Weighted inversion numbers, restricted growth functions, and standard Young tableaux
type: Item
instance of: scholarly article
title: Weighted inversion numbers, restricted growth functions, and standard Young tableaux
zbMATH Open document ID: 0579.05006
DOI: 10.1016/0097-3165(85)90044-5
author: Kevin W. J. Kadell
published in: Journal of Combinatorial Theory. Series A
publication date: 1985-01-01T00:00:00Z
review text: The author proves several results about statistics on words. He first finds a series of statistics that interpolate between the inversion number and the major index. For each j, \(2\leq j\leq\), fix a \(\lambda_ j\) between 1 and j-1, inclusive. Define \(w_{ij}\) to be i if \(i=\lambda_ j\), 1 if \(\lambda_ j<i<j\), 0 otherwise. The statistic defined by the \(\lambda_ j\) on a word \(\sigma\) is the sum of \(w_{ij}\) over all pairs \(i<j\) such that \(\sigma (i)>\sigma (j)\). The author shows that the generating function for words counted by this statistic is independent of the \(\lambda_ j\) and thus the inversion number \((\lambda_ j=1)\) and the major index \((\lambda_ j=j-1)\) have the same generating function. The last two sections consist of partial results in attempts to prove the equality of the generating functions with respect to the inversion number and the z-statistic and to find a q-analog of the hook-length formula for counting standard Young tableaux.
Mathematics Subject Classification ID: 05A15
zbMATH DE Number: 3926924
zbMATH Keywords: statistics on words
zbMATH Keywords: generating function
zbMATH Keywords: inversion number
zbMATH Keywords: major index
zbMATH Keywords: hook-length formula
zbMATH Keywords: Young tableaux
reviewed by: David M. Bressoud
MaRDI profile type: Publication
full work available at URL: 0097-3165(85)90044-5
OpenAlex ID: W2085735971
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description: scientific article; zbMATH DE number 3926924