Determinants of super-Schur functions, lattice paths, and dotted plane partitions
type: Item
instance of: scholarly article
title: Determinants of super-Schur functions, lattice paths, and dotted plane partitions
zbMATH Open document ID: 0796.05090
author: Francesco Brenti
published in: Advances in Mathematics
publication date: 1993-06-29T00:00:00Z
full work available at URL: http://hdl.handle.net/2027.42/30915
review text: This is an important contribution to the combinatorics of plane partitions, of nonintersecting lattice paths, and to combinatorial determinants. A large variety of results is obtained in a uniform way. The subject of this paper are minors of the matrix \[ {\mathcal S} ({\mathbf x}, {\mathbf y}) : =(s_{(k)} (x_ 1,\dots,x_ n/y_ 1,\dots,y_ n))_{n,k \in \mathbb{N}}, \] where \(s_{(k)} (x_ 1, \dots, x_ n/y_ 1, \dots, y_ n) = \sum_{i=0}^{k} h_ j(x_ 1, \dots,x_ n) e_{k-j} (y_ 1,\dots,y_ n)\) with \(h_ j\) and \(e_ j\) denoting the complete homogeneous and elementary symmetric functions, respectively. The \(s_{(k)}\) are special super-Schur functions (see e.g. \textit{V. G. Kac} [Adv. Math. 26, 8-96 (1977; Zbl 0366.17012)]). The focus is on giving interesting combinatorial interpretations for the minors of \({\mathcal S}({\mathbf x},{\mathbf y})\). The most straightforward interpretation is in terms of generating functions for nonintersecting lattice paths with three types of steps. As in the case of two types of steps (see \textit{I. M. Gessel} and \textit{X. G. Viennot}, ``Determinants, paths, and plane partitions'', preprint; or \textit{J. R. Stembridge} [Adv. Math. 83, No. 1, 96-131 (1990; Zbl 0790.05007)]), these families of nonintersecting lattice paths can be translated into certain plane partitions. The plane partitions under consideration in the paper under review are plane partitions whose entries are from two distinct copies of the set of integers. Since the integers from the second copy are written as dotted integers, the author calls his plane partitions dotted plane partitions. Subsequently, several specializations of \({\mathcal S}({\mathbf x},{\mathbf y})\) are considered. Interpretations in terms of generating functions for dotted plane partitions are obtained for many determinants of symmetric functions, of \(q\)-binomial coefficients, and of \(q\)-Stirling numbers of the first and second kind, respectively. Many of these interpretations are new. For the r…
Mathematics Subject Classification ID: 05E05
Mathematics Subject Classification ID: 05A10
Mathematics Subject Classification ID: 05A17
Mathematics Subject Classification ID: 05A20
Mathematics Subject Classification ID: 05E10
Mathematics Subject Classification ID: 11B65
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 11P81
zbMATH DE Number: 221238
zbMATH Keywords: \(q\)-binomial coefficients
zbMATH Keywords: \(q\)-Stirling numbers
zbMATH Keywords: plane partitions
zbMATH Keywords: nonintersecting lattice paths
zbMATH Keywords: symmetric functions
zbMATH Keywords: super-Schur functions
zbMATH Keywords: minors
zbMATH Keywords: generating functions
zbMATH Keywords: set partitions
reviewed by: C. Krattenthaler
MaRDI profile type: Publication
OpenAlex ID: W2050810891
DOI: 10.1006/AIMA.1993.1012
description: scientific article; zbMATH DE number 221238