Bruhat Order on Partial Fixed Point Free Involutions

The order complex of inclusion poset $PF_n$ of Borel orbit closures in skew-symmetric matrices is investigated. It is shown that $PF_n$ is an EL-shellable poset, and furthermore, its order complex triangulates a ball. The rank-generating function of $PF_n$ is computed and the resulting polynomial is contrasted with the Hasse-Weil zeta function of the variety of skew-symmetric matrices over finite fields.


Introduction
This paper is a continuation of our earlier investigations [5], [6] on the Bruhat order on certain sets of involutions, and our notation follows these references closely: C : field of complex numbers, S n : symmetric group of n × n permutation matrices, R n : rook monoid of n × n partial permutation matrices, I n : involutions in S n , F n : fixed-point-free involutions in S n , P I n : partial involutions in R n , Mat n : all n × n matrices over C, Sym n : all n × n symmetric matrices over C, GL n : invertible n × n matrices over C, B n : Borel group of invertible upper triangular matrices from GL n .
In addition to the above list of notation, we consider Skew n , the space of all n×n skewsymmetric matrices over C, and P F n , the set of all fixed-point-free partial involutions. The purpose of this article is to investigate some combinatorial properties of P F n . In some sense, this is the final step of our program for showing that the sets of partial permutations R n , P I n , and P F n all share the same algebraic combinatorial properties.
Let X be a variety on which a Borel group B acts algebraically. Let W denote the set of B-orbits in X, and define the B-ordering ≤ on P by Study of this basic combinatorial set-up is important for group theory. Indeed, suppose G is a linear algebraic group with a Borel subgroup B. Then the double cosets of B in G are equivalent to the orbits of B × B acting on X = G via (g, h) · x = gxh −1 . Furthermore, B × B-orbits in X are parametrized by the 'Weyl group' of G (the Bruhat-Chevalley decomposition). We have a well-known special case, when G = GL n . Then, B n × B n -orbits are parametrized by S n , and the induced partial ordering is the Bruhat-Chevalley ordering on S n . In [17], by generalizing Bruhat-Chevalley decomposition to linear algebraic monoids, Renner constructs a rich family of orbit posets. In particular, among other things, he shows that the orbits of the Borel group action (g, h) · A = gAh −1 , g, h ∈ B n , A ∈ Mat n . ( are parametrized by R n . Basic combinatorial properties of B n × B n -ordering on R n are investigated in [1]. In [18], Richardson and Springer investigate the Borel orbits in the setting of symmetric spaces. In particular, they show that the set of involutions I n of S n parametrizes the Borel orbits in the symmetric space SL n /SO n , and furthermore, the corresponding B n -ordering on I n agrees with the restriction of the Bruhat-Chevalley ordering from S n (see [19]). Here SL n is the special linear group and SO n is its special orthogonal subgroup. Also in [18], they show that B n -orbits in SL n /Sp n are parametrized by F n ⊂ I n .
The monoid of matrices Mat n can be viewed as a partial compactification of GL n , and similarly, the set of all symmetric matrices (respectively, set of all skew-symmetric matrices) can be viewed as a partial compactification of SL n /SO n (respectively, of SL 2n /Sp 2n ). Similar to the construction of R n , by using suitable modifications of the method of Gauss-Jordan elimination, it is shown in [21] for X = Sym n , and in [7] for X = Skew n that the B n -orbits of the action are parametrized by P I n and P F n , respectively. Further combinatorial properties of the B n -ordering on P I n and on P F n are investigated by the second author in the papers [1] (joint with E. Bagno) and [7]. There is an interesting relation between P F n and the set of invertible involutions: Let x ∈ P F n be a partial fixed-point-free involution with determinant 0. We denote byx the completion of x to an involution in I n by adding the missing diagonal entries. For example, Define φ : P F n → I n by setting It is not difficult to check that φ is a bijection between P F n and I n such that φ(x) = x for all x ∈ F n . Now that we have two sets in bijection with corresponding B norderings, it is natural to ask for their comparison. This is one of the goals of our paper.
