Descent c-Wilf Equivalence

Abstract

Let Sn denote the symmetric group. For any σ ∈ Sn, we let des(σ) denote the number of descents of σ, inv(σ) denote the number of inversions of σ, and LRmin(σ) denote the number of left-to-right minima of σ. For any sequence of statistics stat1, . . . , stat k on permutations, we say two permutations α and β in Sj are (stat1, . . . stat k )-c-Wilf equivalent if the generating function of k i=1 x stat i i over all permutations which have no consecutive occurrences of α equals the generating function of k i=1 x stat i i over all permutations which have no consecutive occurrences of β. We give many examples of pairs of permutations α and β in Sj which are des-c-Wilf equivalent, (des, inv)-c-Wilf equivalent, and (des, inv, LRmin)-c-Wilf equivalent. For example, we will show that if α and β are minimally overlapping permutations in Sj which start with 1 and end with the same element and des(α) = des(β) and inv(α) = inv(β), then α and β are (des, inv)-c-Wilf equivalent.