Abstract: The well-known Cauchy Product Identities state that \(\prod _{i,j} (1-x_iy_j)^{-1}= \sum _{\lambda } s_{\lambda }(X) s_{\lambda }(Y)\) and \(\prod _{i,j} (1+x_iy_j)=\sum _{\lambda } s_{\lambda }(X)s_{\lambda '}(Y)\) , where \(s_{\lambda }\) denotes a Schur polynomial. The classical bijective proofs of these identities use the Robinson–Schensted–Knuth (RSK) algorithm to map matrices with entries in \({\mathbb {Z}}_{\ge 0}\) or \(\{0,1\}\) to pairs of semistandard tableaux. This paper gives new involution-based proofs of the Cauchy Product Identities using the abacus model for antisymmetrized Schur polynomials. These proofs provide a novel combinatorial perspective on these formulas in which carefully engineered sign cancellations gradually impose more and more structure on collections of matrices. PubDate: 2019-05-21

Abstract: We generalize a number of works on the zeros of certain level 1 modular forms to a class of weakly holomorphic modular functions whose q-expansions satisfy the following: $$\begin{aligned} f_k(A; \tau )\, {:}{=}\,q^{-k}(1+a(1)q+a(2)q^2+\cdots ) + O(q), \end{aligned}$$ where a(n) are numbers satisfying a certain analytic condition. We show that the zeros of such \(f_k(\tau )\) in the fundamental domain of \(\text {SL}_2(\mathbb {Z})\) lie on \( \tau =1\) and are transcendental. We recover as a special case earlier work of Witten on extremal “partition” functions \(Z_k(\tau )\) . These functions were originally conceived as possible generalizations of constructions in three-dimensional quantum gravity. PubDate: 2019-05-20

Abstract: Affine structures of a group G (=braces with adjoint group G) are characterized equationally without assuming further invertibility conditions. If G is finite, the construction of affine structures is reduced to affine structures of p, q-groups. The delicate relationship between finite solvable groups and involutive Yang–Baxter groups is further clarified by showing that much of an affine structure is already inherent in the Sylow system of the group. Semidirect products of braces are modified (shifted) in two ways to handle affine structures of semidirect products of groups. As an application, two of Vendramin’s conjectures on affine structures of p, q-groups are verified. A further example illustrates what happens beyond semidirect products. PubDate: 2019-05-17

Abstract: We prove various inequalities between the number of partitions with the bound on the largest part and some restrictions on occurrences of parts. We explore many interesting consequences of these partition inequalities. In particular, we show that for \(L\ge 1\) , the number of partitions with \(l-s \le L\) and \(s=1\) is greater than the number of partitions with \(l-s\le L\) and \(s>1\) . Here l and s are the largest part and the smallest part of the partition, respectively. PubDate: 2019-05-14

Abstract: We examine the non-symmetric Macdonald polynomials \(\mathrm {E}_\lambda \) at \(q=1\) , as well as the more general permuted-basement Macdonald polynomials. When \(q=1\) , we show that \(\mathrm {E}_\lambda (\mathbf {x};1,t)\) is symmetric and independent of t whenever \(\lambda \) is a partition. Furthermore, we show that, in general \(\lambda \) , this expression factors into a symmetric and a non-symmetric part, where the symmetric part is independent of t, and the non-symmetric part only depends on \(\mathbf {x}\) , t, and the relative order of the entries in \(\lambda \) . We also examine the case \(q=0\) , which gives rise to the so-called permuted-basement t-atoms. We prove expansion properties of these t-atoms, and, as a corollary, prove that Demazure characters (key polynomials) expand positively into permuted-basement atoms. This complements the result that permuted-basement atoms are atom-positive. Finally, we show that the product of a permuted-basement atom and a Schur polynomial is again positive in the same permuted-basement atom basis. Haglund, Luoto, Mason, and van Willigenburg previously proved this property for the identity basement and the reverse identity basement, so our result can be seen as an interpolation (in the Bruhat order) between these two results. The common theme in this project is the application of basement-permuting operators as well as combinatorics on fillings, by applying results in a previous article by Per Alexandersson. PubDate: 2019-05-11

Abstract: For a given lattice polytope, two fundamental problems within the field of Ehrhart theory are (1) to determine if its (Ehrhart) \(h^*\) -polynomial is unimodal and (2) to determine if its Ehrhart polynomial has only positive coefficients. The former property of a lattice polytope is known as Ehrhart unimodality and the latter property is known as Ehrhart positivity. These two properties are often simultaneously conjectured to hold for interesting families of lattice polytopes, yet they are typically studied in parallel. As to answer a question posed at the 2017 Introductory Workshop to the MSRI Semester on Geometric and Topological Combinatorics, the purpose of this note is to show that there is no general implication between these two properties in any dimension greater than two. To do so, we investigate these two properties for families of well-studied lattice polytopes, assessing one property where previously only the other had been considered. Consequently, new examples of each phenomena are developed, some of which provide an answer to an open problem in the literature. The well-studied families of lattice polytopes considered include zonotopes, matroid polytopes, simplices of weighted projective spaces, empty lattice simplices, smooth polytopes, and \({\varvec{s}}\) -lecture hall simplices. PubDate: 2019-05-10