Recall that the order complex ∆(P ) of a poset P is the abstract simplicial complex consisting of all chains in P . Important topological information on a simplicial complex is hidden in the orderings of its facets (which corresponds to the maximal chains in P ). If the facets are ordered in a way that the intersection of a facet with all the preceding facets is a simplicial subcomplex of codimension 1, then the complex is called shellable. In this case, it is known that the simplicial complex has the homology type of a sphere, or of a ball. For posets, a purely combinatorial criteria for checking the shellability condition is found by Björner in [2], and it is called the "lexicographic shellability" of P .
A finite graded poset P with a maximum and a minimum element is called ELshellable, if there exists a map f = f Γ : C(P ) → Γ from the set of covering relations C(P ) of P into a totally ordered set Γ satisfying 1. in every interval [x, y] ⊆ P of length k > 0 there exists a unique saturated chain such that the entries of the sequence are weakly increasing.
2. The sequence (5) is lexicographically smallest among all sequences of the form In literature there are different versions of this notion and EL-shellability is known to imply the others (see [23]). A brief history of the shellability questions in Borel orbit posets is as follows: In [10], Edelman proves that BC-order on S n is EL-shellable. Shortly after, Proctor in [15] shows that all classical Weyl groups are EL-shellable. Around the same time, in [3], Björner and Wachs show that Bruchat-Chevalley ordering on all Coxeter groups, as well as on all sets of minimal-length coset representatives (quotients) in Coxeter groups are "dual CL-shellable" (a weaker alternative to EL-shellability). A decade after the introduction of CL-shellability, in [9], M. Dyer shows that Bruhat-Chevalley ordering on all Coxeter groups and all quotients are EL-shellable. As an application of EL-shellability, using Dyer's methods, in [24], L. Williams shows that the poset of cells of a cell decomposition for totally non-negative part of a flag variety is EL-shellable. In the papers [12] and [11] A. Hultman, although avoids showing lex. shellability, obtains the same topological consequences for the Bruhat-Chevalley ordering on "twisted involutions" in Coxeter groups.
There are various directions that the results of [3] are extended. For semigroups, in [16], Putcha shows that "J-classes in Renner monoids" are CL-shellable. In [4], the first author shows that for the special Renner monoid R n , not only the J-classes are lex. shellable, but also the whole rook monoid R n is EL-shellable. In [6], the first and the third authors show that P I n is EL-shellable. In [5], we show that F n is also EL-shellable, and furthermore, its order complex is a ball of appropriate dimension. In [13], Incitti shows that I n is EL-shellable, and in [14] he shows that the B-order on involutions in all classical Weyl groups are EL-shellable.
Contributing to the above literature, we show in this paper that P F n is an ELshellable poset. Moreover, we show that the order complex of P F n triangulates a ball of dimension n(n − 1)/2. On the other hand, it is known that the order complex of I n triangulates a sphere of dimension ⌊n/4⌋ (see [13], page 255).
The structure of our paper is as follows. In the next section we introduce basic notation for poset theory. In particular, we recollect some known, basic facts about Bruhat-Chevalley ordering on rooks and partial involutions. In Subsection 2.3, we compare the length functions of P F n and P I n .
Unfortunately, P F n is not a connected subposet of P I n , hence we are not able to directly utilize our earlier results from [6]. Therefore, we devote all of Section 3 for the review of the covering relations of I n , F n , and of P I n in order for describing the covering relations of P F n next.
In Section 4 we present our proof of EL-shellability of P F n . As an application of this result, in Section 5, we determine the homotopy type of the order complex of the proper part of P F n , namely P F n with its smallest and the largest elements excluded.
In the final section of our paper, we investigate the length-generating functions of certain subposets of P F n . In particular, we relate our length generating function computations to the number of rational points of the variety of skew-symmetric matrices of fixed rank defined over a finite field.
Acknowledgement. The first and third authors are partially supported by the Louisiana Board of Regents Research and Development Grant.

Preliminaries
Notation: Let m be a positive integer. We denote the set {1, . . . , m} by [m]. The rank of a matrix x ∈ Mat n is denoted by rk(x).

