Proc. Amer. Math. Soc., 19, 1968, 236-240..
Dominique Foata
On the Netto Inversion Number of a Sequence
Abstract.
Let g=(a(1),a(2),...,a(n)) be an arbitrary finite sequence of real numbers and C the set of all sequences that can be formed from g by permuting the a(i)'s. If h=(b(1),b(2),..,b(n)) is in C, the inversion number S(h) of h is defined to be the number of pairs (j, k) comprised within 1 and n such that j is less than k and b(j) greater than b(k), while the major index of h is defined to be the sum of all integers i comprised between 1 and (n-1) such that b(i) is greater than b(i+1). MacMahon proved that the generating polynomials for the set C by the statistics S and T were equal. The purpose of the paper is construct an explicit bijection F of C onto C such that S(h)=F(T(h)) for each h in C, for any such a set C.
foata@unistra.fr
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