We will study the inversion statistic of $321$-avoiding permutations, and obtain that the number of $321$-avoiding permutations on $[n]$ with $m$ inversions is given by\[|\mathcal {S}_{n,m}(321)|=\sum_{b \vdash m}{n-\frac{\Delta(b)}{2}\choose l(b)}.\]where the sum runs over all compositions $b=(b_1,b_2,\ldots,b_k)$ of $m$, i.e.,\[m=b_1+b_2+\cdots+b_k \quad{\rm and}\quad b_i\ge 1,\]$l(b)=k$ is the length of $b$, and $\Delta(b):=|b_1|+|b_2-b_1|+\cdots+|b_k-b_{k-1}|+|b_k|$. We obtain a new bijection from $321$-avoiding permutations to Dyck paths which establishes a relation on inversion number of $321$-avoiding permutations and valley height of Dyck paths.
@article{10_37236_5451,
author = {Pingge Chen and Zhousheng Mei and Suijie Wang},
title = {Inversion formulae on permutations avoiding 321},
journal = {The electronic journal of combinatorics},
year = {2015},
volume = {22},
number = {4},
doi = {10.37236/5451},
zbl = {1326.05004},
url = {http://geodesic.mathdoc.fr/articles/10.37236/5451/}
}
TY - JOUR
AU - Pingge Chen
AU - Zhousheng Mei
AU - Suijie Wang
TI - Inversion formulae on permutations avoiding 321
JO - The electronic journal of combinatorics
PY - 2015
VL - 22
IS - 4
UR - http://geodesic.mathdoc.fr/articles/10.37236/5451/
DO - 10.37236/5451
ID - 10_37236_5451
ER -
%0 Journal Article
%A Pingge Chen
%A Zhousheng Mei
%A Suijie Wang
%T Inversion formulae on permutations avoiding 321
%J The electronic journal of combinatorics
%D 2015
%V 22
%N 4
%U http://geodesic.mathdoc.fr/articles/10.37236/5451/
%R 10.37236/5451
%F 10_37236_5451
Pingge Chen; Zhousheng Mei; Suijie Wang. Inversion formulae on permutations avoiding 321. The electronic journal of combinatorics, Tome 22 (2015) no. 4. doi: 10.37236/5451
