Related papers: Singular combinatorics
We give a review of modern approaches to constructing formal solutions to integrable hierarchies of mathematical physics, whose coefficients are answers to various enumerative problems. The relationship between these approaches and…
In this paper we study the coefficients of the powers of an ordinary generating function and their properties. A new class of functions based on compositions of an integer $n$ is introduced and is termed composita. We present theorems about…
We introduce a generating function associated to the homogeneous generators of a graded algebra that measures how far is this algebra from being finitely generated. For the case of some algebras of Frobenius endomorphisms we describe this…
This paper presents a detailed symbolic approach to the study of self-similar tilings. It uses properties of addresses associated with graph-directed iterated function systems to establish conjugacy properties of tiling spaces. Tiles may be…
Let F be the quotient of an analytic function with a product of linear functions. Working in the framework of analytic combinatorics in several variables, we compute asymptotic formulae for the Taylor coefficients of F using multivariate…
In this paper, we consider the generating functions of the complete and elementary symmetric functions and provide a new generalization of these classical symmetric functions. Some classical relationships involving the complete and…
We introduce a systematic approach to express generating functions for the enumeration of maps on surfaces of high genus in terms of a single generating function relevant to planar surfaces. Central to this work is the comparison of two…
The paper considers asymptotics of summation functions of additive and multiplicative arithmetic functions. We also study asymptotics of summation functions of natural and prime arguments. Several assertions on this subject are proved and…
Up-down permutations are counted by tangent resp. secant numbers. Considering words instead, where the letters are produced by independent geometric distributions, there are several ways of introducing this concept; in the limit they all…
This work is a study of polynomial compositions having a fixed number of terms. We outline a recursive method to describe these characterizations, give some particular results and discuss the general case. In the final sections, some…
We define matrix models that converge to the generating functions of a wide variety of loop models with fugacity taken in sets with an accumulation point. The latter can also be seen as moments of a non-commutative law on a subfactor planar…
Obtained a new property of superposition of the generating functions ln(1/(1-F(x))), where F(x) - generating function with integer coefficients, which allows the construction a primality tests. The theorem which is based on compositions of…
We analyse q-functional equations arising from tree-like combinatorial structures, which are counted by size, internal path length, and certain generalisations thereof. The corresponding counting parameters are labelled by a positive…
Symmetric functions provide one of the most efficient tools for combinatorial enumeration, in the context of objects that may be acted upon by permutations. Only assuming a basic knowledge of linear algebra, we introduce and describe the…
In this paper, we study a composition of exponential generating functions. We obtain new properties of this composition, which allow to distinguish prime numbers from composite numbers. Using the result of paper we get the known properties…
We compute generating functions for the sum of the real-valued character degrees of the finite general linear and unitary groups, through symmetric function computations. For the finite general linear group, we get a new combinatorial proof…
The theory of quasirandomness has greatly expanded from its inaugural graph theoretical setting to several different combinatorial objects such as hypergraphs, tournaments, permutations, etc. However, these quasirandomness variants have…
Unlike classical and free independence, the boolean and monotone notions of independence lack of the property of independent constants. In the scalar case, this leads to restrictions for the central limit theorems, as observed by F.…
We give a brief account of a construction called tokens here, which is significant in algebra, analysis, combinatorics, and physics. Tokens allow to express a semigroup on one set via a semigroup convolution on another set. Therefore tokens…
The aim of this paper is to construct general forms of ordinary generating functions for special numbers and polynomials involving Fibonacci type numbers and polynomials, Lucas numbers and polynomials, Chebyshev polynomials, Sextet…