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In 2007, Andrews introduced the odd rank of odd Durfee symbols. Let N 0 ( m , n ) denote the number of odd Durfee symbols of n with odd rank m, and N 0 ( r , m ; n ) be the number of odd Durfee symbols of n with odd rank congruent to r modulo m. In this paper, we give the generating functions for N 0 ( r , 12 ; n ) by utilizing some identities involving Appell-Lerch sums m ( x , q , z ) and a universal mock theta function g ( x , q ) . Based on these formulas, we determine the signs of N 0 ( r , 12 ; 4 n + t ) − N 0 ( s , 12 ; 4 n + t ) for all 0 ≤ r , s ≤ 6 and 0 ≤ t ≤ 3 . In particular, we prove that N 0 ( 2 , 12 ; 4 n + 1 ) = N 0 ( 4 , 12 ; 4 n + 1 ) . Moreover, let D k 0 ( n ) denote the number of k-marked odd Durfee symbols of n. Andrews conjectured that D 2 0 ( 8 n + s ) and D 3 0 ( 16 n + t ) are even with s ∈ { 4 , 6 } and t ∈ { 1 , 9 , 11 , 13 } which were confirmed by Wang. In this paper, we found new congruences for D k 0 ( n ) . In particular, for k = 2 or 3, we give characterizations of n such that D k 0 ( n ) is odd and prove that D k 0 ( n ) take even values with probability 1 for n ≥ 0 .

Phylogenetic networks are used to represent evolutionary relationships between species in biology. Such networks are often categorized into classes by their topological features, which stem from both biological and computational motivations. We study two network classes in this paper: tree-based networks and orchard networks. Tree-based networks are those that can be obtained by inserting edges between the edges of an underlying tree. Orchard networks are a recently introduced generalization of the class of tree-child networks. Structural characterizations have already been discovered for tree-based networks; this is not the case for orchard networks. In this paper, we introduce cherry covers—a unifying characterization of both network classes—in which we decompose the edges of the networks into so-called cherry shapes and reticulated cherry shapes. We show that cherry covers can be used to characterize the class of tree-based networks as well as the class of orchard networks. Moreover, we also generalize these results to non-binary networks.

Frieze patterns form a nexus between algebra, combinatorics, and geometry. T-paths with respect to triangulations of surfaces have been used to obtain expansion formulae for cluster variables. This paper will introduce the concepts of weak friezes and T-paths with respect to dissections of polygons. Our main result is that weak friezes are characterised by satisfying an expansion formula which we call the T-path formula. We also show that weak friezes can be glued together, and that the resulting weak frieze is a frieze if and only if so was each of the weak friezes being glued., 17 pages

In this paper, we study the Wilf-type equivalence relations among multiset permutations. We identify all multiset equivalences among pairs of patterns consisting of a pattern of length three and another pattern of length at most four. To establish our results, we make use of a variety of techniques, including Ferrers-equivalence arguments, sorting by minimal/maximal letters, analysis of active sites and direct bijections. In several cases, our arguments may be extended to prove multiset equivalences for infinite families of pattern pairs. Our results apply equally well to the Wilf-type classification of compositions, and as a consequence, we obtain a complete description of the Wilf-equivalence classes for pairs of patterns of type (3,3) and (3,4) on compositions, with the possible exception of two classes of type (3,4)., 26 pages

Grinberg defined Nyldon words as those words which cannot be factorized into a sequence of lexicographically nondecreasing smaller Nyldon words. He was inspired by Lyndon words, defined the same way except with "nondecreasing" replaced by "nonincreasing." Charlier, Philibert, and Stipulanti proved that, like Lyndon words, any word has a unique nondecreasing factorization into Nyldon words. They also show that the Nyldon words form a right Lazard set, and equivalently, a right Hall set. In this paper, we provide a new proof of unique factorization into Nyldon words related to Hall set theory and resolve several questions of Charlier et al. In particular, we prove that Nyldon words of a fixed length form a circular code, we prove a result on factorizing powers of words into Nyldon words, and we investigate the Lazard procedure for generating Nyldon words. We show that these results generalize to a new class of Hall sets, of which Nyldon words are an example, that we name "Nyldon-like sets.", Comment: 27 pages; incorporated reviewer comments

In this paper we consider matrix codes equipped with either the Hamming or the rank distance and study the optimal sizes of such codes with a given minimum distance. We use a maximality argument (the Gilbert-Varshamov argument) to show that, asymptotically, maximum size codes are optimal and have the same size for both rank and Hamming distances.

