A Geometric Interpretation of the Intertwining Number

We exhibit a connection between two statistics on set partitions, the intertwining number and the depth-index. In particular, results link the intertwining number to the algebraic geometry of Borel orbits. Furthermore, by studying the generating polynomials of our statistics, we determine the $q=-1$ specialization of a $q$-analogue of the Bell numbers. Finally, by using Renner's $H$-polynomial of an algebraic monoid, we introduce and study a $t$-analog of $q$-Stirling numbers.


Introduction
This paper is concerned with the intertwining number of a set partition, which is a combinatorial statistic introduced by Ehrenborg and Readdy in [10]. This statistic is among the combinatorial parameters on set partitions whose generating function is an important q-analog of the Stirling numbers of the second kind: S q (n, k) = q k−1 S q (n − 1, k − 1) + [k] q S q (n − 1, k) if n, k ≥ 1; δ n,k if n = 0 or k = 0.
(1.1) nilpotent elements in B n . Then the following simple bijection between B nil n and Π n makes this order into an order of Π n : the matrix corresponding to the set partition A has an entry equal to 1 in row i and and column j if and only if (i, j) is an arc of A.
A purely combinatorial way to describe the above partial order on set partitions was recently introduced in [3], where the rank function of the poset was given by a certain combinatorial statistic (on arc-diagrams), called the depth-index of A and was denoted by t(A). Now we are ready to outline the structure of this article and to mention our results which connect the depth-index to the other statistics.
First of all, in Section 2, we relate the intertwining number to the depth-index; this is our Theorem 1. One the one hand, the depth-index t(A) is the rank function of the Bruhat-Chevalley-Renner order on the set partitions. On the other hand, the poset of doubled Borel orbits ordered by the containment relations on closures is ranked by the dimension function. In other words, the depth-index t(A) gives the dimension of B n τ B n where τ is the upper-triangular partial permutation matrix which corresponds to the set partition A. Thus, it follows from Theorem 1 that the intertwining number i(A) is the rank function of the dual poset, i(A) = codim (B n τ B n ). Notice that from a computational point of view the intertwining number is simpler than the depth index. In Section 3, we give another combinatorial interpretation of the depth-index and of the intertwining number using socalled rank-control matrices. In Section 4, we apply Theorem 1 to compute q-Bell numbers corresponding to the depth-index when q = −1. In Section 5, we use Renner's H-polynomial of an algebraic monoid to introduce and study a new t-analog of q-Stirling numbers.

The intertwining number and the depth-index
From now on, we identify a set partition with its arc-diagram. The blocks of a set partitions are chains in its arc-diagram. We will frequently use the following well-known and very important, albeit rather obvious fact: If A is an arc-diagram on n vertices with k arcs, then the number of chains of A is n − k, while singletons are also considered as chains. In this regard, we will denote by Π n,k (k = 1, . . . , n) the set of set partitions of {1, . . . , n} with k blocks, or equivalently, the set of arc-diagrams on n vertices with k chains.
In this section we exhibit the relationship between Ehrenborg and Readdy's intertwining number, introduced in [10], and the depth index, introduced in [3]. Definition 2.1. Let A be a set partition from Π n,k . Then the intertwining number 2 i(A) of A is the total number of crossings in the extended arc diagram of A. More formally, for a pair of disjoint sets B and C of integers, the intertwining number is the cardinality of the set and the intertwining number of a set partition A is the sum of intertwining numbers of all pairs of blocks of A.

