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We define two-parameter families of noncommutative symmetric functions and quasi-symmetric functions, which appear to be the proper analogues of the Macdonald symmetric functions in these settings.
We state combinatorial formulas for hyperoctahedral group ($\mathfrak B_n$) character evaluations of the form $\chi( {{\widetilde C}_w}^{\negthickspace\negthickspace BC}\negthickspace(1))$, where ${{\widetilde…
Let $\mathfrak{g}$ be a Lie algebra all of whose regular subalgebras of rank 2 are type $A_{1}\times A_{1}$, $A_{2}$, or $C_{2}$, and let $B$ be a crystal graph corresponding to a representation of $\mathfrak{g}$. We explicitly describe the…
This paper presents a set of general strategies for the analysis of structure in amorphous materials and a general approach to assessing the utility of a selected structural description. Measures of structural diversity and utility are…
We define three combinatorial models for \hat{sl(n)} crystals, parametrized by partitions, configurations of beads on an `abacus', and cylindric plane partitions, respectively. These are reducible, but we can identify an irreducible…
We construct a non-commutative, non-cocommutative, graded bialgebra $\mathbf{\Pi}$ with a basis indexed by the permutations in all finite symmetric groups. Unlike the formally similar Malvenuto-Poirier-Reutenauer Hopf algebra, this…
A method for calculating pair correlation functions in a crystal is developed. The method is based on separating the one- and two- particle correlation functions into the symmetry conserving and the symmetry broken parts. The conserving…
This paper surveys some combinatorial aspects of Smith normal form, and more generally, diagonal form. The discussion includes general algebraic properties and interpretations of Smith normal form, critical groups of graphs, and Smith…
We develop a combinatorial model of networks on orientable surfaces, and study weight and homology generating functions of paths and cycles in these networks. Network transformations preserving these generating functions are investigated.…
We introduce Schur multiple zeta functions which interpolate both the multiple zeta and multiple zeta-star functions of the Euler-Zagier type combinatorially. We first study their basic properties including a region of absolute convergence…
Analogues of classical combinatorial identities for elementary and homogeneous symmetric functions with coefficients in Yanigian are discussed. As a corollary, similar relations are deduced for shifted Schur functions.
A combinatorial methods are used to investigate some properties of certain generalized Stirling numbers, including explicit formula and recurrence relations. Furthermore, an expression of these numbers with symmetric function is deduced.
We introduce affine Stanley symmetric functions for the special orthogonal groups, a class of symmetric functions that model the cohomology of the affine Grassmannian, continuing the work of Lam and Lam, Schilling, and Shimozono on the…
We discuss three distinct topics of independent interest; one in enumerative combinatorics, one in symmetric function theory, and one in algebraic geometry. The topic in enumerative combinatorics concerns a q-analog of a generalization of…
We introduce an epsilon system on a geometric crystal of type $A_n$, which is a certain set of rational functions with some nice properties. We shall show that it is equipped with a product structure and that it is invariant under the…
A well-known open problem in graph theory asks whether Stanley's chromatic symmetric function, a generalization of the chromatic polynomial of a graph, distinguishes between any two non-isomorphic trees. Previous work has proven the…
We discuss the role of the symmetries in photonic crystals and classify them according to the Cartan-Altland-Zirnbauer scheme. Of particular importance are complex conjugation C and time-reversal T, but we identify also other significant…
We introduce two types bilateral zeta functions, which are related to the primitive and normalized multiple sine functions respectively. Further, we establish their main properties, that is, Fourier expansions, analytic continuations,…
Crystal skeletons were introduced by Maas-Gari\'epy in 2023 by contracting quasi-crystal components in a crystal graph. On the representation theoretic level, crystal skeletons model the expansion of Schur functions into Gessel's…
One important class of tools in the study of the connections between algebraic and topological structures are the "Banach-Stone type theorems", which describe algebraic isomorphisms of algebras (or groups, lattices, etc.) of functions in…