Poset terminology
All of our posets are assumed to be finite, graded, and furthermore, they are assumed to possess a minimal and a maximal element, denoted by0 and1, respectively. We reserve the letter P as the name of a generic such poset and denote by ℓ : P → N (or, by ℓ P , if needed) the length function on P . The set of all covering relations in P is denoted by C(P ). If (x, y) ∈ C(P ), then we write y → x to mean that y covers x.
Recall that the Möbius function of P is defined recursively by the formula for all x ≤ y in P . As customary, we denote by ∆(P ) the order complex of P . It is well known that µ(0,1) is equal to the "reduced Euler characteristic" χ(∆(P )) of the topological realization of ∆(P ). See Proposition 3.8.6 in [20].
Let Γ denote a finite totally ordered poset and let g be a Γ -valued function defined on C(P ). Then g is called an R-labeling of P , if for every interval [x, y] in P , there exists a unique chain x = x 1 ← x 2 ← · · · ← x n−1 ← x n = y such that Thus, P is EL-shellable, if it has an R-labeling g : C(P ) → Γ such that for each interval [x, y] in P the sequence (6) is lexicographically smallest among all sequences of the form , by P S we denote the subset P S = {x ∈ P : ℓ(x) ∈ S}, and denote by µ S the Möbius function of the posetP S that is obtained from P S by adjoining a smallest and a largest element, if they are missing. For an R-labeling g : C(P ) → Γ of P , it is well known that the quantity (−1) |S|−1 µ S (0P S ,1P S ) is equal to the number of maximal chains x 0 =0 ← x 1 ← · · · ← x n =1 in P for which the sequence (g(x 0 , x 1 ), · · · , g(x n−1 , x n ) has descent set S, that is to say, for which

B-order on partial involutions
The notation F n , I n , P I n , R n , S n , Skew n , and Sym n are as in the introduction.
Recall that R n parameterizes the B n × B n -orbits in Mat n . For the purposes of this paper, it is more natural for us to look at the inclusion poset of B ⊤ n × B n -orbit closures in R n , which we denote by (R n , ≤ Rook ). Here B ⊤ n is the Borel subgroup of all lower triangular matrices from GL n .
In [7], Cherniavsky shows that the Borel orbits in Skew n are parametrized by those elements x ∈ Skew n such that 3. in every row and column of x there exists at most one non-zero entry.
Note that when -1's in x are replaced by +1's, the resulting matrixx is a partial involution with no diagonal entry. In other words,x is a fixed-point-free partial involution. It is easy to check that this correspondence is a bijection, hence P F n parameterizes the Borel orbits in Skew n .
Containment relations among the closures of Borel orbits in Skew n define a partial ordering on P F n . We denote its dual by ≤ Skew . Similarly, on P I n we have the dual of the partial ordering induced from the containment relations among the Borel orbit closures in Sym n . We denote this dual partial ordering by ≤ Sym .
2.3 Combinatorial approach to the posets R n , P I n , P F n .
There is a combinatorial method for deciding when two elements x and y from (R n , ≤ Rook ) (respectively, from (P I n , ≤ Sym ), or from(P F n , ≤ Skew )) are comparable with respect to ≤ Rook (respectively, with respect to ≤ Sym , or ≤ Skew ). We denote by Rk(x) the matrix whose i, j-th entry is the rank of the upper left i × j submatrix of x. Hence, Rk(x) is an n × n matrix with non-negative integer coordinates. We call Rk(x), the rank-control matrix of x.
Let A = (a i,j ) and B = (b i,j ) be two matrices of the same size with real number entries. We write A ≤ B if a i,j ≤ b i,j for all i and j. Then The same criterion holds for the posets ≤ Sym and ≤ Skew . We recall some fundamental facts about the covering relations of ≤ Sym and ≤ Skew . Our references are [1] and [7]. Let Rk(x) = (r i,j ) m i,j=1 denote the rank-control matrix of an m × m matrix x. As a notation we set r 0,i = 0 for i = 0, . . . , m and define Then the length function ℓ P F n of the poset P F n is equal to the restriction of ρ < to P F n . Furthermore, x covers y if and only if Rk(x) Rk(y) and ℓ P Fn (x)−ℓ P Fn (y) = 1. The length function of P F n differs from the length function of P I n in two ways: The ranks of two matrices y < x in P F n differ by a multiple of 2, and the smallest element in P I n is the identity matrix, which is not in P F n . The minimal element in P F n is given by the matrix with the largest rank-control matrix. This means that in the case when n is even so that the length function increases only by 1 if the rank drops by 2 and we subtract n 2 because the minimal element has to have length zero. Similarly, when n is odd we have to subtract n−1−rk(x) 2 and n+1 2 . Summarizing, we see that for all n the length function ℓ P Fn (x) of P F n is given by Example 2.1. When n = 6, the smallest element is 3 An EL-labeling of P F n We recall some results on the covering relations of I n , F n , and of P I n [13, 5, 6].

