Remixed Eulerian numbers

Abstract

Remixed Eulerian numbers are a polynomial q-deformation of Postnikov’s mixed Eulerian numbers. They arose naturally in previous work by the authors concerning the permutahedral variety and subsume well-known families of polynomials such as q-binomial coefficients and Garsia–Remmel’s q-hit numbers. We study their combinatorics in more depth. As polynomials in q, they are shown to be symmetric and unimodal. By interpreting them as computing success probabilities in a simple probabilistic process we arrive at a combinatorial interpretation involving weighted trees. By decomposing the permutahedron into certain combinatorial cubes, we obtain a second combinatorial interpretation. At $q=1$ , the former recovers Postnikov’s interpretation whereas the latter recovers Liu’s interpretation, both of which were obtained via methods different from ours.

The Permutahedral Variety, Mixed Eulerian Numbers, and Principal Specializations of Schubert Polynomials

“…For instance, they will help us determine the harmonic polynomials for the ideal QSym + n . The Ω • -correspondence plays a key rule in determining these harmonics and so does the cubical subdivision of the permutahedron discussed in [NT22].…”

“…So while we are able to infer positivity of a w via an explicit formula, the expression itself involves summing rational numbers. This aspect motivated a deeper study of mixed Eulerian numbers; see [NT21b,NT22] where a q-analogue is introduced. There has been other recent work utilizing mixed Eulerians in a crucial way; see [BST20,Hor21].…”