The Springer numbers, introduced by Arnold, are generalizations of Euler numbers in the sense of Coxeter groups. They appear as the row sums of a double triangular array $(v_{n,k})$ of integers, $1\leq|k|\leq n$, defined recursively by a boustrophedon algorithm. We say a sequence of combinatorial objects $(X_{n,k})$ is an Arnold family if $X_{n,k}$ is counted by $v_{n,k}$. A polynomial refinement $V_{n,k}(t)$ of $v_{n,k}$, together with the combinatorial interpretations in several combinatorial structures was introduced by Eu and Fu recently. In this paper, we provide three new Arnold families of combinatorial objects, namely the cycle-up-down permutations, the valley signed permutations and Knuth's flip equivalences on permutations. We shall find corresponding statistics to realize the refined polynomial arrays.
@article{10_37236_11988,
author = {Sen-Peng Eu and Louis Kao},
title = {Three new refined {Arnold} families},
journal = {The electronic journal of combinatorics},
year = {2023},
volume = {30},
number = {4},
doi = {10.37236/11988},
zbl = {1532.05004},
url = {http://geodesic.mathdoc.fr/articles/10.37236/11988/}
}
TY - JOUR
AU - Sen-Peng Eu
AU - Louis Kao
TI - Three new refined Arnold families
JO - The electronic journal of combinatorics
PY - 2023
VL - 30
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/11988/
DO - 10.37236/11988
ID - 10_37236_11988
ER -
%0 Journal Article
%A Sen-Peng Eu
%A Louis Kao
%T Three new refined Arnold families
%J The electronic journal of combinatorics
%D 2023
%V 30
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/11988/
%R 10.37236/11988
%F 10_37236_11988
Sen-Peng Eu; Louis Kao. Three new refined Arnold families. The electronic journal of combinatorics, Tome 30 (2023) no. 4. doi: 10.37236/11988
