Remixed Eulerian numbers
Forum of Mathematics, Sigma, Tome 11 (2023)
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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.
@article{10_1017_fms_2023_57,
author = {Philippe Nadeau and Vasu Tewari},
title = {Remixed {Eulerian} numbers},
journal = {Forum of Mathematics, Sigma},
publisher = {mathdoc},
volume = {11},
year = {2023},
doi = {10.1017/fms.2023.57},
language = {en},
url = {http://geodesic.mathdoc.fr/articles/10.1017/fms.2023.57/}
}
Philippe Nadeau; Vasu Tewari. Remixed Eulerian numbers. Forum of Mathematics, Sigma, Tome 11 (2023). doi: 10.1017/fms.2023.57
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