Authors:Abdulaziz A. Alanazi; Stephen M. Gagola; Augustine O. Munagi Abstract: We provide a combinatorial proof of the inequality \({p(a)p(b) > p(a+b)}\) , where p(n) is the partition function and a, \({b > 1}\) are integers satisfying \({a+b > 9}\) . This problem was posed by Bessenrodt and Ono who used the inequality to study a new multiplicative property of an extended partition function [Ann. Combin. 20, 59–64 (2016)]. PubDate: 2017-07-21 DOI: 10.1007/s00026-017-0358-9

Authors:Alexander Isaev Abstract: For zero-dimensional complete intersections with homogeneous ideal generators of equal degrees over an algebraically closed field of characteristic zero, we give a combinatorial proof of the smoothness of the corresponding catalecticant schemes along an open subset of a particular irreducible component. PubDate: 2017-07-14 DOI: 10.1007/s00026-017-0357-x

Authors:Ekaterina Vassilieva Abstract: This paper is devoted to the explicit computation of some generating series for the connection coefficients of the double cosets of the hyperoctahedral group that arise in the study of the spectra of normally distributed random matrices. Aside their direct algebraic and combinatorial interpretations in terms of factorizations of permutations with specific properties, these connection coefficients are closely linked to the theory of zonal spherical functions and zonal polynomials. As shown by Hanlon, Stanley, Stembridge (1992), their generating series in the basis of power sum symmetric functions is equal to the mathematical expectation of the trace of (XUYU t ) n where X and Y are given symmetric matrices, U is a random real valued square matrix of standard normal distribution and n a non-negative integer. We provide the first explicit evaluation of these series in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some decorated forests. As a corollary we provide a simple explicit evaluation of a similar generating series that gives the mathematical expectation of the trace of (XUYU*) n when U is complex valued and X and Y are given hermitian matrices and recover a former result by Morales and Vassilieva (2009). PubDate: 2017-07-07 DOI: 10.1007/s00026-017-0356-y

Authors:Luisa Carini Abstract: We determine all the shapes \({\lambda}\) such that the plethysms \({p_2}\) [ \({s_\lambda}\) ](x) of the power symmetric function \({p_2}\) (x) and the Schur function \({s_\lambda}\) (x) are multiplicity-free. PubDate: 2017-07-06 DOI: 10.1007/s00026-017-0354-0

Authors:Zekiye Sahin Eser; Laura Felicia Matusevich Abstract: We provide explicit combinatorial descriptions of the primary components of codimension two lattice basis ideals. As an application, we compute the set of parameters for which a bivariate Horn system of hypergeometric differential equations is holonomic. PubDate: 2017-07-06 DOI: 10.1007/s00026-017-0355-z

Authors:Shouta Tounai Abstract: Let Λ be a submonoid of the additive monoid \({\mathbb{N}}\) . There is a natural order on Λ defined by \({\lambda \leq \lambda +\mu}\) for \({\lambda,\mu \in \Lambda}\) . A Frobenius complex of Λ is defined to be the order complex of an open interval of Λ. Suppose \({r \geq 2}\) and let \({\rho}\) be a reducible element of Λ. We construct the additive monoid \({\Lambda[\rho/r]}\) obtained from Λ by adjoining a solution to the equation \({r\alpha=\rho}\) . We show that any Frobenius complex of \({\Lambda[\rho/r]}\) is homotopy equivalent to a wedge of iterated suspensions of Frobenius complexes of Λ. As a consequence, we derive a formula for the multi-graded Poincaré series associated to \({\Lambda[\rho/r]}\) . As an application, we determine the homotopy types of the Frobenius complexes of some additive monoids. For example, we show that if Λ is generated by a finite geometric sequence, then any Frobenius complex of Λ is homotopy equivalent to a wedge of spheres. PubDate: 2017-05-17 DOI: 10.1007/s00026-017-0353-1

Authors:Jay Pantone Abstract: We find the generating function for the class of all permutations that avoid the patterns 3124 and 4312 by showing that it is an inflation of the union of two geometric grid classes. PubDate: 2017-05-12 DOI: 10.1007/s00026-017-0352-2

Authors:Harrie Hendriks; Martien C. A. van Zuijlen Abstract: For a fixed unit vector \({a = (a_1, a_2,..., a_n) \in S^{n-1}}\) , that is, \({\sum^n_{i=1} a^2_1 = 1}\) , we consider the 2 n signed vectors \({\varepsilon = (\varepsilon_1, \varepsilon_2,..., \varepsilon_n) \in \{-1, 1\}^n}\) and the corresponding scalar products \({a \cdot \varepsilon = \sum^n_{i=1} a_i \varepsilon_i}\) . In [3] the following old conjecture has been reformulated. It states that among the 2 n sums of the form \({\sum \pm a_i}\) there are not more with \({ \sum^n_{i=1} \pm a_i > 1}\) than there are with \({ \sum^n_{i=1} \pm a_i \leq 1}\) . The result is of interest in itself, but has also an appealing reformulation in probability theory and in geometry. In this paper we will solve an extension of this problem in the uniform case where \({a_1 = a_2 = \cdot\cdot\cdot = a_n = n^{-1/2}}\) . More precisely, for S n being a sum of n independent Rademacher random variables, we will give, for several values of \({\xi}\) , precise lower bounds for the probabilities $$P_n: = \mathbb{P} \{-\xi \sqrt{n} \leq S_n \leq \xi \sqrt{n}\}$$ or equivalently for $$Q_n: = \mathbb{P} \{-\xi \leq T_n \leq \xi \},$$ where \({T_n}\) is a standardized binomial random variable with parameters n and \({p = 1/2}\) . These lower bounds are sharp and much better than for instance the bound that can be obtained from application of the Chebyshev inequality. In case \({\xi = 1}\) Van Zuijlen solved this problem in [5]. We remark that our bound will have nice applications in probability theory and especially in random walk theory (cf. [1, 2]). PubDate: 2017-05-11 DOI: 10.1007/s00026-017-0351-3

Authors:Héctor Blandin Abstract: This work enrols the research line of M. Haiman on the Operator Theorem (the former Operator Conjecture). Given a \({\mathfrak{S}_n}\) -stable family F of homogeneous polynomials in the variables \({x_i j}\) with \({1 \leq i \leq \ell}\) and \({1 \leq j \leq n}\) . We define the polarization module generated by the family F, as the smallest vector space closed under taking partial derivatives and closed under the action of polarization operators that contains F. These spaces are representations of the direct product \({\mathfrak{S}_n \times GL_\ell(\mathbb{C})}\) . We compute the graded Frobenius characteristic of these modules. We use some basic tools to study these spaces and give some in-depth calculations of low degree examples of a family or a single symmetric polynomial. PubDate: 2017-04-28 DOI: 10.1007/s00026-017-0350-4

Authors:Felix Breuer; Zafeirakis Zafeirakopoulos 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: 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: 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: 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: 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: 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: 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: 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: 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:Henning Conrad; Gerhard Röhrle 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:Michael A. Henning; Alister J. Marcon 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