Abstract: Abstract Motivated by recent work of Bessenrodt, Olsson, and Sellers on unique path partitions, we consider partitions of an integer n wherein the parts are all powers of a fixed integer \({m \geq 2}\) and there are no "gaps" in the parts; that is, if \({m^i}\) is the largest part in a given partition, then \({m^j}\) also appears as a part in the partition for each \({0 \leq j < i}\) . Our ultimate goal is to prove an infinite family of congruences modulo powers of m which are satisfied by these functions. PubDate: 2017-09-08 DOI: 10.1007/s00026-017-0369-6

Authors:Ethan Alwaise; Robert Dicks; Jason Friedman; Lianyan Gu; Zach Harner; Hannah Larson; Madeline Locus; Ian Wagner; Josh Weinstock Abstract: Abstract The partition function p(n), which counts the number of partitions of a positive integer n, is widely studied. Here, we study partition functions pS(n) that count partitions of n into distinct parts satisfying certain congruence conditions. A shifted partition identity is an identity of the form \({{p_{S_1}} (n - H) = {p_{S_2}} (n)}\) for all n in some arithmetic progression. Several identities of this type have been discovered, including two infinite families found by Alladi. In this paper, we use the theory of modular functions to determine the necessary and sufficient conditions for such an identity to exist. In addition, for two specific cases, we extend Alladi’s theorem to other arithmetic progressions. PubDate: 2017-08-28 DOI: 10.1007/s00026-017-0360-2

Authors:Antonio Bernini; Luca Ferrari Abstract: Abstract We introduce vincular pattern posets, then we consider in particular the quasiconsecutive pattern poset, which is defined by declaring σ ≤ τ whenever the permutation τ contains an occurrence of the permutation σ in which all the entries are adjacent in τ except at most the first and the second. We investigate the Möbius function of the quasi-consecutive pattern poset and we completely determine it for those intervals [σ, τ] such that σ occurs precisely once in τ. PubDate: 2017-08-21 DOI: 10.1007/s00026-017-0364-y

Authors:Mikhail Mazin Abstract: Abstract Pak and Stanley introduced a labeling of the regions of a k-Shi arrangement by k-parking functions and proved its bijectivity. Duval, Klivans, and Martin considered a modification of this construction associated with a graph G. They introduced the G-Shi arrangement and a labeling of its regions by G-parking functions. They conjectured that their labeling is surjective, i.e., that every G-parking function appears as a label of a region of the G-Shi arrangement. Later Hopkins and Perkinson proved this conjecture. In particular, this provided a new proof of the bijectivity of Pak-Stanley labeling in the k = 1 case. We generalize Hopkins-Perkinson’s construction to the case of arrangements associated with oriented multigraphs. In particular, our construction provides a simple straightforward proof of the bijectivity of the original Pak-Stanley labeling for arbitrary k. PubDate: 2017-08-21 DOI: 10.1007/s00026-017-0368-7

Authors:Alexey Garber Abstract: Abstract We show that every four-dimensional parallelohedron P satisfies a recently found condition of Garber, Gavrilyuk & Magazinov sufficient for the Voronoi conjecture being true for P. Namely, we show that for every four-dimensional parallelohedron P the one-dimensional homology group of its \({\pi}\) -surface is generated by half-belt cycles. PubDate: 2017-08-21 DOI: 10.1007/s00026-017-0366-9

Authors:Renrong Mao Abstract: Abstract In this paper, we introduce the symmetrized positive rank and crank moments of overpartitions and obtain some inequalities between them. These results enable us to settle a conjecture of Andrews, Chan, Kim, and Osburn on inequalities between ordinary rank and crank moments of overpartitions. PubDate: 2017-08-21 DOI: 10.1007/s00026-017-0367-8

Authors:Miklós Bóna; Marie-Louise Lackner; Bruce E. Sagan Abstract: Abstract Let \({\pi}\) be a permutation of [n] = {1, . . . , n} and denote by \({\ell(\pi)}\) the length of a longest increasing subsequence of \({\pi}\) . Let \({\ell_n,k}\) be the number of permutations \({\pi}\) of [n] with \({\ell(\pi) = k}\) . Chen conjectured that the sequence \({\ell_{n,1}, \ell_{n,2}, . . . , \ell_{n,n}}\) is log concave for every fixed positive integer n. We conjecture that the same is true if one is restricted to considering involutions and we show that these two conjectures are closely related. We also prove various analogues of these conjectures concerning permutations whose output tableaux under the Robinson-Schensted algorithm have certain shapes. In addition, we present a proof of Deift that part of the limiting distribution is log concave. Various other conjectures are discussed. PubDate: 2017-08-19 DOI: 10.1007/s00026-017-0365-x

Authors:G. Lunardon; G. Marino; O. Polverino; R. Trombetti Abstract: Abstract In this paper, elaborating on the link between semifields of dimension n over their left nucleus and \({\mathbb{F}s}\) -linear sets of rank en disjoint from the secant variety \({\Omega(\mathcal{S}_{n,n})}\) of the Segre variety \({\mathcal{S}_{n,n}}\) of \({PG(n^2-1, q), q=s^e}\) , we extend some operations on semifield whose definition relies on dualising the relevant linear set. PubDate: 2017-08-07 DOI: 10.1007/s00026-017-0362-0

Authors:Steven Kelk; Mareike Fischer Abstract: Abstract Within the field of phylogenetics there is great interest in distance measures to quantify the dissimilarity of two trees. Recently, a new distance measure has been proposed: the Maximum Parsimony (MP) distance. This is based on the difference of the parsimony scores of a single character on both trees under consideration, and the goal is to find the character which maximizes this difference. Here we show that computation of MP distance on two binary phylogenetic trees is NP-hard. This is a highly nontrivial extension of an earlier NP-hardness proof for two multifurcating phylogenetic trees, and it is particularly relevant given the prominence of binary trees in the phylogenetics literature. As a corollary to the main hardness result we show that computation of MP distance is also hard on binary trees if the number of states available is bounded. In fact, via a different reduction we show that it is hard even if only two states are available. Finally, as a first response to this hardness we give a simple Integer Linear Program (ILP) formulation which is capable of computing the MP distance exactly for small trees (and for larger trees when only a small number of character states are available) and which is used to computationally verify several auxiliary results required by the hardness proofs. PubDate: 2017-08-07 DOI: 10.1007/s00026-017-0361-1

Authors:Olivia Beckwith; Michael H. Mertens Abstract: Abstract Improving upon previous work [3] on the subject, we use Wright’s Circle Method to derive an asymptotic formula for the number of parts in all partitions of an integer n that are in any given arithmetic progression. PubDate: 2017-08-05 DOI: 10.1007/s00026-017-0363-z

Authors:Chris Jennings-Shaffer Abstract: Abstract We introduce two new integer partition functions, both of which are the number of partition quadruples of n with certain size restrictions. We prove both functions satisfy Ramanujan-type congruences modulo 3, 5, 7, and 13 by use of generalized Lambert series identities and q-series techniques. PubDate: 2017-07-26 DOI: 10.1007/s00026-017-0359-8

Authors:Abdulaziz A. Alanazi; Stephen M. Gagola; Augustine O. Munagi Abstract: 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: 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: 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: 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: 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: 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: 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: 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: 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