EL-labeling of I n
For a permutation σ ∈ S n , a rise of σ is a pair of indices 1 ≤ i 1 , i 2 ≤ n such that For σ ∈ S n , define its fixed point set, its exceedance set and its defect set to be respectively. Given a rise (i 1 , i 2 ) of σ, its type is defined to be the pair (a, b), if i 1 ∈ I a (σ) and i 2 ∈ I b (σ), for some a, b ∈ {f, e, d}. We call a rise of type (a, b) an ab-rise. On the other hand, two kinds of ee-rises have to be distinguished from each other; an The rise (i 1 , i 2 ) of an involution σ ∈ I n is called suitable if it is free and if its type is one of the following: (f, f ), (f, e), (e, f ), (e, e), (e, d).
A covering transformation, denoted ct (i 1 ,i 2 ) (σ), of a suitable rise (i 1 , i 2 ) of σ is the involution obtained from σ by moving the 1's from the black dots to the white dots as depicted in Figure 1.
It is shown in [13] that if τ and σ are two involutions in I n , then Let Γ denote the totally ordered set [n] × [n] with respect to lexicographic ordering.
In the same paper, Incitti shows that the labeling defined by is an EL-labeling, hence, (I n , ≤ Sym ) is an EL-shellable poset.

EL-labeling of F 2n
Recall that F 2n is a connected graded subposet of I 2n . Therefore, its covering relations are among the covering relations of I 2n . On the other hand, within F 2n we use two types of covering transformations, only: a non-crossing ee-rise and an ed-rise. These moves correspond to the items numbered 4 and 6 in Table 1 of [13]. It is shown in [5] that these covering labels is an EL-labeling for F 2n .

EL-labeling of P I n
When two partial involutions x and y have the same zero rows and zero columns, the covering relation x → y is not different than the invertible case.
Note that x → y if and only if the invertible involutionx, that is obtained from x by removing the rows and columns of x with no non-zero entries, covers the invertible involutionỹ that is obtained from y by removing its rows and columns with zeros only.
f f -rise: Non-crossing ee-rise: Crossing ee-rise:    In the light of the above examples, we label a covering relation x → y in P I n as follows.

