Authors:Jason Fulman; Robert Guralnick Pages: 295 - 316 Abstract: Abstract Feit and Fine derived a generating function for the number of ordered pairs of commuting \(n{\times}n\) matrices over the finite field \({\mathbb{F}_{q}}\) . This has been reproved and studied by Bryan and Morrison from the viewpoint of motivic Donaldson-Thomas theory. In this note, we give a new proof of the Feit-Fine result, and generalize it to the Lie algebra of finite unitary groups and to the Lie algebra of odd characteristic finite symplectic groups. We extract some asymptotic information from these generating functions. Finally, we derive generating functions for the number of commuting nilpotent elements for the Lie algebras of the finite general linear and unitary groups, and of odd characteristic symplectic groups. PubDate: 2018-06-01 DOI: 10.1007/s00026-018-0390-4 Issue No:Vol. 22, No. 2 (2018)

Authors:Andrew Morrison; Frank Sottile Pages: 363 - 375 Abstract: Abstract The Murnaghan-Nakayama rule expresses the product of a Schur function with a Newton power sum in the basis of Schur functions. We establish a version of the Murnaghan- Nakayama rule for Schubert polynomials and a version for the quantum cohomology ring of the Grassmannian. These rules compute all intersections of Schubert cycles with tautological classes coming fromthe Chern character. Like the classical rule, both rules are multiplicity-free signed sums. PubDate: 2018-06-01 DOI: 10.1007/s00026-018-0387-z Issue No:Vol. 22, No. 2 (2018)

Authors:Dong Quan Ngoc Nguyen Abstract: Abstract For a nonnegative integer n, and a prime \({\mathcal{P}}\) in \({\mathbb{F}_{q}[T]}\) , we prove a result that provides a method for computing the number of integers m with \({0 \leq m \leq n}\) for which the Carlitz binomial coefficients \({(_{m}^{n})_{C}}\) fall into each of the residue classes modulo \({\mathcal{P}}\) . Our main result can be viewed as a function field analogue of the Garfield-Wilf theorem. PubDate: 2018-07-27 DOI: 10.1007/s00026-018-0400-6

Authors:Nate Harman Abstract: Abstract We consider the following counting problem related to the card game SET: how many k-element SET-free sets are there in an n-dimensional SET deck' Through a series of algebraic reformulations and reinterpretations, we show the answer to this question satisfies two polynomiality conditions. PubDate: 2018-07-27 DOI: 10.1007/s00026-018-0401-5

Authors:Samuel D. Judge; William J. Keith; Fabrizio Zanello Abstract: Abstract The purpose of this note is to introduce a new approach to the study of one of the most basic and seemingly intractable problems in partition theory, namely, the conjecture that the partition function p(n) is equidistributed modulo 2. Our main result will relate the densities, say, \({\delta_t}\) , of the odd values of the t-multipartition functions \({p_t(n)}\) , for several integers t. In particular, we will show that if \({\delta_t > 0}\) for some \({t \in \{5, 7, 11, 13, 17, 19, 23, 25\}}\) , then (assuming it exists) \({\delta_1 > 0}\) ; that is, p(n) itself is odd with positive density. Notice that, currently, the best unconditional result does not even imply that p(n) is odd for \({\sqrt{x}}\) values of \({n \leq x}\) . In general, we conjecture that \({\delta_t = 1/2}\) for all t odd, i.e., that similarly to the case of p(n), all multipartition functions are in fact equidistributed modulo 2. Our arguments will employ a number of algebraic and analytic methods, ranging from an investigation modulo 2 of some classical Ramanujan identities and several other eta product results, to a unified approach that studies the parity of the Fourier coefficients of a broad class of modular form identities recently introduced by Radu. PubDate: 2018-07-24 DOI: 10.1007/s00026-018-0397-x

Authors:George E. Andrews Abstract: Abstract The paper begins with a study of a couple of classes of partitions in which each even part is smaller than each odd. In one class, a Dyson-type crank exists to explain a mod 5 congruence. The second part of the paper treats the arithmetic and combinatorial properties of the third order mock theta function \({\nu(q)}\) and relates the even part of \({\nu(q)}\) to the partitions initially considered. We also consider a surprisingly simple combinatorial relationship between the cranks and the ranks of the partition of n. PubDate: 2018-07-16 DOI: 10.1007/s00026-018-0398-9

Authors:Kevin Grace; Stefan H. M. van Zwam Abstract: Abstract Geelen, Gerards, and Whittle [3] announced the following result: let \({q = p^k}\) be a prime power, and let \({\mathcal{M}}\) be a proper minor-closed class of GF(q)-representable matroids, which does not contain PG(r − 1, p) for sufficiently high r. There exist integers k, t such that every vertically k-connected matroid in \({\mathcal{M}}\) is a rank- \({(\leq t)}\) perturbation of a frame matroid or the dual of a frame matroid over GF(q). They further announced a characterization of the perturbations through the introduction of subfield templates and frame templates. We show a family of dyadic matroids that form a counterexample to this result. We offer several weaker conjectures to replace the ones in [3], discuss consequences for some published papers, and discuss the impact of these new conjectures on the structure of frame templates. PubDate: 2018-07-16 DOI: 10.1007/s00026-018-0396-y

