Related papers: Jack characters and enumeration of maps
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.
Even with the introduction of supercharacter theories, the representation theory of many unipotent groups remains mysterious. This paper constructs a family of supercharacter theories for normal pattern groups in a way that exhibit many of…
Higher Lie characters form a distinguished family of symmetric group characters, which appear in many areas of algebra and combinatorics. An old open problem of Thrall is to decompose them into irreducibles. We propose a novel asymptotic…
Using a notation of corner between edges when graph has a fixed rotation, i.e. cyclical order of edges around vertices, we define combinatorial objects - combinatorial maps as pairs of permutations, one for vertices and one for faces.…
We present applications of rectangular matrix models to various combinatorial problems, among which the enumeration of face-bicolored graphs with prescribed vertex degrees, and vertex-tricolored triangulations. We also mention possible…
Multiple harmonic-like numbers are studied using the generating function approach. A closed form is stated for binomial sums involving these numbers and two additional parameters. Several corollaries and examples are presented which are…
The family of cycle completable graphs has several cryptomorphic descriptions, the equivalence of which has heretofore been proven by a laborious implication-cycle that detours through a motivating matrix completion problem. We give a…
We give some formulas of the James-Hopf maps by using combinatorial methods. An application is to give a product decomposition of the spaces $\Omega\Sigma^2(X)$.
We present a versatile construction allowing one to obtain pairs of integer sets with infinite symmetric difference, infinite intersection, and identical representation functions.
We prove a closed formula for the derivative, of any order, of a implicit function, in terms of some binomial building blocks, and explain the combinatorics behind the coefficients appearing in the formula.
We introduce jacobian graphs, which are explicit families of regular graphs that are spectrally indistinguishable from random graphs, but whose local structure is very different from that of random graphs. The construction relies on the…
We establish a Springer theory for classical symmetric pairs. We give an explicit description of character sheaves in this setting. In particular we determine the cuspidal character sheaves.
Index maps taking values in the $K$-theory of a mapping cone are defined and discussed. The resulting index theorem can be viewed in analogy with the Freed-Melrose index theorem. The framework of geometric $K$-homology is used in a…
We consider the generating series of oriented and non-oriented hypermaps with controlled degrees of vertices, hyperedges and faces. It is well known that these series have natural expansions in terms of Schur and Zonal symmetric functions,…
We strengthen the standard bifurcation theorems for saddle-node, transcritical, pitchfork, and period-doubling bifurcations of maps. Our new formulation involves adding one or two extra terms to the standard truncated normal forms with…
We apply matrix methods to arithmetic functions by associating matrices to the functions in a manner drawn from the theory of symmetric functions. Then we study the characteristic polynomials of the associated matrices.
We introduce several commutative rings, the snake rings, that have strong connections to cluster algebras. The elements of these rings are residue classes of unions of certain labeled graphs that were used to construct canonical bases in…
In this article we determine the coefficient bounds for functions in certain subclasses of analytic functions defined by subordination which are related to the well-known classes of starlike and convex functions. The main results deal with…
Some Dirichlet-like functions, attached to a pair (periodic function, polynomial) are introduced and studied. These functions generalize the standard Dirichlet L-functions of Dirichlet characters. They have similar properties, being…
In this paper we introduce doubly symmetric functions, arising from the equivalence of particular linear combinations of Schur functions and hook Schur functions. We study algebraic and combinatorial aspects of doubly symmetric functions,…