Let us denote by Arcs(A) the set of arcs of
where depth(v), which is called the depth of a vertex v, is the number of arcs (i, j) ∈ Arcs(A) with 1 ≤ i < v < j ≤ n, and depth(α), which is called the depth of an arc α = (u, v), is the number of arcs (i, j) ∈ Arcs(A) with 1 ≤ i < u < v < j ≤ n.
The intertwining number is best understood by visualizing it on the extended arc diagram of the set partition. An example is depicted in Figure 2  Moreover, the sum α∈Arcs(A) depth(α) is just the number of nestings 5 of A. Note that the number of nestings is equidistributed with the number of crossings 6 . This statistic and the dimension exponent both occur in the theory of supercharacters. More precisely, for any supercharacter indexed by a set partition A, the dimension is given by the dimension exponent of A, and the scalar product with a supercharacter indexed by the same set partition equals the number of crossings of A, see [1].
Our main result is the following: Theorem 1. For any set partition A ∈ Π n , we have For the proof it will be convenient to refine the intertwining number and the depth index as follows. The partner u of an element is 0 if v is the minimal element of its block, otherwise it is the largest element in the same block smaller than v, that is, (u, v) is an arc in the arc diagram.
The partial intertwining number, denoted by i v (A), is the number of crossings of the arc (or half-arc) ending in v with arcs or half-arcs whose smaller vertex i is between u and v.
The partial depth index, denoted by t v (A), is the sum of the number of (proper) arcs (i, j) with u < i < j < v and the number u, which is the partner of v.
In Figure 2.1, the partial intertwining numbers are written on the second line, below the elements of the set partition. Since every crossing is counted precisely once, the sum of these numbers is the intertwining number of A. The partial depth indices are written below, on the third line.
It is clear that the sum of partial intertwining numbers is the intertwining number of the set partition. The corresponding statement for the depth index is also true: Lemma 2.5. The sum of the partial depth indices of a set partition is equal to its depth index.
Before we prove this lemma, let us note a second useful fact. Clearly, Theorem 1 follows at once from Lemma 2.5 and Lemma 2.6.
Proof of 2.6. Let u be the partner of v. Then any arc (i, j) with u < i < v either satisfies u < i < j < v, and thus contributes +1 to the partial depth index, or it contributes precisely one crossing to i v (A).
Proof of 2.5. Let A be a set partition of {1, . . . , n} and let v ∈ {1, . . . , n}. Let A ′ be the set partition obtained from A by removing the last vertex with label n.
We will now use induction, so, we proceed with the assumption that t( If n is a singleton block of A, then t(A) is obtained from t(A ′ ) by adding the number of arcs of A. Otherwise, we assume that (m, n) is an arc in A, and there are ν arcs (i, j) in A with m < i < j < n. Then since the new arc (m, n) contributes n − 1 to the first sum in the definition of the depth index, and the new arc increases the depth of each of the vertices between m and n by 1.
In both of these cases, t(A) = t(A ′ ) + m + ν = t(A ′ ) + t n (A), hence the proof is finished.

The intertwining number and the rank control matrix
We start with setting up our notation.
Definition 3.1. For an n × n matrix X, let X k,ℓ denote the lower-left k × ℓ submatrix of X. Then the rank control matrix R(X) = (r k,ℓ ) n k,ℓ=1 is the n × n matrix with entries defined by r k,ℓ := rank (X k,ℓ ).
As far as we know, the rank control matrix R(X) was introduced by Melnikov in [12]. A closely related version is used in [16] and in [13] for describing the Bruhat-Chevalley order on symmetric groups. Incitti [11] used it in his study of the Bruhat order on involutions. After Incitti's work, the rank control matrix is used in [2], [9], [5], [4] for studying Bruhat orders on partial involutions and partial fixed-point-free involutions. See [6] for related work on the rook monoid.
Next, we introduce the "inequalities statistic" of an arc-diagram.
is an arc in A; 0 otherwise.
In this notation, the inequalities statistic of A, denoted by D(A), is defined by Here, r i,j 's are the entries of rank control matrix R(M(A)) = (r i,j ) n i,j=1 . In other words, M(A) is the adjacency matrix of A, regarded as the directed graph with edges directed towards the vertices with bigger labels. The statistic D(A) is the total number of inequalities along antidiagonals from south-west to north-east of the rank control matrix R(M(A)). Proposition 1. Let A be a set partition of {1, . . . , n} and let (r i,j ) n i,j=1 denote the rank control matrix of M(A). In this case, the entry r i,j (i, j ∈ {1, . . . , n}) is equal to the number of arcs (k, l) in A such that i ≤ k < l ≤ j.
The proof of Proposition 1 follows from the definition of rank control matrix of M(A), so we omit writing it. Nevertheless, we give an example that explains it. This is not a coincidence as we will show in our next result.
Theorem 2. Let A be an arc-diagram from Π n . By viewing A as a directed graph we let R = (r i,j ) n i,j=1 denote the rank-control matrix of the adjacency matrix of A. In this case, the following relations hold true: We divide the proof into two propositions. Proof. For a fixed j with 2 ≤ j ≤ n, we set D j (A) = |{i | 2 ≤ i ≤ n, and r i,j = r i+1,j−1 }|. We will prove that t j (A) = D j (A) for every j, 2 ≤ j ≤ n, where t j (A) is the partial depth index defined as in Definition 2.4.
If the vertex j is not the endpoint of any arc, hence, the partner of j is 0, or, equivalently, If the j-th column of M(A) has a 1 at the position (w, j), then (w, j) is an arc in A for some w (w < j) and furthermore there are w inequalities of the form r i,j > r i+1,j−1 for each i such that 1 ≤ i ≤ w. Also, it is easily seen that each arc (u, v), where w < u < v < j, contributes an inequality of the form r u,j > r u+1,j−1 . Therefore, D j (A) = w + |{(u, v) | w < u < v < j , m u,v = 1}|. Since w is the partner of the vertex j, we see that t j (A) = w + |{(u, v) | w < u < v < j , (u, v) is an arc in A}|. Therefore, in this case we have t j (A) = D j (A) also. Now, since by Lemma 2.5, t(A) = n j=2 t j (A), and D(A) = n j=2 D j (A), the proof is finished.
By combining Theorem 1 and Proposition 2 we are able to express the intertwining number in terms of the number of equalities in anti-diagonals of the rank control matrix. To this end, if A is an arc-diagram from Π n , then let us denote by E(A) the following statistic: where, as before, r i,j 's are the entries of the rank control matrix R(M(A)). Proof of Theorem 2. This is a combination of Propositions 2 and 3.
Before proceeding to the next section, we briefly discuss the algebraic geometric significance of the equalities t

