The Complete Cd-index of Boolean Lattices

Let [u, v] be a Bruhat interval of a Coxeter group such that the Bruhat graph BG(u, v) of [u, v] is isomorphic to a Boolean lattice. In this paper, we provide a combinatorial explanation for the coefficients of the complete cd-index of [u, v]. Since in this case the complete cd-index and the cd-index of [u, v] coincide, we also obtain a new combinatorial interpretation for the coefficients of the cd-index of Boolean lattices. To this end, we label an edge in BG(u, v) by a pair of nonnegative integers and show that there is a one-to-one correspondence between such sequences of nonnegative integer pairs and Bruhat paths in BG(u, v). Based on this labeling, we construct a flip F on the set of Bruhat paths in BG(u, v), which is an involution that changes the ascent-descent sequence of a path. Then we show that the flip F is compatible with any given reflection order and also satisfies the flip condition for any cd-monomial M. Thus by results of Karu, the coefficient of M enumerates certain Bruhat paths in BG(u, v), and so can be interpreted as the number of certain sequences of nonnegative integer pairs. Moreover, we give two applications of the flip F. We enumerate the number of cd-monomials in the complete cd-index of [u, v] in terms of Entringer numbers, which are refined enumerations of Euler numbers. We also give a refined enumeration of the coefficient of d n in terms of Poupard numbers, and so obtain new combinatorial interpretations for Poupard numbers and reduced tangent numbers.


Introduction
Let W be a Coxeter group and u, v ∈ W such that u < v in the Bruhat order.The complete cd-index ψ u,v (c, d) of the interval [u, v] is a nonhomogeneous polynomial in the noncommuting variables c and d, which was introduced by Billera and Brenti [2] and conjectured to have nonnegative coefficients, see also Billera [1].The complete cdindex ψ u,v (c, d) encodes compactly the ascent-descent sequences of the Bruhat paths in the Bruhat graph BG(u, v) of [u, v], and its combinatorial invariance is equivalent to the combinatorial invariance of the celebrated Kazhdan-Lusztig and R-polynomials, see [2,12].
In [2], Billera and Brenti also showed that ψ u,v (c, d) is a generalization of the cd-index ψ u,v (c, d) of [u, v] in the sense that ψ u,v (c, d) is the sum of the highest degree terms of ψ u,v (c, d).For the definition of the cd-index of an Eulerian poset, see, e.g., Stanley [16].Purtill [15] gave a combinatorial interpretation for the cd-index ψ Bn (c, d) of the Boolean lattice B n by showing that ψ Bn (c, d) is the sum of the cd-variation monomials of augmented André permutations on [n] := {1, 2, . . ., n}, and then derived a recursive formula for ψ Bn (c, d).Besides, ψ Bn (c, d) is also a refined enumeration of simsun permutations, which were first introduced by Simion and Sundaram [17,18], see also Hetyei [9].
In this paper, we give a combinatorial interpretation for the coefficients of the complete cd-index of [u, v], where BG(u, v) is isomorphic to the Boolean lattice B n .Since in this case all the edges in BG(u, v) are covering relations, ψ u,v (c, d) has no lower degree terms.Hence we have Thus we also obtain a new combinatorial interpretation for the coefficients of the cd-index of the Boolean lattice B n .
To this end, we label a directed edge in BG(u, v), say x → y, by a pair of nonnegative integers (i, j), where i (resp.j) is the number of edges y → z (y < z v) such that the reflection y −1 z is larger (resp.smaller) than the reflection x −1 y in a given reflection order O. Then we show that there is a one-to-one correspondence between the Bruhat paths in BG(u, v) and the sequences of nonnegative integer pairs ((i 1 , j 1 ), (i 2 , j 2 ), . . ., (i n , j n )), such that i k + j k = n − k for 1 k n.
Based on this labeling, we construct a flip F on the set of Bruhat paths in BG(u, v), which is an involution that changes the ascent-descent sequence of a path.We show that the flip F is compatible with the reflection order O and also satisfies the flip condition for any cd-monomial M defined by Karu [11].Then the coefficient of M enumerates certain Bruhat paths in BG(u, v).Such paths are called valid paths and their corresponding sequences are called valid sequences.Therefore the coefficient of M is the number of certain valid paths in BG(u, v) or certain valid sequences of nonnegative integer pairs.
We give two applications of the flip F.
Purtill [15] showed that the number of cd-monomials in ψ Bn (c, d) is the Euler number E n , that is, We give a refined enumeration of ψ u,v (1,1) in terms of the Entringer numbers E n (k).To be more specific, we show that the number of valid sequences of length n beginning with (n − k, k − 1) or the number of valid paths in BG(u, v) with first edge labeled by As the second application, we give a refined enumeration of the coefficient of d n in ψ u,v (c, d) in terms of the Poupard numbers P n (k) (1 k 2n + 1), where BG(u, v) is isomorphic to B 2n+1 .The paths in BG(u, v) corresponding to the monomial d n are called alternating paths.We show that the number of alternating paths with first edge labeled by (2n − k + 1, k − 1) is equal to the Poupard number P n (k).Since by [8], the reduced tangent number t n satisfies t n = 2n+1 k=1 P n (k), we deduce that the coefficient of d n is the reduced tangent number t n .Therefore we obtain new combinatorial interpretations for the Poupard numbers and reduced tangent numbers.