Authors:L. Bossinger; X. Fang; G. Fourier; M. Hering; M. Lanini Abstract: Abstract We establish an explicit bijection between the toric degenerations of the Grassmannian Gr(2, n) arising from maximal cones in tropical Grassmannians and the ones coming from plabic graphs corresponding to Gr(2, n). We show that a similar statement does not hold for Gr(3, 6). PubDate: 2018-07-16 DOI: 10.1007/s00026-018-0395-z

Authors:Ben Anzis; Shuli Chen; Yibo Gao; Jesse Kim; Zhaoqi Li; Rebecca Patrias Abstract: Abstract In this paper, we work toward answering the following question: given a uniformly random algebra homomorphism from the ring of symmetric functions over \({\mathbb{Z}}\) to a finite field \({\mathbb{F}_{q}}\) , what is the probability that the Schur function \({s_{\lambda}}\) maps to zero' We show that this probability is always at least 1/q and is asymptotically 1/q. Moreover, we give a complete classification of all shapes that can achieve probability 1/q. In addition, we identify certain families of shapes for which the events that the corresponding Schur functions are sent to zero are independent. We also look into the probability that Schur functions are mapped to nonzero values in \({\mathbb{F}_{q}}\) . PubDate: 2018-07-16 DOI: 10.1007/s00026-018-0399-8

Abstract: Abstract In this paper, we provide combinatorial proofs for certain partition identities which arise naturally in the context of Langlands’ beyond endoscopy proposal. These partition identities motivate an explicit plethysm expansion of \({{\rm Sym}^j}\) \({{{\rm Sym}^{k}}V}\) for \({{\rm GL}_2}\) in the case k = 3. We compute the plethysm explicitly for the cases k = 3, 4. Moreover, we use these expansions to explicitly compute the basic function attached to the symmetric power L-function of \({{\rm GL}_2}\) for these two cases. PubDate: 2018-06-06 DOI: 10.1007/s00026-018-0391-3

Abstract: Abstract Alternating sign matrices and totally symmetric self-complementary plane partitions are equinumerous sets of objects for which no explicit bijection is known. In this paper, we identify a subset of totally symmetric self-complementary plane partitions corresponding to permutations by giving a statistic-preserving bijection to permutation matrices, which are a subset of alternating sign matrices. We use this bijection to define a new partial order on permutations, and prove this new poset contains both the Tamari lattice and the Catalan distributive lattice as subposets. We also study a new partial order on totally symmetric self-complementary plane partitions arising from this perspective and show that this is a distributive lattice related to Bruhat order when restricted to permutations. PubDate: 2018-06-06 DOI: 10.1007/s00026-018-0394-0

Authors:Ginji Hamano; Takayuki Hibi; Hidefumi Ohsugi Abstract: Abstract The fractional stable set polytope FRAC(G) of a simple graph G with d vertices is a rational polytope that is the set of nonnegative vectors (x1, . . . , x d ) satisfying x i + xj \({\leq}\) 1 for every edge (i, j) of G. In this paper we show that (i) the \({\delta}\) -vector of a lattice polytope 2FRAC(G) is alternatingly increasing, (ii) the Ehrhart ring of FRAC(G) is Gorenstein, (iii) the coefficients of the numerator of the Ehrhart series of FRAC(G) are symmetric, unimodal and computed by the \({\delta}\) -vector of 2FRAC(G). PubDate: 2018-06-05 DOI: 10.1007/s00026-018-0392-2

Authors:Hugo Parlier; Lionel Pournin Abstract: Abstract We explore several families of flip-graphs, all related to polygons or punctured polygons. In particular, we consider the topological flip-graphs of once punctured polygons which, in turn, contain all possible geometric flip-graphs of polygons with a marked point as embedded sub-graphs. Our main focus is on the geometric properties of these graphs and how they relate to one another. In particular, we show that the embeddings between them are strongly convex (or, said otherwise, totally geodesic). We find bounds on the diameters of these graphs, sometimes using the strongly convex embeddings and show that the topological flip-graph is Hamiltonian. These graphs relate to different polytopes, namely to type D associahedra and a family of secondary polytopes which we call pointihedra. PubDate: 2018-06-05 DOI: 10.1007/s00026-018-0393-1