As in Example 3.1, if the covering relation
x → y is derived from the covering relationx →ỹ of invertible involutions that are obtained from x and y, respectively, then we use the labelingx →ỹ as defined in [13] The corresponding labeling here is (3,5).   It is shown in [6] that the covering labelings defined in Definition 3.9 is an ELlabeling for P I n .
4 An EL-labeling of P F n Covering relations of F n are covering relations in I n , as well. Unfortunately, this is not the case for P F n relative to P I n . In other words, as a subposet of P I n , P F n is not connected. For example, when n = 2, there are only two partial fixed-point-free involutions: x = 0 0 0 0 and y = 0 1 1 0 , hence x covers y as a partial fixed-pointfree involution. However, viewed as a partial involution x does not cover y since Then either x covers y as an element of P I n , or there exists z ∈ P I n such that x → z by an d-cover as an element of P I n , and z → y by an r-cover in P I n , where at each step the rank drops by 1. Furthermore, in the first case, there are two possibilities: 1.
x → y is an r-cover in P I n , or 2.
x → y is a c-cover corresponding to a non-crossing ee, or to an ed-rise in P I n .
Proof. Obviously, if x covers y in P I n and if both x and y are members of P F n , then x covers y in P F n , also. Thus, the last assertion follows from Lemma 3. 10 We proceed with the assumption that x, y ∈ P F n but x does not cover y in P I n . Towards a contradiction, assume that there does not exists z ∈ P I n as in the conclusion of the lemma. This means that the open interval (y, x) = {z ∈ P I n : y < z < x} lies in P I n \ P F n . In other words, any z ∈ (y, x) has to have a non-zero diagonal entry. This eliminates the possibility of z → y being a c-cover (see Figure 1). Clearly, z → y cannot be a d-cover, neither.
We continue with the assumption that z is obtained from y by an r-move, which places two symmetric entries on the diagonal. In this case, another r-move is possible in y involving the same 1's. (To construct an example to this situation, start with y as in Example 3.8.) Let z 1 denote this new element from P F n . By comparing their rank-control matrices, we see that Rk(x) < Rk(z 1 ), hence y < z 1 < x. This contradicts with our assumption that the interval (y, x) lies in P I n \ P F n . Therefore, z covers y by an r-move, by deleting a 1 from y and placing another to diagonal. Then by a d-move removing this diagonal 1 we obtain x. Thus we obtain a contradiction to our initial assumption.
Remark 4.2. Let x and y be two elements from P F n such that x covers y by an rmove. Let x = (x 1 , . . . , x n ) and y = (y 1 , . . . , y n ) denote x and y in one-line notation. Then exactly one of the following statements is true: 1. x is obtained from y by replacing exactly two entries of y = (y 1 , . . . , y n ) by 0's.

There exists i ∈ [n]
such that x is obtained from y by replacing y i by the number x i , setting y i -th entry of y to 0 and replacing the x i -th entry of y (which is a 0) by i.
In the light of Lemma 4.1 we make the following definition. 1. If the covering relation is derived from a c-move, then we use the labeling as defined in [6] and transform this label (i, j) into (n − i, n − j).

