In 2008, Chebikin considered the $cd$-index of $\mathfrak{S}_n$ with respect to the alternating descent statistic and asked for a combinatorial interpretation for its coefficients. In this paper, we provide an answer to Chebikin's open problem in terms of permutations without double descents and ending with an ascent, with respect to a new statistic defined on these permutations. Additionally, we demonstrate a $cd$-index approach to proving the gamma-expansions of alternating Eulerian polynomials. Furthermore, we offer a direct combinatorial interpretation for their gamma-coefficients.
@article{10_37236_12446,
author = {Qiongqiong Pan and Cheng Qian},
title = {On the \(cd\)-index for alternating descents},
journal = {The electronic journal of combinatorics},
year = {2024},
volume = {31},
number = {4},
doi = {10.37236/12446},
zbl = {1551.05012},
url = {http://geodesic.mathdoc.fr/articles/10.37236/12446/}
}
TY - JOUR
AU - Qiongqiong Pan
AU - Cheng Qian
TI - On the \(cd\)-index for alternating descents
JO - The electronic journal of combinatorics
PY - 2024
VL - 31
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/12446/
DO - 10.37236/12446
ID - 10_37236_12446
ER -
%0 Journal Article
%A Qiongqiong Pan
%A Cheng Qian
%T On the \(cd\)-index for alternating descents
%J The electronic journal of combinatorics
%D 2024
%V 31
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/12446/
%R 10.37236/12446
%F 10_37236_12446
Qiongqiong Pan; Cheng Qian. On the \(cd\)-index for alternating descents. The electronic journal of combinatorics, Tome 31 (2024) no. 4. doi: 10.37236/12446
