Macmahon's statistics properties on sets of words

Properties of MacMahon's statistics of maj and inv are considered on three sets of words over {1,..., 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 (des, maj) and (des, inv) are obtained on sets 1 and 3; the corresponding recursive descriptions on the set 2 are only obtained for (des, maj) and for statistics 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 maj and inv. On the set 2, the statistics of fas and cas are defined as special average values of a symbol in a word, fas and des are equally distributed, and the theorem of coincidence of distributions of couples (fas, maj) and (fas, inv), and also of couples (cas, maj) and (cas, inv) is proved.

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