MacMahon's statistics properties on sets of words

L. N. Bondarenko ; M. L. Sharapova

Prikladnaya Diskretnaya Matematika. Supplement, no. 8 (2015), pp. 6-8

Résumé

Properties of MacMahon's statistics of $\mathrm{maj}$ and $\mathrm{inv}$ are considered on three sets of words over $\{1,\dots,n\}$: 1) permutations of degree $n$; 2) all words of length $n$; 3) concave permutations of degree $n$. New recursive descriptions of the generating polynomials of couples $\mathrm{(des,maj)}$ and $\mathrm{(des,inv)}$ are obtained on sets 1 and 3; the corresponding recursive descriptions on the set 2 are only obtained for $\mathrm{(des,maj)}$ and for statistics $\mathrm{inv}$. On the sets 1 and 2, these recursive descriptions are used for another proof of the known MacMahon's theorem about the coincidence of distributions of $\mathrm{maj}$ and $\mathrm{inv}$. On the set 2, the statistics of $\mathrm{fas}$ and $\mathrm{cas}$ are defined as special average values of a symbol in a word, $\mathrm{fas}$ and $\mathrm{des}$ are equally distributed, and the theorem of coincidence of distributions of couples $\mathrm{(fas,maj)}$ and $\mathrm{(fas,inv)}$, and also of couples $\mathrm{(cas,maj)}$ and $\mathrm{(cas,inv)}$ is proved.

Détail

Keywords: MacMahon's statistics, generating polynomial, recursive description, Euler's statistics.