(A) = D(A) and i(A) = E(A).
We already mentioned that the doubled Borel group B n × B n acts on matrices via and the action (3.4) restricts to give an action on B n . Proposition 4. Let X and Y be two matrices from Mat n such that Y = BXC, where B and C are from B n . If X kℓ and Y kℓ denotes the lower-left k × ℓ submatrices of X and Y , respectively, then rank (X kℓ ) = rank (Y kℓ ) for all 1 ≤ k, ℓ ≤ n.
Proof. Let us write the matrix BXC in block form: and therefore, Y kℓ = B ′ X kℓ C ′ . Matrices B ′ and C ′ are invertible (upper-triangular k × k and ℓ × ℓ submatrices of B and C respectively), which implies that Y kℓ and X kℓ have equal ranks.
Proposition 4 shows that the rank control matrix R is an invariant of a B n × B n -orbit. Let N n denote the n 2 dimensional affine space of upper triangular n × n matrices which are nilpotent. If Z is from N n , then the closure B n ZB n in B n is an affine algebraic subvariety of N n . As we mentioned in the introduction, such orbit closures are parametrized by the arc-diagrams, and furthermore, the Bruhat-Chevalley-Renner order can be interpreted in a combinatorial way on the arc-diagrams. All of this is recorded in [3].
Remark 3.5. There is another combinatorial method to determine the relation between two elements in the Bruhat-Chevalley-Renner order. Namely, it is sufficient to compare the corresponding rank control matrices componentwise. More precisely, for two k × ℓ matrices X = (x i,j ), Y = (y i,j ) let us define X Y if x i,j ≤ y i,j for all i, j, 1 ≤ i ≤ k, 1 ≤ j ≤ ℓ. It can be shown (see [2]) that B n Z 1 B n ⊆ B n Z 2 B n if and only if R(Z 1 ) R(Z 2 ). Now, if A and B are two arc-diagrams, then we write that A B if the corresponding nilpotent partial permutation matrices σ A and σ B satisfy σ A σ B . In this notation, A B if and only if R(M(A)) R(M(B)).
Recall that the depth index statistic gives us the dimensions of the B n × B n -orbits, therefore, by Theorem 1, the intertwining number of an arc-diagram gives the codimension of the corresponding orbit closure in N n . Note also that the dimension of the variety B n ZB n equals dim N n minus the number of algebraically independent polynomial relations that defines the affine variety B n ZB n . Therefore, by using Proposition 3, we obtain the following result.
Proposition 5. Let B n ZB n (Z ∈ N n ) be a doubled Borel subgroup orbit. Then the parameter E(R(Z)) which counts the number of equalities in the anti-diagonals in the upper triangle of R(Z) actually counts the number of algebraically independent polynomial equations that define the variety B n ZB n .
There is a proof of Proposition 5 that does not use Theorem 1. However, this direct proof is somewhat long and it is very similar to the proof of Theorem 7.6 of [2], so, we omit it.

