On an open problem of Green and Losonczy: exact enumeration of freely braided permutations

Recently, Green and Losonczy [5, 6] introduced freely braided permutations as a special class of restricted permutations that has arisen in the study of Schubert varieties. They suggest as an open problem to enumerate the number of freely braided permutations in S n . In this paper, we prove that the generating function for the number of freely braided permutations in S n is given by


Introduction
Let α ∈ S n and τ ∈ S k be two permutations.Then α contains τ if there exists a subsequence 1 ≤ i 1 < i 2 < • • • < i k ≤ n such that (α i 1 , . . ., α i k ) is order-isomorphic to τ; in such a context τ is usually called a pattern; α avoids τ, or is τ-avoiding, if α does not contain such a subsequence.The set of all τ-avoiding permutations in S n is denoted by S n (τ).For a collection of patterns T , α avoids T if α avoids all τ ∈ T ; the corresponding subset of S n is denoted by S n (T ).
Recently, a special class of restricted permutations has arisen in the study of Schubert varieties.Green and Losonczy [5] defined, for any simply laced Coxeter group, a subset of "freely braided elements" (for details, see [5] and [6]), and they suggest as an open problem to enumerate the number of freely-braided permutations in S n .In this paper we give a complete answer for the number of freely-braided permutations in F n .The main result of this paper can be formulated as follows.
Theorem 1.3.The generating function for the number of freely-braided permutations in F n is given by is the generating function for the Catalan numbers It is natural to extend f n to the case m = 0 by setting f n (∅) = f n = #F n .The following properties of the numbers f n (b 1 , . . ., b m ) can be deduced easily from the definitions.Lemma 2.1.
Proof.For (1), observe that if π ∈ S n is such that π 1 . . .Using similar arguments as in the proof of (3) we get that (4) and ( 5) hold.
Next we introduce objects A m (n), B m (n) and C m (n) which organize suitably the information about the numbers f n (b 1 , b 2 , ..., b m ) and play an important role in the proof of the main result.
As before, this definition is extended to the case m = 0 by setting A 0 (n) = C 0 (n) = f n and B 0 (n) = 0.
In the following two subsections we derive expressions for A m (n) and B 1 (n), which are used in subsection 2.3 to complete the proof of Theorem 1.3.

A recurrence for the f j , also involving B 1 (n)
In the following result we derive an expression for A m (n).

Proposition 2.3.
(1) For all n ≥ 2, (2) For all n ≥ 3, Proof.For (1), Definition 2.2 for A 1 (n) gives that Observe that if π ∈ S n is such that π 1 = n − 1 or π 1 = n, then no occurrence of the patterns 1234, 1243, 1324, 2134 in π can involve the entry π 1 .So we get that the number of permutations in F n starting with n Using Lemma 2.1, parts (1) and ( 5), and Definition 2.2 we obtain that Hence, by using the proof of (1) and Definition 2.2 we get the desired result.
Using Lemma 2.1, parts ( 2) and ( 3), we have Definition 2.2 and Lemma 2.1 (4) give Hence, by induction on m together with (1) we get the desired result.
We next find an explicit expression for with the usual convention that a b = 0 if a < b or a < 0.
or equivalently, for all n ≥ 4. Multiplying by x n and summing over all n ≥ 4 together with using Lemma 2.8 we arrive at or equivalently, .
The rest is easy to check.

Definition 1.1. A permutation
π is said to be freely-braided if and only if π avoids each of the four patterns 1234, 1243, 1324, and 2134.We denote the set of all freely-braided permutations in S n by F n , i.e., F n = S n (1234, 1243, 1324, 2134).