Combinatorial Interpretation of General Eulerian Numbers

Since the 1950s, mathematicians have successfully interpreted the traditional Eulerian numbers and -Eulerian numbers combinatorially. In this paper, the authors give a combinatorial interpretation to the general Eulerian numbers defined on general arithmetic progressions . 1. Introduction Definition 1. Given a positive integer , define as the set of all permutations of . For a permutation , is called an ascent of if ; is called a weak exceedance of if . It is well known that a traditional Eulerian number is the number of permutations that have weak exceedances [1, page 215]. And satisfies the recurrence: , , , Besides the recursive formula (1), can be calculated directly by the following analytic formula [2, page 8]: Definition 2. Given a permutation , define functions Since the 1950s, Carlitz [3, 4] and his successors have generalized Euler’s results to -sequences . Under Carlitz’s definition, the -Eulerian numbers are given by where functions are as defined in Definition 2. In [5], instead of studying -sequences, the authors have generalized Eulerian numbers to any general arithmetic progression Under the new definition, and given an arithmetic progression as defined in (5), the general Eulerian numbers can be calculated directly by the following equation [5, Lemma 2.6]: Interested readers can find more results about the general Eulerian numbers and even general Eulerian polynomials in [5]. 2. Combinatorial Interpretation of General Eulerian Numbers The following concepts and properties will be heavily used in this section. Definition 3. Let be the set of -permutations with weak exceedances. Then . Furthermore, given a permutation , let , where . Given a permutation , it is known that can be written as a one-line form like , or can be written in a disjoint union of distinct cycles. For written in a cycle form, we can use a standard representation by writing (a) each cycle starting with its largest element and (b) the cycles in increasing order of their largest element. Moreover, given a permutation written in a standard representation cycle form, define a function as to be the permutation obtained from by erasing the parentheses. Then is known as the fundamental bijection from to itself [6, page 30]. Indeed, the inverse map of the fundamental bijection function is also famous in illustrating the relation between the ascents and weak exceedances as follows [2, page 98]. Proposition 4. The function gives a bijection between the set of permutations on with ascents and the set . Example 5. The standard representation of permutation is , and ; ; has