Authors:Lior Fishman; Keith Merrill; David Simmons Abstract: Abstract Intrinsic Diophantine approximation on fractals, such as the Cantor ternary set, was undoubtedly motivated by questions asked by K. Mahler (1984). One of the main goals of this paper is to develop and utilize the theory of infinite de Bruijn sequences in order to answer closely related questions. In particular, we prove that the set of infinite de Bruijn sequences in \({k \geq 2}\) letters, thought of as a set of real numbers via a decimal expansion, has positive Hausdorff dimension. For a given k, these sequences bear a strong connection to Diophantine approximation on certain fractals. In particular, the optimality of an intrinsic Dirichlet function on these fractals with respect to the height function defined by symbolic representations of rationals follows from these results. PubDate: 2018-04-27 DOI: 10.1007/s00026-018-0384-2

Authors:Patrick G. Cesarz; Robert S. Coulter Abstract: Abstract Neo-difference sets arise in the study of projective planes of Lenz-Barlotti types I.3 and I.4. In the course of their proof that an abelian neo-difference set of order 3n satisfies either n = 1 or 3 n, Ghinelli and Jungnickel produce a Wilbrink-like equation for neo-difference sets of order 3n. In this note we generalise that part of their proof to produce a version of this equation that holds for all neo-difference sets. PubDate: 2018-04-23 DOI: 10.1007/s00026-018-0382-4

Authors:Cristian-Silviu Radu Abstract: Abstract In this paper, we present an algorithm which can prove algebraic relations involving \({\eta}\) -quotients, where \({\eta}\) is the Dedekind eta function. PubDate: 2018-04-23 DOI: 10.1007/s00026-018-0388-y

Authors:Shu Xiao Li Abstract: Abstract The immaculate functions, \({\mathfrak{S}_a}\) , were introduced as a Schur-like basis for NSym, the ring of noncommutative symmetric functions. We investigate their structure constants. These are analogues of Littlewood-Richardson coefficents. We will give a new proof of the left Pieri rule for the \({\mathfrak{S}_a}\) , a translation invariance property for the structure coefficients of the \({\mathfrak{S}_a}\) , and a counterexample to an \({\mathfrak{S}_a}\) -analogue of the saturation conjecture. PubDate: 2018-04-23 DOI: 10.1007/s00026-018-0386-0

Authors:Guo-Niu Han; Huan Xiong Abstract: Abstract Motivated by the Nekrasov-Okounkov formula on hook lengths, the first author conjectured that the Plancherel average of the 2k-th power sum of hook lengths of partitions with size n is always a polynomial of n for any \({k \in \mathbb{N}}\) . This conjecture was generalized and proved by Stanley (Ramanujan J. 23(1–3), 91–105 (2010)). In this paper, inspired by the work of Stanley and Olshanski on the differential poset of Young lattice, we study the properties of two kinds of difference operators D and \({D^{-}}\) defined on functions of partitions. Even though the calculations for higher orders of D are extremely complex, we prove that several wellknown families of functions of partitions are annihilated by a power of the difference operator D. As an application, our results lead to several generalizations of classic results on partitions, including the marked hook formula, Stanley Theorem, Okada-Panova hook length formula, and Fujii-Kanno-Moriyama-Okada content formula. We insist that the Okada constants K r arise directly from the computation for a single partition \({\lambda}\) , without the summation ranging over all partitions of size n. PubDate: 2018-04-23 DOI: 10.1007/s00026-018-0385-1

Authors:Bart De Bruyn Abstract: Abstract We characterize the valuations of the near polygon \({\mathbb{H}_n}\) that are induced by classical valuations of the dual polar space \({DW(2n-1, 2)}\) into which it is isometrically embeddable. An application to near 2n-gons that contain \({\mathbb{H}_n}\) as a full subgeometry is given. PubDate: 2018-04-23 DOI: 10.1007/s00026-018-0383-3

Authors:Jakob Ablinger; Carsten Schneider Abstract: Abstract Indefinite nested sums are important building blocks to assemble closed forms for combinatorial counting problems or for problems that arise, e.g., in particle physics. Concerning the simplicity of such formulas an important subtask is to decide if the arising sums satisfy algebraic relations among each other. Interesting enough, algebraic relations of such formal sums can be derived from combinatorial quasi-shuffle algebras. We will focus on the following question: can one find more relations if one evaluates these sums to sequences and looks for relations within the ring of sequences. In this article we consider the sequences of the rather general class of (cyclotomic) harmonic sums and show that their relations coincide with the relations found by their underlying quasi-shuffle algebra. In order to derive this result, we utilize the quasi-shuffle algebra and construct a difference ring with the following property: (1) the generators of the difference ring represent (cyclotomic) harmonic sums, (2) they generate within the ring all (cyclotomic) harmonic sums, and (3) the sequences produced by the generators are algebraically independent among each other. This means that their sequences do not satisfy any polynomial relations. The proof of this latter property is obtained by difference ring theory and new symbolic summation results. As a consequence, any sequence produced by (cyclotomic) harmonic sums can be formulated within our difference ring in an optimal way: there does not exist a subset of the arising sums in which the sequence can be formulated as polynomial expression. PubDate: 2018-04-23 DOI: 10.1007/s00026-018-0381-5