Latin tableaux are a generalization of Latin squares, which first appeared in the early 2000's in a paper of Chow, Fan, Goemans, and Vondr\'{a}k. Here, we extend the notion of isotopy, a permutation group action, from Latin squares to Latin tableaux. We define isotopy graphs for Latin tableaux, which encode the structure of orbits under the isotopy action, and investigate the relationship between the shape of a Latin tableau and the structure of its isotopy graph. Our main result shows that for any positive integer $d$, there is a Latin tableau whose isotopy graph is a $d$-dimensional cube. We show that most isotopy graphs are triangle-free, and we give a characterization of all the Latin tableaux for which the isotopy graph contains a triangle. We also give a formula for the degree of a vertex of each component of an isotopy graph which depends on both the shape of the Latin Tableaux and the filling., Comment: 19 Pages, 8 Figures Replacement: Referee's suggestions added. Fixed a number of typos and sentence-level issues. Clarified exposition in several places. Corrected exposition about history of the Alon-Tarsi conjecture; see Remark 1.2 in current version. Fixed typos in appendix, which made several tableaux not Latin.Second Replacement: fixed typos in appendix, conjecture 1.1

In this paper we present some new identities for multiple polylogarithms (abbr. MPLs) and multiple harmonic star sums (abbr. MHSSs) by using the methods of iterated integral computations of logarithm functions. Then, by applying these formulas obtained, we establish some explicit relations between Kaneko-Yamamoto type multiple zeta values (abbr. K-Y MZVs), multiple zeta values (abbr. MZVs) and MPLs. Further, we find some explicit relations between MZVs and multiple zeta star values (abbr. MZSVs). Furthermore, we define an Apery-type variant of MZSVs ζ B ⋆ ( k ) (called multiple zeta B-star values, abbr. MZBSVs) which involve MHSSs and central binomial coefficients, and establish some explicit connections among MZVs, alternating MZVs and MZBSVs by using the method of iterated integrals. Finally, some interesting consequences and illustrative examples are presented.

Let G R be the set of real points of a complex linear reductive group and G ˆ λ , its classes of irreducible admissible representations with infinitesimal integral regular character λ. In this case, each cell of representations is associated to a special nilpotent orbit. This helps organize the corresponding set of irreducible Harish-Chandra modules. The goal of this paper is to bypass the need for the character table of the Weyl group associated with G ˆ λ in the Springer correspondence, by describing a new and less computationally intensive parametrization of irreducible representations of simple complex Weyl groups.

The polygon valued Minkowski valuations in R 2 intertwining with the special linear group are completely classified in this paper without any additional assumptions. Our results characterize projection bodies and difference bodies in R 2 . There are also some SL ( 2 ) intertwining Minkowski valuations that can only be defined in R 2 . It somehow suggests why such classification problems are more complicated than higher dimensional spaces.

In this paper, we introduce the concept of weakly increasing trees on a multiset M, which is an extension of plane trees and increasing trees on the set { 0 , 1 , … , n } . We define the M-Eulerian–Narayana polynomial for weakly increasing trees on a multiset M, which interpolates between the Eulerian polynomial and the Narayana polynomial. We obtain a compact product formula for the number of weakly increasing trees on a general multiset. Inspired by some remarkable equidistributions between multipermutations and s-inversion sequences, we establish two connections between our M-Eulerian–Narayana polynomials for the multiset M = { 1 2 , 2 2 , … , n 2 } (resp. M = { 1 2 , 2 2 , … , ( n − 1 ) 2 , n } ) and Savage and Schuster's s-Eulerian polynomials for the sequence s = ( 1 , 1 , 3 , 2 , 5 , 3 , … , 2 n − 1 , n , n + 1 ) (resp. s = ( 1 , 1 , 3 , 2 , 5 , 3 , … , 2 n − 1 , n ) ). We also derive equidistributions that involve some natural tree statistics among weakly increasing trees on different multisets. Via introducing a group action on weakly increasing trees, we prove combinatorially the γ-positivity of the M-Eulerian–Narayana polynomials for a general multiset M. As an application of this γ-positivity result, we obtain a combinatorial interpretation of γ-coefficients of descent polynomials of permutations on the multiset { 1 2 , 2 2 , … , n 2 } in terms of weakly increasing trees.

A clique partition e of graph G is a set of cliques such that each edge of G belongs to exactly one clique, and the total size of e is the sum of cardinalities of all elements in e. The e-degree of a vertex u is the number of cliques in e containing u. We say that e is a k-restricted clique partition if each vertex has e-degree at least k. The (k-restricted) clique partition number of G is the smallest cardinality of a (k-restricted) clique partition of G. In this paper, we obtain eigenvalue bounds for e-degrees, clique partition number and restricted clique partition number of a graph. As applications, we derive the De Bruijn-Erdős Theorem from our eigenvalue bounds, obtain accurate estimation of the 2-restricted clique partition number of line graphs, and give spectral lower bounds for the minimum total size of clique partitions of a graph.