Weighted inversion numbers, restricted growth functions, and standard Young tableaux
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.
scientific article; zbMATH DE number 3926924