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Register a new userGeneralized permutations related to the degenerate Eulerian numbers
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Authors
Orli Herscovici
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In this work we propose a combinatorial model that generalizes the standard definition of permutation. Our model generalizes the degenerate Eulerian polynomials and numbers of Carlitz from 1979 and provides missing combinatorial proofs for some relations on the degenerate Eulerian numbers.
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A ballot permutation is a permutation {pi} such that in any prefix of {pi} the descent number is not more than the ascent number. In this article, we obtained a formula in close form for the multivariate generating function of {A(n,d,j)}, which denote the number of permutations of length n with d descents and j as the first letter. Besides, by a series of calculations with generatingfunctionology, we confirm a recent conjecture of Wang and Zhang for ballot permutations.
Recently, Nunge studied Eulerian polynomials on segmented permutations, namely emph{generalized Eulerian polynomials}, and further asked whether their coefficients form unimodal sequences. In this paper, we prove the stability of the generalized Eulerian polynomials and hence confirm Nunges conjecture. Our proof is based on Brandens stable multivariate Eulerian polynomials. By acting on Brandens polynomials with a stability-preserving linear operator, we get a multivariate refinement of the generalized Eulerian polynomials. To prove Nunges conjecture, we also develop a general approach to obtain generalized Sturm sequences from bivariate stable polynomials.
We extend an observation due to Stong that the distribution of the number of degree $d$ irreducible factors of the characteristic polynomial of a random $n times n$ matrix over a finite field $mathbb{F}_{q}$ converges to the distribution of the number of length $d$ cycles of a random permutation in $S_{n}$, as $q rightarrow infty$, by having any finitely many choices of $d$, say $d_{1}, dots, d_{r}$. This generalized convergence will be used for the following two applications: the distribution of the cokernel of an $n times n$ Haar-random $mathbb{Z}_{p}$-matrix when $p rightarrow infty$ and a matrix version of Landaus theorem that estimates the number of irreducible factors of a random characteristic polynomial for large $n$ when $q rightarrow infty$.
Recently, Bagno, Garber and Mansour studied a kind of excedance number on the complex reflection groups and computed its multidistribution with the number of fixed points on the set of involutions in these groups. In this note, we consider the similar problems in more general cases and make a correction of one result obtained by them.
The Euler numbers occur in the Taylor expansion of $tan(x)+sec(x)$. Since Stieltjes, continued fractions and Hankel determinants of the even Euler numbers, on the one hand, of the odd Euler numbers, on the other hand, have been widely studied separately. However, no Hankel determinants of the (mixed) Euler numbers have been obtained and explicitly calculated. The reason for that is that some Hankel determinants of the Euler numbers are null. This implies that the Jacobi continued fraction of the Euler numbers does not exist. In the present paper, this obstacle is bypassed by using the Hankel continued fraction, instead of the $J$-fraction. Consequently, an explicit formula for the Hankel determinants of the Euler numbers is being derived, as well as a full list of Hankel continued fractions and Hankel determinants involving Euler numbers. Finally, a new $q$-analog of the Euler numbers $E_n(q)$ based on our continued fraction is proposed. We obtain an explicit formula for $E_n(-1)$ and prove a conjecture by R. J. Mathar on these numbers.
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