If the covering relation
x → y results from an r-move, then we define the label to be (i + n, j), where x > y results from y by moving the 1 in column i to row j. If the 1 is pushed out of the matrix, then we set j = n + 1.
In the case of invertible fixed-point-free involutions we show in [5] that the lexicographically largest chain is the only decreasing chain. Since the label is transformed from (i, j) to (n − i, n − j) now the lexicographically smallest chain is increasing. The reason the label of r-moves is shifted by n in the first coordinate is to ensure that every r-cover has a bigger label than any c-cover. In Figure 2, we illustrate the Definition 4.3. Proposition 4.4. Let y < x be two partial fixed-point-free involutions from P F n , and let c : x = x 1 < x 2 < · · · < x s+1 = y denote the maximal chain whose sequence of labels f (c), as defined in Definition 4.3, is lexicographically smallest among all such sequences. Then f (c) is a weakly increasing sequence.
In each of these cases, we either produce an immediate contradiction by showing that the two moves are interchangeable (hence c is not the smallest chain), or we construct an element z ∈ [x, y] ∩ P F n which covers x t−1 , and such that f ((x t−1 , z)) < f ((x t−1 , x t )). Since we assume that f (c) is the lexicographically smallest Jordan-Hölder sequence, the existence of such an element z is a contradiction, also.
To this end, suppose that the label of the first move (x t → x t−1 ) is (i, j), and the second move (x t+1 → x t ) is labeled by (k, l).
Case 1: Follows from the proof for invertible fixed-point-free involutions. Case 2: If i = k, then l > j. In this case, we interchange the two moves to obtain our desired contradiction. Therefore we continue with assuming k < i. If k − n = j then j < i−n and (m+ n, l) is possible in x t−1 with m < j < i, where (m, i−n) is the position of the 1 in x t−1 . If k − n = j then either the two moves are interchangeable, or (k, l) removes a suitable rise in x t−1 which corresponds to a move with a smaller label than (i, j).
Case 3: This case is impossible since every c-move has a smaller label than any r-move.
Case 4: If the r-cover labeled (i, j) is the covering relation with the lexicographically smallest label then there is no suitable rise in x t−1 . The c-move has to involve one of the moved 1's since otherwise there is a suitable rise in x t−1 . For this, one of the moved 1's has to have a 1 to the upper left or the lower right in x t that was not to the upper left or lower right of it in x t−1 . Since the 1's are moved right and down respectively, it is impossible that there is a 1 to the lower right in x t that is not to the lower right in x t−1 . If the c-cover corresponds to the suitable rise (m, i − n) (with label (n − m, i)), then (i, j) is not the r-move with the smallest label in x t−1 since in this case (m + n, j) is possible in x t−1 with (n + m, j) < (i, j). If the c-cover corresponds to the rise (m, j), then the r-move (m + n, i − n) is possible in x t−1 which again has a smaller label than (i, j). Proof. We use induction on the length s + 1 of the interval [y, x] to prove that no other chain is lexicographically increasing. Clearly, if x covers y, there is nothing to prove, so, we assume that for any interval of length k ≤ s there exists a unique increasing maximal chain.
Case 1: Done in the proof for the invertible case. Case 2: It is impossible for i = k since there is only one r-move for each 1. Therefore assume that i < k. Let the moved 1's be on the symmetric positions (i − n, m) and (m, i − n) in x 0 . If k = m + n then (l + n, j) is possible in x ′ 1 with (l + n, j) < (k, l). If k = m then either the two moves are interchangeable or the suitable rise (n − i, n − k) is possible in x ′ 1 . Case 3: Since no r-move can remove a suitable rise, there exists a legal c-move in x ′ 1 . But this c-move has a smaller label than (k, l) which is our desired contradiction. Case 4: This case is not possible because every c-move has a smaller label than any r-move.
Combining previous two propositions, we have our first main result: Theorem 4.6. The poset P F n is an EL-shellable poset. 5 The order complex of P F n In [5], it is shown that the order complex ∆(F n ) of fixed-point-free involutions triangulates a ball of dimension n 2 − n − 2. In this section we obtain a similar result for P F n .
We continue by analyzing the intervals of length two.
Lemma 5.2. Each length two interval [y, x] ⊆ P F n has at most four, at least three elements.
In the first case, [y, x] is isomorphic to an interval in F m for some m ≤ n, and therefore, it has at most 4 elements (since F m is a connected subposet of I m , which is Eulerian).
In the second case, we look at the one-line notations of y and x. See ??. If z is obtained from y by setting two non-zero entries of y to 0's, and if, at the same time, x is obtained from z by setting two non-zero entries of z to 0's, then y and x differ at exactly 4 entries. Therefore, [y, x] contains at most one other element other than z, which is obtained from y by setting two entries of y to 0's. If z is obtained by increasing the i-th entry y i of y to z i , and if, at the same time, x is obtained from z by increasing the i-th entry z i of z to x i , then [y, x] has exactly 3 elements. If z is obtained from y by increasing the i-th entry y i of y to z i , and if x is obtained from z with no overlap with the replaced/increased entries of y, then [y, x] has exactly 4 elements. Finally, if z is obtained by increasing the i-th entry y i of y to z i , and x is obtained from z by replacing the z i -th entry of z by 0, then y and x differ at exactly at 4 positions. Therefore, the interval [y, x] have at most 4 elements.
Since the arguments of Case 3 and Case 4 are identical, we handle Case 3 only. Suppose that there exist more than 4 elements in [y, x]. Since one of the elements y < z < x is obtained from y by a c-move, the covering type of any other y < z 1 < x is not of type c. Otherwise, to obtain x from z we need to apply another c-move to z. But then the matrix ranks of y and x would be the same. Therefore, we conclude that if z 1 = z and y < z 1 < x, then z 1 is obtained from y by an r-move, and x is obtained from z 1 by a c-move. Now it is clear that it is impossible to have another element y < z 2 < x such that z 2 covers y by an r-move and z 2 / ∈ {z, z 1 }. Therefore [y, x] have exactly 4 elements and the proof is complete.
We know from [8] that a pure, shellable simplicial complex ∆ of which every dim ∆ − 1 face is contained in at most two facets is homeomorphic to either a ball, or a sphere. By Lemma 5.2, we see that ∆(P F n ) satisfies this property. Theorem 5.3. Let P F n denote the proper part of P F n , namely the subposet obtained from P F n by removing its smallest and the largest elements. For n ≥ 3, the order complex ∆( P F n ) triangulates a ball of dimension dim ∆(P F n ) − 2 = n 2 − 2. Proof. By the discussion above, it is enough to show that the reduced Euler characteristic of ∆( P F n ) is 0.
Let P F * n denote the dual of P F n . By abuse of notation we use0 for the smallest element of P F * n although it is1 of P F n . Similarly, we denote the largest element of P F * n by1. Now, since µ P Fn ([0,1]) = µ P F * n ([0,1]), we are going to show that the later value is 0.
For simplicity, let us denote µ P F * n by µ, and denote the length function ℓ P F * n by ℓ. We prove by induction that µ([0, z]) = 0 for all z with ℓ(z) > 1. Our base case is when ℓ(z) = 2. In this case, [0, z] is a chain of length 2 by the discussion in the previous paragraph, and hence, the corresponding value is 0. Now assume that µ([0, z]) = 0 for all z with 2 ≤ ℓ(z) ≤ s, and let z ′ ∈ P F * n be an element with ℓ(z ′ ) = s + 1. Since the proof is complete.