Abstract: Fraenkel and Peled have defined the minimal excludant or “ \({{\,\mathrm{mex}\,}}\) ” function on a set S of positive integers is the least positive integer not in S. For each integer partition \(\pi \) , we define \({{\,\mathrm{mex}\,}}(\pi )\) to be the least positive integer that is not a part of \(\pi \) . Define \(\sigma {{\,\mathrm{mex}\,}}(n)\) to be the sum of \({{\,\mathrm{mex}\,}}(\pi )\) taken over all partitions of n. It will be shown that \(\sigma {{\,\mathrm{mex}\,}}(n)\) is equal to the number of partitions of n into distinct parts with two colors. Finally the number of partitions \(\pi \) of n with \({{\,\mathrm{mex}\,}}(\pi )\) odd is almost always even. PubDate: 2019-05-03

Abstract: In Haglund et al. (Trans. Amer. Math. Soc. 370(6):4029–4057, 2018), Haglund, Remmel and Wilson introduce a conjecture which gives a combinatorial prediction for the result of applying a certain operator to an elementary symmetric function. This operator, defined in terms of its action on the modified Macdonald basis, has played a role in work of Garsia and Haiman on diagonal harmonics, the Hilbert scheme, and Macdonald polynomials (Garsia and Haiman in J. Algebraic Combin. 5:191–244, 1996; Haiman in Invent. Math. 149:371–407, 2002). The Delta Conjecture involves two parameters q, t; in this article we give the first proof that the Delta Conjecture is true when \(q=0\) or \(t=0\) . PubDate: 2019-05-03

Abstract: There have been a number of papers on partitions in which the parity of parts plays a central role. In this paper, the parts of partitions are separated by parity, either all odd parts are smaller than all even parts or vice versa. This concept first arose in a study related to the third-order mock theta function \(\nu (q)\) . The current study also leads back to one of the Ramanujan’s more mysterious functions. PubDate: 2019-05-02

Abstract: We investigate combinatorial properties of a family of probability distributions on finite abelian p-groups. This family includes several well-known distributions as specializations. These specializations have been studied in the context of Cohen–Lenstra heuristics and cokernels of families of random p-adic matrices. PubDate: 2019-04-27

Abstract: An arithmetic matroid is weakly multiplicative if the multiplicity of at least one of its bases is equal to the product of the multiplicities of its elements. We show that if such an arithmetic matroid can be represented by an integer matrix, then this matrix is uniquely determined. This implies that the integral cohomology ring of a centered toric arrangement whose arithmetic matroid is weakly multiplicative is determined by its poset of layers. This partially answers a question asked by Callegaro–Delucchi. PubDate: 2019-04-26

Abstract: In 1997 Oda conjectured that every smooth lattice polytope has the integer decomposition property. We prove Oda’s conjecture for centrally symmetric 3-dimensional polytopes, by showing they are covered by lattice parallelepipeds and unimodular simplices. PubDate: 2019-03-11

Abstract: We generalize the generating series of the Dyson ranks and \(M_2\) -ranks of overpartitions to obtain k-fold variants and give a combinatorial interpretation of each. The k-fold generating series correspond to the full ranks of two families of buffered Frobenius representations, which generalize Lovejoy’s first and second Frobenius representations of overpartitions, respectively. PubDate: 2019-02-28

Authors:Richard Ehrenborg; Alex Happ Abstract: We give a short proof that the f-vector of the descent polytope \({{\,\mathrm{DP}\,}}_{\mathbf {v}}\) is componentwise maximized when the word \(\mathbf {v}\) is alternating. Our proof uses an f-vector analog of the boustrophedon transform. PubDate: 2019-02-12 DOI: 10.1007/s00026-019-00422-1

Authors:Geoffrey Pearce; Cheryl E. Praeger Abstract: A graph is Cartesian decomposable if it is isomorphic to a Cartesian product of strictly smaller graphs, each of which has more than one vertex and admits no such decomposition. These smaller graphs are called the Cartesian-prime factors of the Cartesian decomposition, and were shown, by Sabidussi and Vizing independently, to be uniquely determined up to isomorphism. We characterise by their parameters those generalised Paley graphs which are Cartesian decomposable, and we prove that for such graphs, the Cartesian-prime factors are themselves smaller generalised Paley graphs. This generalises a result of Lim and the second author which deals with the case where all the Cartesian-prime factors are complete graphs. These results contribute to the determination, by parameters, of generalised Paley graphs with automorphism groups larger than the one-dimensional affine subgroups used to define them. They also shed light on the structure of primitive cyclotomic association schemes. PubDate: 2019-02-10 DOI: 10.1007/s00026-019-00423-0

