper bound for the total number of all Nash equilibria is given. PubDate: 2017-02-02 DOI: 10.1007/s00026-017-0344-2

Authors:Benjamin Braun; Robert Davis Pages: 705 - 717 Abstract: Abstract An interesting open problem in Ehrhart theory is to classify those lattice polytopes having a unimodal h*-vector. Although various sufficient conditions have been found, necessary conditions remain a challenge. In this paper, we consider integrally closed reflexive simplices and discuss an operation that preserves reflexivity, integral closure, and unimodality of the h*-vector, providing one explanation for why unimodality occurs in this setting. We also discuss the failure of proving unimodality in this setting using weak Lefschetz elements. PubDate: 2016-12-01 DOI: 10.1007/s00026-016-0337-6 Issue No:Vol. 20, No. 4 (2016)

Authors:Takayuki Hibi; Liam Solus Pages: 815 - 829 Abstract: Abstract Let k, n, and r be positive integers with k < n and \({r \leq \lfloor \frac{n}{k} \rfloor}\) . We determine the facets of the r-stable n, k-hypersimplex. As a result, it turns out that the r-stable n, k-hypersimplex has exactly 2n facets for every \({r < \lfloor \frac{n}{k} \rfloor}\) . We then utilize the equations of the facets to study when the r-stable hypersimplex is Gorenstein. For every k > 0 we identify an infinite collection of Gorenstein r-stable hypersimplices, consequently expanding the collection of r-stable hypersimplices known to have unimodal Ehrhart \({\delta}\) -vectors. PubDate: 2016-12-01 DOI: 10.1007/s00026-016-0325-x Issue No:Vol. 20, No. 4 (2016)

Authors:Matthew Hyatt Pages: 869 - 881 Abstract: Abstract We introduce new recurrences for the type B and type D Eulerian polynomials, and interpret them combinatorially. These recurrences are analogous to a well-known recurrence for the type A Eulerian polynomials. We also discuss their relationship to polynomials introduced by Savage and Visontai in connection to the real-rootedness of the corresponding Eulerian polynomials. PubDate: 2016-12-01 DOI: 10.1007/s00026-016-0327-8 Issue No:Vol. 20, No. 4 (2016)

Authors:Mitch Phillipson; Catherine H. Yan Pages: 883 - 897 Abstract: Abstract In this paper we give simple bijective proofs that the number of fillings of layer polyominoes with no northeast chains is the same as the number with no southeast chains. We consider 01-fillings and \({{\mathbb{N}}}\) -fillings and prove the results for both strong chains where the smallest rectangle containing the chain is also in the polyomino, and for regular chains where only the corners of the smallest rectangle containing the chain are required to be in the polyomino. PubDate: 2016-12-01 DOI: 10.1007/s00026-016-0326-9 Issue No:Vol. 20, No. 4 (2016)

Abstract: Abstract A 0-Hecke algebra is a deformation of the group algebra of a Coxeter group. Based on work of Norton and Krob-Thibon, we introduce a tableau approach to the representation theory of 0-Hecke algebras of type A, which resembles the classic approach to the representation theory of symmetric groups by Young tableaux and tabloids. We extend this approach to types B and D, and obtain a correspondence between the representation theory of 0-Hecke algebras of types B and D and quasisymmetric functions and noncommutative symmetric functions of types B and D. Other applications are also provided. PubDate: 2016-11-01 DOI: 10.1007/s00026-016-0338-5

Authors:Henning Conrad; Gerhard Röhrle Abstract: Abstract The reflection arrangement of a Coxeter group is a well-known instance of a free hyperplane arrangement. In 2002, Terao showed that equipped with a constant multiplicity each such reflection arrangement gives rise to a free multiarrangement. In this note we show that this multiarrangment satisfies the stronger property of inductive freeness in case the Coxeter group is of type A. PubDate: 2016-10-22 DOI: 10.1007/s00026-016-0335-8