The organization of this paper is as follows.In Section 2, we give some basic notation and definitions on Coxeter groups and the complete cd-index.We also recall some results of Karu.In Section 3, we first construct a flip F on [u, v] and show that this flip is compatible with the given reflection order and satisfies the flip condition.Then we provide a combinatorial interpretation for the coefficient of ψ u,v (c, d).In Section 4, We give a refined enumeration of the number of cd-monomials in ψ u,v (c, d).Finally, in Section 5, we give a refined enumeration of the coefficient of d n in ψ u,v (c, d).

Preliminary
Let (W, S) be a Coxeter system, and let T = {wsw −1 | s ∈ S, w ∈ W } be the set of reflections, see, e.g., Humphreys [10].We use (w) to denote the length of w ∈ W .For u, v ∈ W , we say that u v in the Bruhat order if there exists a sequence of reflections w v} be the interval formed by u and v in the Bruhat order.The atoms of [u, v] are the elements w ∈ [u, v] such that w covers u.
The Bruhat graph BG(W ) of the Coxeter group W is a directed graph whose vertices are the elements of W and there is a directed edge from u to v, denoted by u → v, if v = ut for some reflection t ∈ T and (u) such that t i = x −1 i−1 x i ∈ T for 1 i n.We call (t 1 , t 2 , . . ., t n ) the reflection sequence of x, and call the first edge u → x 1 of x a first edge of the interval [u, v].Let B k (u, v) denote the set of Bruhat paths of length k from u to v, and let the electronic journal of combinatorics 22(2) (2015), #P2.45 Recall that a reflection order (O, ≺) is a total order defined on the set of reflections, see Dyer [4].The reverse of the order O, denoted by O, is also a reflection order.Throughout this paper, we shall always use a given reflection order (O, ≺).We say that the path x in (1) is increasing if Dyer [4] showed that each Bruhat interval [u, v] is shellable.That is, there is a unique increasing (resp.decreasing) path of length n, say x (resp.y), and the reflection sequence of x (resp.y) is the lexicographically smallest (resp.largest) among all the reflection sequences of paths in B n (u, v).
The following result is due to Dyer [5].
Then we have The ascent-descent sequence of the Bruhat path x is a monomial in the noncommuting variables a and b defined by where  [2] showed that φ u,v (a, b) can be expressed in terms of c and d: It can be shown that ψ u,v (c, d) does not depend on the reflection order.Now we proceed to recall some definitions and results in [11].
For an ab-monomial M , denote by M the ab-monomial obtained by exchanging a and b in M .This operator is an involution on the noncommutative ring Z a, b .
the electronic journal of combinatorics 22(2) (2015), #P2.45 Fix a flip for every interval in the Bruhat graph of W .Let 1 m n and be a path in B(u, v).After applying the flip F xm,v to x, we obtain Let the variables a, b, c have degree 1, and let the variable d have degree 2. Given a cd-monomial M (c, d) of degree n, we can obtain a unique ab-monomial M (a, ba) of degree n by substituting a for c and ba for d in M (c, d).Clearly, this is a one-to-one correspondence between cd-monomials and ab-monomials in which every b is followed by an a.
Note that s m,γm (x), and hence s M (x), depend on both the reflection order and the given flip.Denote by s m,γm (x) the value of s m,γm (x) by using the reverse reflection order O, and let s M (x) = n m=1 s m,γm (x).Definition 4. A flip F is said to be compatible with the reflection order O if for any interval [u, v], any path x ∈ B(u, v) and any cd-monomial M .
Theorem 5. Assume that the flip F is compatible with the reflection order O.For any cd-monomial M of degree n, the coefficient of the electronic journal of combinatorics 22(2) (2015), #P2.45 If −1 does not appear in the above sum, then the coefficient of M is clearly nonnegative.Therefore Karu [11] introduced the following flip condition.Definition 6.The flip condition holds for the interval [u, v] and monomial M if for every path x ∈ B(u, v) the following is satisfied.If s m,γm (x) = −1 for some m, then there exists k > m such that s k,γ k (x) = 0.
From Theorem 5 we have In [11], Karu proved that when M contains at most one d, that is, M = c i or M = c i dc j (i, j 0), the flip condition holds by Theorem 1. Then the coefficient of M is nonnegative by Corollary 8. Recently, the authors showed that when M = dc i dc j (i, j 0), the coefficient of M is also nonnegative, see [7].