q = −1 specialization
The next result of our article builds on the connections between various set partition statistics that we established so far. The n-th Bell number, denoted by B n is the sum B n := k S(n, k). It follows from Theorem 1 that the following polynomial is a q-analog of the Bell numbers: Theorem 3. Let X n (q) denote the rank generating series of the depth-index statistic as in (4.1). Then if n ≡ 0 or 1 or 3 or 10 (mod 12); −1 if n ≡ 4 or 6 or 7 or 9 (mod 12); 0 if n ≡ 2 (mod 3).
As we hinted at in the introduction, the intertwining number on Π n,k has the q-Stirling numbers as its generating series, Another statistic with the same generating function is introduced by Milne in [14].
. . |A k be a set partition from Π n,k . The dual major index 7 of A is defined by The nomenclature is due to Sagan [19]. Although it is not referred to as the 'dual major index' in [21], maj is the second of the three statistics that are studied by Wagner. In [20], Steingrímsson refers to maj as "LOS", the left opener statistic.
A bijection φ : Π n,k → Π n,k satisfying maj(φ(A)) = i(A), and thus proving the equidistribution of the intertwining number and the dual major index was found by Parviainen in [15]. It is defined as follows. As before, let i v (A) be the number of crossings of the arc (or line) ending in v with arcs or lines whose smaller vertex i is between the partner of v and v. Then the blocks of the set partition φ(A) are the sets of elements sharing the same number i v (A). For example, the image of the set partition A = 18|2569|37|4 from Figure 2.1 is φ(A) = 16|2|3579|48 and the dual major index of φ(A) is given by maj(φ(A)) = 0 · 2 + 1 · 1 + 2 · 4 + 3 · 2 = 15.
We are now ready to prove our third result.
Proof of Theorem 3. Let X n (q) denote the generating function of the depth index t and Y n (q) the generating function of the intertwining number i, By Theorem 1, we know that Following Wagner, we denote the generating function of maj by B n (q), Then Y n (q) = B n (q) .
Wagner in [21] proved that Notice that n 2 = n(n−1) 2 is even when n ≡ 0 or 1 (mod 4), and it is odd when n ≡ 2 or 3 (mod 4). Now, we have Obviously, n ≡ 2 (mod 3) if and only if n ≡ 2 or 5 or 8 or 11 ( mod 12). This finishes the proof.
In this section, by relating the depth index and the intertwining number to the numbers of rational points of B n−1 over finite fields, we prove the following result.
where a(σ) is the dimension of the unipotent part of the orbit BσB and b(σ) is the dimension of the diagonalizable part of BσB.
One can think of the H-polynomial of R as a transformed "Hasse-Weil motivic zeta function" of the projectivization P(M − {0}) of M. Indeed, by treating q as a power of a prime number, for all sufficiently large q, (H(R, q) − 1)/(q − 1) is equal to the number of rational points over F q (the finite field with q elements) of P(M − {0}). See [18,Remark 3.2]. For an application of this idea to the rook theory, we recommend [7].
We will consider the simplest example, where M is Mat n , the monoid of all n×n matrices. Clearly, the projectivization of Mat n is the n 2 − 1 dimensional projective space, hence its number of F q -rational points is given by (q n 2 − 1)/(q − 1). The Renner monoid of Mat n is the rook monoid R n . Furthermore, if σ ∈ R n , then a(σ) = ℓ(σ) − rank(σ), and b(σ) = rank(σ), where rank(σ) is the rank of σ as a matrix. Therefore, the H-polynomial of R n becomes q n 2 = H(R n , q) = σ∈Rn q ℓ(σ)−rank(σ) (q − 1) rank(σ) . (5.1) An important consequence of the formula in (5.1) is that if X denotes any B n × B n -stable smooth and irreducible subvariety of Mat n , then H(X, q) gives the number of F q -rational points of X. In particular, we apply this observation to N n , the semigroup of nilpotent elements in B n . Note that, there is a natural algebraic variety isomorphism between N n and B n−1 . The isomorphism is given by the deleting of the first column and the last row of the elements of N n . Moreover, under this isomorphism, B n × B n -orbits in N n are isomorphically mapped onto B n−1 × B n−1 -orbits in B n−1 . Therefore, there is no loss of information to work with B n−1 instead of N n . Now, on one hand we have the number of points of B n−1 over F q , |B n−1 | = q ( n 2 ) . where k ∈ {1, . . . , n}.