We prove various formulas which express exponential generating functions counting permutations by the peak number, valley number, double ascent number, and double descent number statistics in terms of the exponential generating function for Chebyshev polynomials, as well as cyclic analogues of these formulas for derangements. We give several applications of these results, including formulas for the (-1)-evaluation of some of these distributions. Our proofs are combinatorial and involve the use of monomino-domino tilings, the modified Foata-Strehl action (a.k.a. valley-hopping), and a cyclic analogue of this action due to Sun and Wang.
@article{10_37236_8413,
author = {Jordan O. Tirrell and Yan Zhuang},
title = {Hopping from {Chebyshev} polynomials to permutation statistics},
journal = {The electronic journal of combinatorics},
year = {2019},
volume = {26},
number = {3},
doi = {10.37236/8413},
zbl = {1418.05012},
url = {http://geodesic.mathdoc.fr/articles/10.37236/8413/}
}
TY - JOUR
AU - Jordan O. Tirrell
AU - Yan Zhuang
TI - Hopping from Chebyshev polynomials to permutation statistics
JO - The electronic journal of combinatorics
PY - 2019
VL - 26
IS - 3
UR - http://geodesic.mathdoc.fr/articles/10.37236/8413/
DO - 10.37236/8413
ID - 10_37236_8413
ER -
%0 Journal Article
%A Jordan O. Tirrell
%A Yan Zhuang
%T Hopping from Chebyshev polynomials to permutation statistics
%J The electronic journal of combinatorics
%D 2019
%V 26
%N 3
%U http://geodesic.mathdoc.fr/articles/10.37236/8413/
%R 10.37236/8413
%F 10_37236_8413
Jordan O. Tirrell; Yan Zhuang. Hopping from Chebyshev polynomials to permutation statistics. The electronic journal of combinatorics, Tome 26 (2019) no. 3. doi: 10.37236/8413
