Some experiments with complete and elementary symmetric functions
type: Item
instance of: scholarly article
title: Some experiments with complete and elementary symmetric functions
zbMATH Open document ID: 1340.11025
author: Mircea Merca
published in: Periodica Mathematica Hungarica
publication date: 2016-01-08T00:00:00Z
review text: Using computer calculations the author notes some nice determinantal formulas for the classical Stirling numbers of the first and second kind, like \[ \det \left(\begin{bmatrix} i+k\cr j\end{bmatrix}\right)_{1\leq i,j\leq n}=k!^n,\qquad \det \left(\left\{\begin{matrix} i+k\cr j\end{matrix}\right\}\right)_{1\leq i,j\leq n}=n!^k, \] or \[ \det\left(\begin{bmatrix} i+k\cr j+k-1\end{bmatrix}\right)_{1\leq i,j\leq n}=\left\{\begin{matrix} n+k\cr k\end{matrix}\right\},\qquad \det \left(\left\{\begin{matrix} i+k\cr j+k-1\end{matrix}\right\}\right)_{1\leq i,j\leq n}=\begin{bmatrix} n+k\cr k\end{bmatrix}. \] He proves that these are particular cases of general formulas for complete and elementary symmetric functions.
Mathematics Subject Classification ID: 11B73
Mathematics Subject Classification ID: 05E05
zbMATH DE Number: 6528019
zbMATH Keywords: symmetric functions
zbMATH Keywords: Stirling numbers
zbMATH Keywords: determinants
reviewed by: István Mező
describes a project that uses: OEIS
MaRDI profile type: Publication
full work available at URL: s10998-014-0034-3
OpenAlex ID: W2010067893
cites work: Schur functions: Theme and variations
cites work: Q4328336
cites work: A convolution for complete and elementary symmetric functions
cites work: Combinatorial interpretations of particular evaluations of complete and elementary symmetric functions
cites work: The On-Line Encyclopedia of Integer Sequences
DOI: 10.1007/S10998-014-0034-3
description: scientific article; zbMATH DE number 6528019