The \({1/k}\)-Eulerian polynomials
The electronic journal of combinatorics, Tome 19 (2012) no. 1
We use the theory of lecture hall partitions to define a generalization of the Eulerian polynomials, for each positive integer $k$. We show that these ${1}/{k}$-Eulerian polynomials have a simple combinatorial interpretation in terms of a single statistic on generalized inversion sequences. The theory provides a geometric realization of the polynomials as the $h^*$-polynomials of $k$-lecture hall polytopes. Many of the defining relations of the Eulerian polynomials have natural ${1}/{k}$-generalizations. In fact, these properties extend to a bivariate generalization obtained by replacing ${1}/{k}$ by a continuous variable. The bivariate polynomials have appeared in the work of Carlitz, Dillon, and Roselle on Eulerian numbers of higher order and, more recently, in the theory of rook polynomials.
@article{10_37236_16,
author = {Carla D. Savage and Gopal Viswanathan},
title = {The {\({1/k}\)-Eulerian} polynomials},
journal = {The electronic journal of combinatorics},
year = {2012},
volume = {19},
number = {1},
doi = {10.37236/16},
zbl = {1243.05022},
url = {http://geodesic.mathdoc.fr/articles/10.37236/16/}
}
Carla D. Savage; Gopal Viswanathan. The \({1/k}\)-Eulerian polynomials. The electronic journal of combinatorics, Tome 19 (2012) no. 1. doi: 10.37236/16
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