Authors:Jason Fulman; Robert Guralnick Pages: 295 - 316 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: 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)

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: 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: 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: 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: 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:A. Stoimenow Abstract: In relation to a conjecture of Hoste on the roots of the Alexander polynomial of alternating knots, we prove that any root z of the Alexander polynomial of a 2-bridge (rational) knot or link satisfies \({ z^{1/2}-z^{-1/2} < 2}\) . We extend our result to properties of zeros for some Montesinos knots, and to an analogous statement about the skein polynomial. A similar estimate is derived for alternating 3-braid links. PubDate: 2018-04-27 DOI: 10.1007/s00026-018-0389-x

Authors:Patrick G. Cesarz; Robert S. Coulter 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: 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: 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: 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: 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: 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

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: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