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Coefficients of the Inflated Eulerian Po...
Auli, Juan S....

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Coefficients of the Inflated Eulerian Polynomial by Auli, Juan S. ( Author )
Lemonjar Open Access
N.A
05-04-2015
It follows from work of Chung and Graham that for a certain family of polynomials Tn(x), derived from the descent statistic on permutations, the coefficient sequence of Tn−1(x) coincides with that of the polynomial Tn(x)/(1+x+⋯+xn−1). We observed computationally that the inflated s-Eulerian polynomial Q(s)n(x), which satisfies Q(s)n(x)=Tn(x) when s=(1,2,…,n), also satisfies this property for many sequences s. In this work we characterize those sequences s for which the coefficient sequence of Q(s)n−1(x) coincides with that of the polynomial Q(s)n(x)/(1+x+⋯+xsn−1). In particular, we show that all nondecreasing sequences satisfy this property. We also settle a conjecture of Pensyl and Savage by showing that the inflated s-Eulerian polynomials are unimodal for all choices of positive integer sequences s. In addition, we determine when these polynomials are palindromic and show our characterization is equivalent to another of Beck, Braun, Köppe, Savage, and Zafeirakopoulos.
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English
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MYR 0.00
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http://arxiv.org/abs/1504.01089
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