Authors:Mingjia Yang Abstract: Enumeration problems related to words avoiding patterns as well as permutations that contain the pattern 123 exactly once have been studied in great detail. However, the problem of enumerating words that contain the pattern 123 exactly once is new and will be the focus of this paper. Previously, Zeilberger provided a shortened version of Burstein’s combinatorial proof of Noonan’s theorem which states that the number of permutations with exactly one 321 pattern is equal to \(\frac{3}{n} \left( {\begin{array}{c}2n\\ n+3\end{array}}\right) \) . Surprisingly, a similar method can be directly adapted to words. We are able to use this method to find a formula enumerating the words with exactly one 123 pattern. Further inspired by Shar and Zeilberger’s work on generating functions enumerating 123-avoiding words with r occurrences of each letter, we examine the algebraic equations for generating functions for words with r occurrences of each letter and with exactly one 123 pattern. PubDate: 2019-02-08 DOI: 10.1007/s00026-019-00416-z

Authors:Sami Mustapha Abstract: We prove that the sequence \(\left( e_n^{\Gamma }\right) _{n\in \mathbb {N}}\) of numbers of excursions in a three-quadrant cone corresponding to a non-singular step set \(\Gamma \subset \{0, \pm 1\}^2\) with an infinite group does not satisfy any nontrivial linear recurrence with polynomial coefficients. This gives a positive answer to a question asked by M. Bousquet-Mélou about D-finiteness of the trivariate generating function of the numbers of walks with given length and prescribed ending point. In the process, we determine the asymptotics of \(e_n^{\Gamma }\) , \(n\rightarrow \infty \) , for the 74 non-singular two-dimensional models, giving the first complete computation of excursions asymptotics in a non-convex cone. Moreover, using potential theoretic comparison arguments, we give the asymptotics of the number of walks avoiding the negative quadrant and of length n for all non-singular step sets having zero drift. PubDate: 2019-02-06 DOI: 10.1007/s00026-019-00413-2

Authors:Oliver Roche-Newton; Igor E. Shparlinski; Arne Winterhof Abstract: Balog and Wooley have recently proved that any subset \({\mathcal {A}}\) of either real numbers or of a prime finite field can be decomposed into two parts \({\mathcal {U}}\) and \({\mathcal {V}}\) , one of small additive energy and the other of small multiplicative energy. In the case of arbitrary finite fields, we obtain an analogue that under some natural restrictions for a rational function f both the additive energies of \({\mathcal {U}}\) and \(f({\mathcal {V}})\) are small. Our method is based on bounds of character sums which leads to the restriction \(\# {\mathcal {A}}> q^{1/2}\) , where q is the field size. The bound is optimal, up to logarithmic factors, when \(\# {\mathcal {A}}\ge q^{9/13}\) . Using \(f(X)=X^{-1}\) we apply this result to estimate some triple additive and multiplicative character sums involving three sets with convolutions \(ab+ac+bc\) with variables a, b, c running through three arbitrary subsets of a finite field. PubDate: 2019-02-02 DOI: 10.1007/s00026-019-00420-3

Authors:Gene B. Kim Abstract: The distribution of descents in certain conjugacy classes of \(S_n\) has been previously studied, and it is shown that its moments have interesting properties. This paper provides a bijective proof of the symmetry of the descents and major indices of matchings (also known as fixed point free involutions) and uses a generating function approach to prove an asymptotic normality theorem for the number of descents in matchings. PubDate: 2019-02-01 DOI: 10.1007/s00026-019-00414-1

Authors:Benny Chor; Péter L. Erdős; Yonatan Komornik Abstract: Given two binary trees on N labeled leaves, the quartet distance between the trees is the number of disagreeing quartets. By permuting the leaves at random, the expected quartet distance between the two trees is \(\frac{2}{3}\left( {\begin{array}{c}N\\ 4\end{array}}\right) \) . However, no strongly explicit construction reaching this bound asymptotically was known. We consider complete, balanced binary trees on \(N=2^n\) leaves, labeled by n bits long sequences. Ordering the leaves in one tree by the prefix order, and in the other tree by the suffix order, we show that the resulting quartet distance is \(\left( \frac{2}{3} + o(1)\right) \left( {\begin{array}{c}N\\ 4\end{array}}\right) \) , and it always exceeds the \(\frac{2}{3}\left( {\begin{array}{c}N\\ 4\end{array}}\right) \) bound. PubDate: 2019-01-05 DOI: 10.1007/s00026-018-0411-3