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Real-rootedness of variations of Eulerian polynomials. (English) Zbl 1415.05016

Summary: The binomial Eulerian polynomials, introduced by A. Postnikov et al. [Doc. Math. 13, 207–273 (2008; Zbl 1167.05005)] are \(\gamma\)-positive polynomials and can be interpreted as \(h\)-polynomials of certain flag simplicial polytopes. Recently, C. A. Athanasiadis [“Binomial Eulerian polynomials for colored permutations ”, Preprint, arXiv:1812.00434] studied analogs of these polynomials for colored permutations. In this paper, we generalize them to s-inversion sequences and prove that these new polynomials have only real roots by the method of interlacing polynomials. Three applications of this result are presented. The first one is to prove the real-rootedness of binomial Eulerian polynomials, which confirms a conjecture of J. Ma et al. [“Recurrence relations for binomial-Eulerian polynomials”, Preprint, arXiv:1711.09016]. The second one is to prove that the symmetric decomposition of binomial Eulerian polynomials for colored permutations is real-rooted. Thirdly, our polynomials for certain s-inversion sequences are shown to admit a similar geometric interpretation related to edgewise subdivisions of simplexes.

MSC:

05A15 Exact enumeration problems, generating functions
26C10 Real polynomials: location of zeros
52B05 Combinatorial properties of polytopes and polyhedra (number of faces, shortest paths, etc.)
11B68 Bernoulli and Euler numbers and polynomials

Citations:

Zbl 1167.05005
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References:

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