Authors:Wenwen Fan; Cai Heng Li; Hai Peng Qu Abstract: We show that a complete bipartite graph \({{\bf K}_{{p^e}, p_{f}}}\) , where p is an odd prime, has an edge-transitive embedding in an orientable surface with all faces bounded by simple cycles if and only if e = f. There are exactly \({p^{2(e-1)}}\) such embeddings up to isomorphism. Among them, \({p^{e-1}}\) are orientably regular, one of which is reflexible and \({p^{e-1} -1}\) form chiral pairs. The remaining \({p^{2(e-1)} - p^{e-1}}\) embeddings are non-regular (not arc-transitive). All of these embeddings have genus \({\frac{1}{2} (p^{e}-1) (p^{e}-2)}\) . PubDate: 2018-02-05 DOI: 10.1007/s00026-018-0373-5

Authors:Padraig Ó Catháin Abstract: In this short note we construct codes of length 4n with 8n+8 codewords and minimum distance 2n−2 whenever 4n+4 is the order of a Hadamard matrix. This generalises work of Constantine who obtained a similar result in the special case that n is a prime power. PubDate: 2018-02-05 DOI: 10.1007/s00026-018-0379-z

Authors:Igor E. Shparlinski Abstract: We give efficient constructions of reasonably small dominating sets of various types in a circulant graph on n notes and k distinct chord lengths. The structure of a cyclic group underlying circulant graph makes them suitable for applications of methods of analytic number theory. In particular, our results are based on bounds on some double exponential sums. PubDate: 2018-02-05 DOI: 10.1007/s00026-018-0377-1

Authors:Emily Barnard; Emily Meehan; Nathan Reading; Shira Viel Abstract: We construct universal geometric coefficients for the cluster algebra associated to the four-punctured sphere and obtain, as a by-product, the g-vectors of cluster variables. We also construct the rational part of the mutation fan. These constructions rely on a classification of the allowable curves (the curves which can appear in quasi-laminations). The classification allows us to prove the Null Tangle Property for the four-punctured sphere, thus adding this surface to a short list of surfaces for which this property is known. The Null Tangle Property then implies that the shear coordinates of allowable curves are the universal coefficients. We compute shear coordinates explicitly to obtain universal geometric coefficients. PubDate: 2018-02-05 DOI: 10.1007/s00026-018-0378-0

Authors:Soojin Cho; Kyoungsuk Park Abstract: A symmetry of (t, q)-Eulerian numbers of type B is combinatorially proved by defining an involution preserving many important statistics on the set of permutation tableaux of type B, which solves the problem suggested by Corteel in [12]. This involution also proves a symmetry of the generating polynomial \({\hat{D}_{n,k} (p, q, r)}\) of the numbers of crossings and alignments, and hence q-Eulerian numbers of type A defined by Lauren K.Williams. By considering a restriction of our bijection, we were led to define a new statistic on the permutations of type D and (t, q)-Eulerian numbers of type D, which is proved to have a particular symmetry as well. We conjecture that our new statistic is in the family of Eulerian statistics for the permutations of type D. PubDate: 2018-02-05 DOI: 10.1007/s00026-018-0372-6

Authors:Rebecca L. Jayne; Kailash C. Misra Abstract: For \({\ell \geq 1}\) and \({k \geq 2}\) , we consider certain admissible sequences of k−1 lattice paths in a colored \({\ell \times \ell}\) square. We show that the number of such admissible sequences of lattice paths is given by the sum of squares of the number of standard Young tableaux of shape \({\lambda \vdash \ell}\) with \({l(\lambda) \leq k}\) , which is also the number of (k + 1)k··· 21-avoiding permutations in \({S_\ell}\) . Finally, we apply this result to the representation theory of the affine Lie algebra \({\widehat{sl}(n)}\) and show that this gives the multiplicity of certain maximal dominant weights in the irreducible highest weight \({\widehat{sl}(n)}\) -module \({V(k \Lambda_0)}\) . PubDate: 2018-02-02 DOI: 10.1007/s00026-018-0374-4

Authors:David B. Leep; Claus Schubert Abstract: We calculate the dimensions of the intersections of maximal subspaces of zeros of a nonsingular pair of quadratic forms. We then count the number of sets of distinct such subspaces that intersect in a given dimension. PubDate: 2018-02-01 DOI: 10.1007/s00026-018-0375-3

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

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: 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: 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: 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: 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: 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: 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: 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: 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: 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: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