The size of Fulton's essential set

Abstract

The essential set of a permutation was defined by Fulton as the set of southeast corners of the diagram of the permutation. In this paper we determine explicit formulas for the average size of the essential set in the two cases of arbitrary permutations in $S_n$ and $321$-avoiding permutations in $S_n$. Vexillary permutations are discussed too. We also prove that the generalized Catalan numbers ${r+k-1\choose n}-{r+k-1\choose n-2}$ count $r\times k$-matrices dotted with $n$ dots that are extendable to $321$-avoiding permutation matrices.