Inversion-descent enumerators of $\underline{32}1$-avoiding permutations

Abstract

We give the recurrence relation for counting polynomials for permutations avoiding the $\321$ vincular pattern, with respect to inversion numbers and descent numbers, and consequently give the $q$-exponential generating function for these counting polynomials. Furthermore, we give another explanation for these counting polynomials in terms of restricted-growth words.