Inversion Formulae on Permutations Avoiding 321

Abstract

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.

Patterns in random permutations avoiding the pattern 321

“…where the limit random variable is, up to a constant factor, the area under a Brownian excursion e (see eg, [29] for many other results on this random area). See also the related expressions for the distribution of n 21 ( 321,n ) in Chen and coworkers [16]. Apart from (1.8), we do not know any previous result on asymptotic distributions of n ( 321,n ) beyond the expectations in (1.4) to (1.7).…”

“…Moreover, the equivalence given by Cheng and coworkers between n 21 ( π 321 , n ) and the number of certain squares under a Catalan path implies by standard results for the area under the equivalent Dyck paths that, as n → ∞, where the limit random variable is, up to a constant factor, the area under a Brownian excursion e (see eg, for many other results on this random area). See also the related expressions for the distribution of n 21 ( π 321 , n ) in Chen and coworkers .…”

“…[29] for many other results on this random area). See also the related expressions for the distribution of n 21 (π 321,n ) in Chen, Mei and Wang [16].…”