Eulerian Numbers

Binomial Eulerian polynomials for colored permutations

“…Gamma-positivity implies palindromicity and unimodality and, in fact, has developed into a powerful method to prove these properties; see [4] [23,Chapter 4]. There are versions of this notion for nonpalindromic polynomials [4, Section 5.1].…”

“…Its palindromicity and unimodality can be demonstrated combinatorially, for instance via an expansion of the form A n (t) = ⌊(n−1)/2⌋ i=0 γ n,i t i (1 + t) n−1−2i for some nonnegative integers γ n,i , which establishes its γ-positivity, and geometrically (as well as by other approaches), since A n (t) can be interpreted as the h-polynomial of the boundary complex of a simplicial polytope. For an overview of these and other important properties of Eulerian polynomials, see [23] [34,Section 1.4].…”

Generalized Eulerian Numbers

“…The Eulerian numbers are discussed in the remarkable monograph of Riordan [1] from a combinatorial view and are the special topics in the recently published voluminous and versatile monograph of Petersen [2] with huge material and relations to other topics and with a great number of citations. The last contains A. Wünsche also some remarks about the history of these numbers.…”

“…To this time, we looked for references where these numbers which we calculated explicitly are present in literature and found them in the monograph of Riordan [1] about combinatorics 1 . Meanwhile, there appeared the already mentioned monograph of Petersen [2] which provided us a convenient access to this topics. adjacent numbers in a permutation, the first is larger than the second.…”

“…One may also interpolate between integer numbers µ by choosing for it real 2 An interesting analogy exists also to the Stirling numbers ( ) , S k l of second kind (see Section 7).…”