Avoiding maximal parabolic subgroups of Sk

Let [p] = {1, . . . , p} denote a totally ordered alphabet on p letters, and let α = (α1, . . . ,αm) ∈ [p1], β = (β1, . . . ,βm) ∈ [p2]. We say that α is order-isomorphic to β if for all 1≤ i< j ≤ m one has αi < α j if and only if βi < β j. For two permutations π ∈ Sn and τ ∈ Sk, an occurrence of τ in π is a subsequence 1 ≤ i1 < i2 < .. . < ik ≤ n such that (πi1 , . . . ,πik ) is order-isomorphic to τ; in such a context τ is usually called the pattern. We say that π avoids τ, or is τ-avoiding, if there is no occurrence of τ in π. Pattern avoidance proved to be a useful language in a variety of seemingly unrelated problems, from stack sorting [Kn, Ch. 2.2.1] to singularities of Schubert varieties [LS]. A natural generalization of single pattern avoidance is subset avoidance; that is, we say that π ∈ Sn avoids a subset T ⊂ Sk if π avoids any τ ∈ T . The set of all permutations in Sn avoiding T ⊂ Sk is denoted Sn(T ). A complete study of subset avoidance for the case k = 3 is carried out in [SS]. For k> 3 the situation becomes more complicated, as the number of possible cases grows rapidly. Recently, several authors have considered the case of general k when T has some nice algebraic properties. Paper [BDPP] treats the case when T is the centralizer of k−1 and k under the natural action of Sk on [k] (see also Sec. 3 for more detail). In [AR], T is a Kazhdan–Lusztig cell of Sk, or, equivalently, the Knuth equivalence class (see [St, vol. 2, Ch. A1]). In this paper we consider the case when T is a maximal parabolic subgroup of Sk. Let si denote the simple transposition interchanging i and i + 1. Recall that a subgroup of Sk is called parabolic if it is generated by si1 , . . . ,sir . A parabolic subgroup of Sk is called maximal if the number of its generators equals k− 2. We denote by Pl,m the (maximal) parabolic subgroup of Sl+m generated by s1, . . . ,sl−1,sl+1, . . . ,sl+m−1, and by fl,m(n) the number of permutations in Sn avoiding all the patterns in Pl,m. In this note we find an explicit expression for the generating function of the sequence { fl,m(n)}. †Partially supported by HIACS.