Related papers: Enumerative and Algebraic Combinatorics in the 196…
We describe recent advances in the study of random analogues of combinatorial theorems.
Combinatorial enumeration leads to counting generating functions presenting a wide variety of analytic types. Properties of generating functions at singularities encode valuable information regarding asymptotic counting and limit…
We explain connections among several, a priori unrelated, areas of mathematics: combinatorics, algebraic statistics, topology and enumerative algebraic geometry. Our focus is on discrete invariants, strongly related to the theory of…
This is a complement to my previous article "Advanced Determinant Calculus" (S\'eminaire Lotharingien Combin. 42 (1999), Article B42q, 67 pp.). In the present article, I share with the reader my experience of applying the methods described…
Arithmetic combinatorics is often concerned with the problem of bounding the behaviour of arbitrary finite sets in a group or ring with respect to arithmetic operations such as addition or multiplication. Similarly, combinatorial geometry…
This is a survey of results on partially commutative groups and partially commutative algebras.
We consider a general concept of composition and decomposition of objects, and discuss a few natural properties one may expect from a reasonable choice thereof. It will be demonstrated how this leads to multiplication and co- multiplication…
This is the classical monograph on the combinatorial study of Eulerian polynomials, published in 1970. It has been retyped in TeX and made available on the web with the kind permission of Springer-Verlag. This on-line version has an ouput…
The subjects in the title are interwoven in many different and very deep ways. I recently wrote several expository accounts [64-66] that reflect a certain range of developments, but even in their totality they cannot be taken as a…
Many combinatorial problems can be formulated as a polynomial optimization problem that can be solved by state-of-the-art methods in real algebraic geometry. In this paper we explain many important methods from real algebraic geometry, we…
For Boris Zilber on his 75th birthday. I trace the roots of my collaboration with Boris Zilber, which combines categoricity theory, finite model theory, algorithmics, and combinatorics.
A survey of three recent developments in algebraic combinatorics: (1) the Laurent phenomenon, (2) Gromov-Witten invariants and toric Schur functions, and (3) toric h-vectors and intersection cohomology. This paper is a continuation of…
Here we present the new applications of the Egorychev method of coefficients of integral representations and computation of combinatorial sums developed by the author at the end of 1970's and its recent applications to the algebra and the…
We introduce a family of sequence transformations, defined via partial Bell polynomials, that may be used for a systematic study of a wide variety of problems in enumerative combinatorics. This family includes some of the transformations…
We present a complete computational classification of the combinatorial types of hyperplane sections, or slices, of the regular cube up to dimension six. For each dimension, we determine the exact number of distinct combinatorial types.…
Let A be a class of objects, equipped with an integer size such that for all n the number a(n) of objects of size n is finite. We are interested in the case where the generating fucntion sum_n a(n) t^n is rational, or more generally…
A paper of the first author and Zilke proposed seven combinatorial problems around formulas for the characteristic polynomial and the exponents of an isolated quasihomogeneous singularity. The most important of them was a conjecture on the…
Using random variables as motivation, this paper presents an exposition of the formalisms developed by Rota and Taylor for the classical umbral calculus. A variety of examples are presented, culminating in several descriptions of sequences…
These lecture notes where presented as a course of the CIMPA summer school in Manila, July 20-30, 2009, Semidefinite programming in algebraic combinatorics. This version is an update June 2010.
The paper suggests a short survey of integration algorithms which evolved since 1982. These theorems and algorithms form discrete versions of the calculus theorems.