Bijections related to statistics on words
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})\(=m\}\).
scientific article; zbMATH DE number 4027478
scientific article