The ${1/k}$-Eulerian Polynomials

  • Carla D. Savage
  • Gopal Viswanathan


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 Number