Eulerian calculus. I: Univariable statistics. (English) Zbl 0811.05069

Clarke, Robert J.; Foata, Dominique

Eur. J. Comb. 15, No. 4, 345-362 (1994).

The paper is the first one in a series of three articles. Let \(j,k\) be fixed nonnegative integers, \(X\) be an alphabet consisting of the numbers \(1, 2, \dots, r = j + k\). For every word \(w = x_ 1 x_ 2 \dots x_ m\) in this alphabet let \(v = y_ 1 y_ 2 \dots y_ m\) be the nondecreasing arrangement of the letters in \(w\). The following statistics are defined: \(\text{exc}_ k w\) is the number of \(i\) such that either \(x_ i > y_ i\), or \(x_ i = y_ i\) and \(x_ i > j\); \(\text{des}_ k w\) is the number of \(i\) such that either \(x_ i > x_{i + 1}\), or \(x_ i = x_{i + 1}\) and \(x_ i > j\); (here \(x_{m + 1} = j + 1/2)\).
Case \(k = 0\) corresponds to the classic definitions of the exceedances and descents. The main result of the article is the construction of a transformation \(F_ k\), such that \(\text{des}_ k w = \text{exc}_ k F_ k(w)\). Several formulas for related generating functions are obtained. The last section is devoted to the application of those results to the case of permutations (\(r = m\) and all letters in \(w\) are different).

Reviewer: V.A.Ufnarovski (Lund)

MSC:

05E15	Combinatorial aspects of groups and algebras (MSC2010)
05E05	Symmetric functions and generalizations
05A05	Permutations, words, matrices
05A19	Combinatorial identities, bijective combinatorics
20B30	Symmetric groups

Keywords:

MacMahon theorem; bijective proof; fundamental transformation; elementary symmetric functions; Eulerian polynomial; Eulerian calculus; statistics; exceedances; descents; generating functions; permutations