Popularity of patterns over d-equivalence classes of words and permutations

Abstract

Two same length words are d-equivalent if they have same descent set and same underlying alphabet. In particular, two same length permutations are d-equivalent if they have same descent set. The popularity of a pattern in a set of words is the overall number of copies of the pattern within the words of the set. We show the far-fromtrivial fact that two patterns are d-equivalent if and only if they are equipopular over any d-equivalence class, and this equipopularity does not follow obviously from a trivial equidistribution.

Transformation à la Foata for special kinds of descents and excedances

“…The distribution of the number of descents has been widely studied on several classes of combinatorial objects such as permutations [14], cycles [7,8], and words [3,10]. Many interpretations of this statistic appear in several fields as Coxeter groups [4,11] or lattice theory [5,12].…”

Pattern statistics in faro words and permutations

“…UDUD). The popularity of a pattern p in A is the total number of occurrences of p over all objects of A, that is p(A) = a∈A p(a) (see [5,9,16]). For instance, for a dispersed Dyck path P = F F U DF U U DU U U DDDD we have FF(P ) = 1, DDD(P ) = 2, UD(P ) = 3 and UUUU(P ) = 0.…”