Bijections related to statistics on words
type: Item
instance of: scholarly article
title: Bijections related to statistics on words
zbMATH Open document ID: 0632.05007
DOI: 10.1016/0012-365X(88)90038-6
author: B. George
published in: Discrete Mathematics
publication date: 1988-01-01T00:00:00Z
review text: Consider throughout the collection \(M=M(n_ 1,...,n_ k)\) of all words \b{a}\(=a_ 1...a_ n\) containing \(n_ i\) i's, \(i\leq k\); define the inversion number Inv(\b{a}) to be the number of ordered pairs (i,j) such that \(i<j\) and \(a_ i>a_ j\), and the major index Maj(\b{a}) to be the sum of all j such that \(a_ j>a_{j+1}\). One of MacMahon's results was that \(| S| =| T|\) where \(S=\{\underline a:\) Inv(\b{a})\(=m\}\), \(T=\{\underline a:\) Maj(\b{a})\(=m\}\), both being equal to the q-multinomial coefficient \(\left[\begin{matrix} n\\ n_ 1...n_ k\end{matrix} \right]_ q\): his proof was indirect. Now let \b{a}\({}_{ij}\) denote the subword of \b{a} obtained by scoring out all a's not equal to i or j, and let Z(\b{a})\(=\sum_{i<j\leq k}Maj(\underline a_{ij})\). The author gives a combinatorial proof of the result \(\sum q^{z(\underline a)}=\left[ \begin{matrix} n\\ n_ 1...n_ k\end{matrix} \right]_ q\) and uses it to construct bijections between the 3 sets S, T and \(R=\{\underline a:\) Z(\b{a})\(=m\}\).
Mathematics Subject Classification ID: 05A19
zbMATH DE Number: 4027478
zbMATH Keywords: inversion
zbMATH Keywords: major index
MaRDI profile type: Publication
cites work: Q4146776
cites work: A proof of Andrews' \(q\)-Dyson conjecture
cites work: On the Netto Inversion Number of a Sequence
cites work: Method for constructing bijections for classical partition identities
cites work: Bijective proofs of basic hypergeometric series identities
cites work: Q4057549
Wikidata QID: Q127212423
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description: scientific article; zbMATH DE number 4027478