The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

Details The symmetric and unimodal expansion of Eulerian polynomials via continued fractions The symmetric and unimodal expansion of Eulerian polynomials via continued fractions

2010-05-25

arxiv.org/abs/1005.4583v3

Mathematics

Combinatorics

19 pages, 2 figures

Scientific paper

10.1016/j.ejc.2011.08.005

This paper was motivated by a conjecture of Br\"{a}nd\'{e}n (European J. Combin. \textbf{29} (2008), no.~2, 514--531) about the divisibility of the coefficients in an expansion of generalized Eulerian polynomials, which implies the symmetric and unimodal property of the Eulerian numbers. We show that such a formula with the conjectured property can be derived from the combinatorial theory of continued fractions. We also discuss an analogous expansion for the corresponding formula for derangements and prove a $(p,q)$-analogue of the fact that the (-1)-evaluation of the enumerator polynomials of permutations (resp. derangements) by the number of excedances gives rise to tangent numbers (resp. secant numbers). The $(p,q)$-analogue unifies and generalizes our recent results (European J. Combin. \textbf{31} (2010), no.~7, 1689--1705.) and that of Josuat-Verg\`es (European J. Combin. \textbf{31} (2010), no.~7, 1892--1906).

Affiliated with

Also associated with

No associations

Rating

The symmetric and unimodal expansion of Eulerian polynomials via continued fractions does not yet have a rating. At this time, there are no reviews or comments for this scientific paper.

If you have personal experience with The symmetric and unimodal expansion of Eulerian polynomials via continued fractions, we encourage you to share that experience with our LandOfFree.com community. Your opinion is very important and The symmetric and unimodal expansion of Eulerian polynomials via continued fractions will most certainly appreciate the feedback.

Rate now

Comments { 0 }

Profile ID: LFWR-SCP-O-298385