Séminaire Lotharingien de Combinatoire, 80B.57 (2018), 12 pp.
Arthur Nunge
Eulerian Polynomials on Segmented Permutations
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.
Received: November 14, 2017.
Accepted: February 17, 2018.
Final version: April 1, 2018.
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