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

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).


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
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