Year: 2018
Type: preprint
Abstract: We define a generalization of the Eulerian polynomials and the Eulerian numbers by considering a descent statistic on segmented permutations coming from the study of 2-species exclusion processes and a change of basis in a Hopf algebra. We give some properties satisfied by these generalized Eulerian numbers. We also define a $q$-analog of these Eulerian polynomials which gives back usual Eulerian polynomials and ordered Bell polynomials for specific values of its variables. We also define a noncommutative analog living in the algebra of segmented compositions. It gives us an explicit generating function and some identities satisfied by the generalized Eulerian polynomials such as a Worpitzky-type relation.
Source: arXiv (Cornell University)
Author: Nunge, Arthur
Language: English
Subfield: Discrete Mathematics and Combinatorics
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W2801027375