## 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).

### 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
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