Enumeration of skew Ferrers diagrams and basic Bessel functions
type: Item
instance of: scholarly article
title: Enumeration of skew Ferrers diagrams and basic Bessel functions
zbMATH Open document ID: 0771.05008
DOI: 10.1016/0378-3758(93)90037-7
published in: Journal of Statistical Planning and Inference
publication date: 1993-05-16T00:00:00Z
review text: This is a survey paper on the enumeration of skew Ferrers diagrams according to their area and number of columns. The generating function for the problem \[ F(t;q)={\sum^ \infty_{n=0}((-1)^ nq^{{n+1\choose 2}}/(q;q)_ n(q;q)_{n+1})q^{n+1}t^{n+1}\over\sum^ \infty_{n=0}((-1)^ nq^{{n\choose 2}}/(q;q)^ 2_ n)q^ nt^ n}, \] where \((a;q)_ n=(1-a)(1-aq)(1-aq^ 2)\cdots(1-aq^{n-1})\), is related to the quotient of two basic Bessel functions. The combinatorial theories in the background are the \(q\)-analogue of the Schützenberger method, the enumeration of multichains in posets, and the Ehrhart theory of enumeration of integer points in a polytope.
Mathematics Subject Classification ID: 05A15
Mathematics Subject Classification ID: 05A30
Mathematics Subject Classification ID: 33C10
zbMATH DE Number: 168231
zbMATH Keywords: Dyck words
zbMATH Keywords: \(q\)-calculus
zbMATH Keywords: enumeration
zbMATH Keywords: skew Ferrers diagrams
zbMATH Keywords: generating function
zbMATH Keywords: Bessel functions
zbMATH Keywords: Ehrhart theory
author: Jean-Marc Fédou
reviewed by: László A. Székely
MaRDI profile type: Publication
full work available at URL: 0378-3758(93)90037-7
OpenAlex ID: W1996930411
cites work: On the Geometry of Numbers in Elementary Number Theory
cites work: Q3724706
cites work: Some basic hypergeometric orthogonal polynomials that generalize Jacobi polynomials
cites work: A Sequence of Integers Related to the Bessel Functions
cites work: Q4071773
cites work: Planar Maps are Well Labeled Trees
cites work: Introduction aux polyèdres en combinatoire d'après E. Ehrhart et R. Stanley
cites work: Attribute grammars are useful for combinatorics
cites work: Algebraic languages and polyominoes enumeration
cites work: Q5847356
cites work: Polynômes arithmétiques et méthode des polyedres en combinatoire
cites work: Q3660083
cites work: On congruences and continued fractions for some classical combinatorial quantities
cites work: Combinatoire et représentation du groupe symétrique. Actes de la Table Ronde du C.N.R.S. tenue à l'Université Louis-Pasteur de Strasbourg, 26 au 30 avril 1976
cites work: A Noncommutative Generalization and q-Analog of the Lagrange Inversion Formula
cites work: Formal languages and enumeration
cites work: Tiling with polyominoes
cites work: On the number of lattice animals embeddable in the square lattice
cites work: The zeros of basic Bessel functions, the functions J(nu+ax)(x), and associated orthogonal polynomials
cites work: The combinatorics of q-Hermite polynomials and the Askey-Wilson integral
cites work: Semantics of context-free languages
cites work: Q3752390
cites work: On the number of certain lattice polygons
cites work: Contributions to the Cell Growth Problem
cites work: Q5626549
cites work: On context-free languages and push-down automata
cites work: Q4774085
cites work: Ordered structures and partitions
cites work: Q3855186
cites work: Decompositions of Rational Convex Polytopes
cites work: Q3748279
cites work: Q3741080
cites work: Q4856171
cites work: A proof of Andrews' \(q\)-Dyson conjecture
description: scientific article; zbMATH DE number 168231