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Prikladnaya Diskretnaya Matematika. Supplement, 2015, Issue 8, Pages 6–8
DOI: https://doi.org/10.17223/2226308X/8/1
(Mi pdma204)
 

Theoretical Foundations of Applied Discrete Mathematics

MacMahon's statistics properties on sets of words

L. N. Bondarenkoa, M. L. Sharapovab

a Penza State University, Penza
b Faculty of Mechanics and Mathematics, Lomonosov Moscow State University, Moscow
References:
Abstract: 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.
Keywords: MacMahon's statistics, generating polynomial, recursive description, Euler's statistics.
Document Type: Article
UDC: 519.1
Language: Russian
Citation: L. N. Bondarenko, M. L. Sharapova, “MacMahon's statistics properties on sets of words”, Prikl. Diskr. Mat. Suppl., 2015, no. 8, 6–8
Citation in format AMSBIB
\Bibitem{BonSha15}
\by L.~N.~Bondarenko, M.~L.~Sharapova
\paper MacMahon's statistics properties on sets of words
\jour Prikl. Diskr. Mat. Suppl.
\yr 2015
\issue 8
\pages 6--8
\mathnet{http://mi.mathnet.ru/pdma204}
\crossref{https://doi.org/10.17223/2226308X/8/1}
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