Remixed Eulerian numbers Article Swipe
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· 2023
· Open Access
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· DOI: https://doi.org/10.1017/fms.2023.57
· OA: W4291451508
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.