Authors:Jason Fulman; Dennis Stanton Abstract: Abstract Motivated by analogous results for the symmetric group and compact Lie groups, we study the distribution of the number of fixed vectors of a random element of a finite classical group. We determine the limiting moments of these distributions, and find exactly how large the rank of the group has to be in order for the moment to stabilize to its limiting value. The proofs require a subtle use of some q-series identities. We also point out connections with orthogonal polynomials. PubDate: 2016-10-22 DOI: 10.1007/s00026-016-0336-7

Authors:George E. Andrews; Greg Simay Abstract: Abstract The theory of overpartitions is applied to determine formulas for the number of partitions of n where (1) the mth largest part is k and (2) the mth smallest part is k. PubDate: 2016-10-14 DOI: 10.1007/s00026-016-0333-x

Authors:Alin Bostan; Mireille Bousquet-Mélou; Manuel Kauers; Stephen Melczer Abstract: Abstract Many recent papers deal with the enumeration of 2-dimensional walks with prescribed steps confined to the positive quadrant. The classification is now complete for walks with steps in \({\{0, \pm 1\}^{2}}\) : the generating function is D-finite if and only if a certain group associated with the step set is finite. We explore in this paper the analogous problem for 3- dimensional walks confined to the positive octant. The first difficulty is their number: we have to examine no less than 11074225 step sets in \({\{0, \pm 1\}^{3}}\) (instead of 79 in the quadrant case). We focus on the 35548 that have at most six steps. We apply to them a combined approach, first experimental and then rigorous. On the experimental side, we try to guess differential equations. We also try to determine if the associated group is finite. The largest finite groups that we find have order 48 — the larger ones have order at least 200 and we believe them to be infinite. No differential equation has been detected in those cases. On the rigorous side, we apply three main techniques to prove D-finiteness. The algebraic kernel method, applied earlier to quadrant walks, works in many cases. Certain, more challenging, cases turn out to have a special Hadamard structure which allows us to solve them via a reduction to problems of smaller dimension. Finally, for two special cases, we had to resort to computer algebra proofs. We prove with these techniques all the guessed differential equations. This leaves us with exactly 19 very intriguing step sets for which the group is finite, but the nature of the generating function still unclear. PubDate: 2016-10-14 DOI: 10.1007/s00026-016-0328-7

Authors:Michael A. Henning; Alister J. Marcon Abstract: Abstract In this paper, we continue the study of semitotal domination in graphs in [Discrete Math. 324, 13–18 (2014)]. A set \({S}\) of vertices in \({G}\) is a semitotal dominating set of \({G}\) if it is a dominating set of \({G}\) and every vertex in \({S}\) is within distance 2 of another vertex of \({S}\) . The semitotal domination number, \({{\gamma_{t2}}(G)}\) , is the minimum cardinality of a semitotal dominating set of \({G}\) . This domination parameter is squeezed between arguably the two most important domination parameters; namely, the domination number, \({\gamma (G)}\) , and the total domination number, \({{\gamma_{t}}(G)}\) . We observe that \({\gamma (G) \leq {\gamma_{t2}}(G) \leq {\gamma_{t}}(G)}\) . A claw-free graph is a graph that does not contain \({K_{1, \, 3}}\) as an induced subgraph. We prove that if \({G}\) is a connected, claw-free, cubic graph of order \({n \geq 10}\) , then \({{\gamma_{t2}}(G) \leq 4n/11}\) . PubDate: 2016-10-12 DOI: 10.1007/s00026-016-0331-z

Authors:Sylvie Corteel; Sandrine Dasse-Hartaut Abstract: Abstract We give a simple bijection between staircase tableaux and inversion tables. Some nice properties of the bijection allow us to easily compute the generating polynomials of subsets of the staircase tableaux. We also give a combinatorial interpretation of some statistics of these tableaux in terms of permutations. PubDate: 2016-10-12 DOI: 10.1007/s00026-016-0329-6