Involutions containing exactly r pairs of intersecting arcs

Abstract The generating function Fr(x) that counts the involutions on n letters containing exactly r pairs of intersecting arcs in their graphical representation is studied. More precisely, an algorithm that computes the generating function Fr(x) for any given r ≥ 0 is presented. To derive the result for a given r, the algorithm performs certain routine checks on involutions of length 2r + 2 without fixed points. The algorithm is implemented in Maple and yields explicit formulas for 0 ≤ r ≤ 4.


Introduction
In recent years, much attention has been paid to the problem of counting the permutations of length n containing a given number r ≥ 0 of occurrences of a certain pattern. Most of the researchers considered only the case r = 0; namely, studying permutations avoiding a given pattern. Only a few of them considered the case r > 0, usually restricting themselves to the patterns of length 3. For patterns of length 3, there are two different cases τ = 123 and τ = 132 (see Table 1). r Number of permutations in S n containing 123 exactly r times Reference 0 1 n + 1 2n n [6] 1 3 n 2n n − 3 [9] 2 59n 2 + 117n + 100 2n(2n − 1)(n + 5) 2n n − 4 [4] r = 3, 4, . . . , 10 Number of permutations in S n containing 132 exactly r times Reference Let I n denote the set of all involutions in S n , that is, I n = {σ ∈ S n | σ 2 = id}. On I n , the focus of the pattern occurrence counting problem has been on the cases r = 0, 1, and patterns of size at most 4 (for instance, see [3,5] and references therein).
In order to present the main result of this paper, a graphical representation of an involution and the following notation is needed. For σ = σ 1 σ 2 · · · σ n ∈ I n , its graphical representation is a graph with vertices 1, 2, . . . , n on a horizontal line and arcs connecting (i, σ i ) for σ i = i. Henceforth, the involution is identified with its graphical representation. For example, Figure 1 presents the involution 1462(10)37985. r r r r r r r r r r 1 2 3 4 5 6 7 8 9 10 In this paper, we fix the pattern τ to be r r r r 1 2 3 4, or just say τ is the arc-pattern 3412 (where the term arc-pattern refers to the fact that each vertex of τ is a termination point of an arc). We say that an involution σ ∈ I n contains τ if there exist two arcs (a, b) (c, d) in σ where the induced subgraph of σ with vertices a, b, c, d equals τ . In other words, σ contains τ if there exists a pair of arcs (a, b) and (c, d) such that a < c < b < d (i.e., (a, b) intersects (c, d)). We define int(σ) to be the number of occurrences of τ in σ. We denote the set of involutions σ of I n having int(σ) = r by I n,r (see Table 2). We define the generating function for the cardinality of I n,r for a fixed r by F r (x), that is, F r (x) = n≥0 |I n,r |x n . In this paper, we  study the generating function F r (x). More precisely, we present an algorithm that computes the generating function F r (x) for any given r ≥ 0. To obtain the result for a given r, the algorithm performs certain routine checks on members of I 2r+2 without fixed points (recall that i is a fixed point of σ if σ i = i). The algorithm has been implemented in Maple and yields explicit formulas for 0 ≤ r ≤ 4.

Main result
To any involution σ ∈ I n , we assign a bipartite graph G σ as follows. Let V 1 = [n] be the vertices in the first part of G σ and V 4 = {abcd | (a, c) intersects (b, d) in σ} be the vertices in the second part. Entry i ∈ V 1 is connected by an edge to occurrence q ∈ V 4 if i is a letter in q. For example, Figure 2 presents the bipartite graph for the involution 1462(10)37985.  This leads to the following basic lemma.
We denote the maximal connected component of G σ containing the entry 1 by G σ . Clearly, there is no fixed point i = 1 such that i belongs to G σ . Thus, the number of vertices i ∈ [n] belonging to G σ is even, or G σ consists only of the vertex 1.
Let σ = σ i1 σ i2 · · · σ i2t be the entries of σ that belong to G σ and let π σ ∈ I 2t be the corresponding involution. In this context, σ is called the kernel of σ, with π σ referred to as the kernel shape of σ and 2t the size of the kernel shape. The capacity c πσ of σ is defined as the number of pairs of arcs that intersect in π σ .
The following statement is implied immediately by Lemma 2.1.
Theorem 2.1. Let σ ∈ I n,r . Then the size of the kernel shape ρ of σ is at most 2r + 2 and it has no fixed points.
We say that ρ is a kernel involution if it is the kernel shape for some involution σ. Clearly, ρ is a kernel involution if and only if π ρ = ρ. Let ρ ∈ I n be any kernel involution, and we denote the set of involutions of all possible sizes whose kernel shape equals ρ by I(ρ).
Hence, the generating function for the number of involutions σ ∈ I n,r with kernel shape ρ ∈ I 2t is given by Define K r to be the set of all kernel shapes ρ ∈ I 2n having c ρ = r, where 1 ≤ n ≤ r + 1 (see Theorem 2.1). Hence, we can state our main result.
Theorem 2.2. The generating function F r (x) for r ≥ 0 is given by Next, we apply this theorem for 0 ≤ r ≤ 4.

General r
By induction on r, Theorem 2.2, together with the cases r = 0, 1, 2, yields the following result.

Theorem 2.3. The generating function
Moreover, it can be written as

Further results
Theorem 2.2 can be extended as follows. Let F r (x, q) be the generating function for the number of involutions in I n,r according to the number of fixed points. Then, by our structure, we have the following result. For example, Theorem 3.1 gives F 0 (x, q) = 1 + xqF 0 (x, q) + x 2 F 0 (x, q) and F 1 (x, q) = xqF 1 (x, q) + 2x 2 F 1 (x, q)F 0 (x, q) + x 4 (F 0 (x, q)) 4 . Hence, We conclude this paper by referring the reader to [2] for the q = 0 case of Theorem 3.1.