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 Annals of Combinatorics   [SJR: 0.849]   [H-I: 15]   [3 followers]  Follow         Hybrid journal (It can contain Open Access articles)    ISSN (Print) 0219-3094 - ISSN (Online) 0218-0006    Published by Springer-Verlag  [2354 journals]
• Syzygies on Tutte Polynomials of Freedom Matroids
• Authors: Joseph P.S. Kung
Abstract: It follows from a theorem of Derksen [J. Algebraic Combin., 30 (2009) 43–86] that the Tutte polynomial of a rank-r matroid on an n-set is “naturally” a linear combination of Tutte polynomials of rank-r size-n freedom matroids. However, the Tutte polynomials of rank-r size-n freedom matroids are not linearly independent. We construct two natural bases for these polynomials and as a corollary, we prove that the Tutte polynomials of rank-r matroids of size-n span a subspace of dimension $${r(n-r)+1}$$ . We also find a generating set for the linear relations between Tutte polynomials of freedom matroids. This generating set is indexed by a pair of intervals, one of size 2 and one of size 4, in the weak order of freedom matroids. This weak order is a distributive lattice and a sublattice of Young’s partition lattice.
PubDate: 2017-10-07
DOI: 10.1007/s00026-017-0370-0

• Arithmetic Properties of m -ary Partitions Without Gaps
• 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

• Shifted Distinct-part Partition Identities in Arithmetic Progressions
• Authors: Ethan Alwaise; Robert Dicks; Jason Friedman; Lianyan Gu; Zach Harner; Hannah Larson; Madeline Locus; Ian Wagner; Josh Weinstock
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

• Vincular Pattern Posets and the Möbius Function of the
Quasi-Consecutive Pattern Poset
• Authors: Antonio Bernini; Luca Ferrari
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

• Multigraph Hyperplane Arrangements and Parking Functions
• Authors: Mikhail Mazin
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

• On $${\pi}$$ π -Surfaces of Four-Dimensional Parallelohedra
• Authors: Alexey Garber
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

• Inequalities Between Odd Moments of Rank and Crank for Overpartitions
• Authors: Renrong Mao
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

• Longest Increasing Subsequences and Log Concavity
• Authors: Miklós Bóna; Marie-Louise Lackner; Bruce E. Sagan
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

• The Generalized Translation Dual of a Semifield
• Authors: G. Lunardon; G. Marino; O. Polverino; R. Trombetti
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

• On the Complexity of Computing MP Distance Between Binary Phylogenetic
Trees
• Authors: Steven Kelk; Mareike Fischer
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

• On the Number of Parts of Integer Partitions Lying in Given Residue
Classes
• Authors: Olivia Beckwith; Michael H. Mertens
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

• Two Partition Functions with Congruences Modulo 3,5,7, and 13
• Authors: Chris Jennings-Shaffer
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

• Combinatorial Proof of a Partition Inequality of Bessenrodt-Ono
• 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

• A Combinatorial Proof of the Smoothness of Catalecticant Schemes
Associated to Complete Intersections
• 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

• Moments of Normally Distributed Random Matrices Given by Generating Series
for Connection Coefficients — Explicit Bijective Computation
• 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

• On the Multiplicity-Free Plethysms p 2 [ $${s_\lambda}$$ s λ ]
• 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

• Primary Components of Codimension Two Lattice Basis Ideals
• 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

• Homotopy Types of Frobenius Complexes
• 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

• The Enumeration of Permutations Avoiding 3124 and 4312
• 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

• Sharp Concentration Inequalities for Deviations from the Mean for Sums of
• 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

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