Rook polynomials to and from permanents
type: Item
instance of: scholarly article
title: Rook polynomials to and from permanents
zbMATH Open document ID: 1019.15001
DOI: 10.1016/S0012-365X(02)00547-2
author: Gi-Sang Cheon
author: Suk-Geun Hwang
author: Seok-Zun Song
published in: Discrete Mathematics
publication date: 2003-04-09T00:00:00Z
review text: Let \(A\) be an \(m \times n\) (0,1) matrix, and let \(\sigma_k(A), 0 \leq k \leq m,\) denote the sum of the permanents of all \(k\)-square submatrices of \(A.\) The \textit{rook vector} of \(A\) is defined by \((\sigma_0(A), \sigma_1(A), \dots, \sigma_m(A))^T;\) its components serve as coefficients of the \textit{rook polynomial} \(r_A(x) = \sigma_0(A) + \sigma_1(A) x + \dots + \sigma_m(A) x^m.\) This paper discusses (i) how the permanent of a matrix is related to the rook vectors of certain submatrices of it, and (ii) how to find the rook polynomial of a matrix in terms of the permanents of some other associated matrices. The results are used to evaluate the permanent of certain types of \(n\)-square (0,1) Toeplitz matrices with band width \(\geq n-1.\)
Mathematics Subject Classification ID: 15A15
Mathematics Subject Classification ID: 15B57
zbMATH DE Number: 1896020
zbMATH Keywords: rook polynomials
zbMATH Keywords: rook vectors
zbMATH Keywords: permanents
zbMATH Keywords: Pascal matrices
zbMATH Keywords: Stirling numbers of the second kind
zbMATH Keywords: Ferrers matrices
zbMATH Keywords: Toeplitz matrices
zbMATH Keywords: sparse matrices
Wikidata QID: Q57702741
MaRDI profile type: Publication
full work available at URL: s0012-365x(02)00547-2
OpenAlex ID: W1969751368
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description: scientific article; zbMATH DE number 1896020