Length-generating functions
Recall that the standard form of an involution π ∈ I n is a product of transpositions of the form where for all 1 t m, i t < j t and i 1 < i 2 < · · · < i m . We call the transpositions appearing in (12) as arcs. Using bijection (4) from the Introduction section, we identify the elements of P F n as involutions in S n . With this identification, let us denote by I(n, k) the set of involutions of S n having k arcs, and define its length generating function by i q (n, k) := π∈I(n,k) q ℓ P Fn (π) .
Recall also that the q-analog of a natural number n ∈ N is the polynomial [n] q = 1 + q + · · · + q n−1 .
On the other hand, since the ranks of π and σ differ by 2, and their sizes differ by 2, by the formula (10), we see that See Example 6.2 for an illustration. Now, in the light of these observations, we derive the desired recurrence: = q n i q (n, k) + (1 + q + q 2 + · · · + q n−1 )i q (n − 1, k − 1) = q n i q (n, k) + [n] q i q (n − 1, k − 1).
6.1 An explicit formula for i q (n, k).
Let π ∈ P F n be a partial involution and let π = (i 1 , j 1 ) (i 2 , j 2 ) · · · (i m , j m ) denote its standard form viewed as an involution in I n via bijection (4). It follows from the proof of Proposition 6.2 of [7] that the following equality is true: where inv(π) is the "modified inversion number," which is equal to the number of inversions in the word i 1 j 1 i 2 j 2 · · · i m j m .
6.2 Length generating function of P F n Next, we look at the length generating function of P F n more closely.

Skew-symmetric matrices over F q
There is an interesting similarity between the rank generating function i q (n, k) and the number of F q -rational points of rank 2k, n × n skew symmetric matrices, which we denote by Skew 2k n . Here F q is the finite field with q elements. It is well known that the number of F q -rational points of the general linear group GL n and the symplectic group Sp n (n = 2m) are given by The group G = GL n acts Skew 2k n transitively. A simple matrix computation shows that |G x | Fq = |GL n−2k | Fq |Sp 2k | Fq |Mat n−2k,2k | Fq , where Mat n−2k,2k is the space of 2k × (n − 2k) matrices. Thus, , which simplifies as follows In other words, |Skew 2k n | Fq = i q (n, k)q 2( k 2 )−( n−2k 2 ) (q − 1) k .