On the difference of successive Gaussian polynomials
type: Item
instance of: scholarly article
title: On the difference of successive Gaussian polynomials
zbMATH Open document ID: 0770.11044
DOI: 10.1016/0378-3758(93)90030-A
author: George E. Andrews
published in: Journal of Statistical Planning and Inference
publication date: 1993-05-16T00:00:00Z
review text: Let \(M,N\in\mathbb{Z}\); as usual, define the ``Gaussian polynomials'' by \({N\brack M}=\prod^ M_ 1(1-q^{N+1-j})\) \((1-q^ j)^{-1}\), if \(0\leq M\leq N\). Otherwise, put \({N\brack M}=0\). According to the author's introduction, ``the object in this paper is to relate differences of Gaussian polynomials to partitions through the use of Frobenius symbols''. [See Section 2 of the author's work Generalized Frobenius partitions (Mem. Am. Math. Soc. 301, 1984; Zbl 0544.10010)]. The Frobenius symbol of any partition \(\pi\) (of a positive integer) ``is constructed as follows: In the Ferrers graph of \(\pi\) delete the main diagonal (of say \(r\) nodes); then create a two-line array of integers wherein the upper row consists of the cardinalities of the \(r\) rows in the Ferrers graph to the right of the main diagonal and the lower row consists of the cardinalities of the \(r\) columns in the Ferrers graph below the main diagonal. For example, if \(\pi\) is the partition 5+4+4+2, then the ... Frobenius symbol is \({4 2 1\choose 3 2 0}\).'' The author proves the theorem: For \(0\leq j\leq N\), \(q^{- j}\bigl({N+j\brack j}-{N+j\brack j-1}\bigr)\) is the generating function for partitions \(\pi\) with Frobenius symbols \({a_ 1\cdots a_ r\choose b_ 1\cdots b_ r}\) satisfying \(a_ 1<N-1\), \(b_ 1<j-1\) and \(a_ i- b_ i<N-j\), for \(1\leq i\leq r\). Setting \(j=N\) in the theorem yields an interpretation of the \(q\)-Catalan numbers, \((1-q)(1-q^{N+1})^{- 1}{2N\brack N}\).
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 168224
zbMATH Keywords: \(q\)-Catalan numbers
zbMATH Keywords: differences of Gaussian polynomials
zbMATH Keywords: partitions
zbMATH Keywords: Frobenius symbols
zbMATH Keywords: Ferrers graph
MaRDI profile type: Publication
cites work: Sieves in the Theory of Partitions
cites work: Q4146776
cites work: Generalized Frobenius partitions
cites work: Extension of the partition sieve
cites work: A Unimodality Result in the Enumeration of Subgroups of a Finite Abelian Group
cites work: q-Catalan numbers
cites work: Q3947818
Recommended article: Q3781819
Recommended article: Q5492602
Recommended article: Q4056080
Recommended article: Q5461456
Recommended article: Q3976657
Recommended article: An overpartition analogue of the \(q\)-binomial coefficients
Recommended article: Q3822240
Recommended article: Q4206804
Recommended article: Identities containing Gaussian binomial coefficients
Recommended article: Q3562146
description: scientific article; zbMATH DE number 168224