Combinatorial interpretation of ψ u,v (c, d)
In this section, we first give a labeling scheme for the edges in BG(u, v) and then construct a flip F on the set of paths in B(u, v) based on the labels of the paths.By using the flip F, we provide a combinatorial interpretation for the coefficients of ψ u,v (c, d).
Let [u, v] be a Bruhat interval such that BG(u, v) is isomorphic to the Boolean lattice B n .In the following, we shall always refer to [u, v] as such an interval if there is no further notification.Note that every edge in BG(u, v) is a covering relation.In fact, if there is an edge u 1 → u 2 in BG(u, v) such that u 2 = u 1 t for some t ∈ T and (u 2 ) − (u 1 ) > 1, then we must have (u 2 ) − (u 1 ) = 2k + 1 for some k 1.This implies BG(u 1 , u 2 ) would not be isomorphic to a Boolean lattice.Hence B(u, v) = B n (u, v), i.e., all the paths from u to v have length n.
Suppose that Moreover, the label of u → u r is lexicographically smaller than the label of u → u k if and only if u −1 u r u −1 u k .
Proof.We make induction on n.By Theorem 1, it is easy to see that the proposition holds for n = 2. Now assume that n > 2. Since [u, v] is shellable, there is a unique increasing (resp.decreasing) path x (resp.y), and the reflection sequence of x (resp.y) is the lexicographically smallest (resp.largest).Without loss of generality, we can assume Therefore, the label of the edge u → u 1 is (n − 1, 0) and the label of the edge u → u n is (0, n − 1).
Notice that u and the atoms u 2 , u 3 , . . ., u n−1 determine a Boolean lattice B n−2 .By induction, the set of labels of the first edges u → u k (2 k n − 1) in the Boolean lattice That is to say, if the edge u → u k has label (s, r) in B n−2 , then its label would be (s + 1, r + 1) in B n .Therefore, in the Boolean lattice B n , the set of labels of the first edges is It is clear that u −1 u 1 (resp.u −1 u n ) is the minimum (resp.maximum) among all the first edges of [u, v] in the reflection order O, and the label of u → u 1 (resp.u → u n ) is the largest (resp.smallest) in the lexicographic order.By induction, for 2 k, r n − 1, the label of u → u k is lexicographically smaller than the label of u → u r if and only Consequently, for 1 k, r n, the label of u → u k is lexicographically smaller than the label of u → u r if and only if u −1 u k u −1 u r in B n .This completes the proof.
Remark 10.According to Proposition 9, if we arrange the labels of the first edges u → u 1 , . . ., u → u n of [u, v] decreasingly in the lexicographic order, then the corresponding reflections u −1 u 1 , . . ., u −1 u n are arranged increasingly in the reflection order.Thus, without loss of generality, we can require the edge u → u k to have label (n − k, k − 1) for 1 k n.
Corollary 11.There is a bijection between the Bruhat paths in B(u, v) and sequences of nonnegative integer pairs ((i 1 , j 1 ), . . ., (i n , j n )) such that i k + j k = n − k for 1 k n.In other words, a label sequence determines a unique path in B(u, v) and vice versa.Proposition 12. Let u k−1 → u k → u k+1 be two adjacent edges in B(u, v) such that the edge u k−1 → u k has label (i k , j k ) and the edge u k → u k+1 has label (i k+1 , j k+1 ).Then (1) (1) Since there are i k edges among the first edges of [u k , v] which are larger than u −1 k−1 u k , by Remark 10, we see that the labels of these i k edges are (i, j) such that 0 . Now we can define a flip on B(u, v) according to the labels of the edges.
By Proposition 12, it is easy to see that F is a flip on B(u, v).For example, let [u, v] be an interval such that BG(u, v) is isomorphic to B 4 , see Figure 1.By Remark 10, we can label the first edges u → u 1 , u → u 2 , u → u 3 , u → u 4 of [u, v] by (3, 0), (2, 1), (1, 2), (0, 3) respectively.The first edges u are labeled by (2, 0), (1, 1), (0, 2) respectively.The images of the flip F on some paths in B(u, v) are listed below.Proof.We first show that the flip F satisfies the flip condition for any cd-monomial M .We claim that s m,b (x) = −1 for any m, any cd-monomial M and any path x in B(u, v).Suppose to the contrary that there exists a path x in B(u, v) such that s m,b (x) = −1 for some integer m.Let Since each b follows by an a in M (a, ba), by the definitions of s m,b (x) and s m,a (x), we can assume that , let the label of the edge x k−1 → x k be (i k , j k ).Then by Proposition 12, we have i m > i m+1 > i m+2 .
It is not hard to check that the paths F(p ) and F qm,v (F(p)) have the same label sequence.Therefore, That is, if we flip the path p at p m to obtain p , then under the flip F, we shall flip F(p) at q m to obtain F(p ).  4 Refined enumeration of ψ u,v (1,1) In this section, we give a refined enumeration of the number of cd-monomials in ψ u,v (c, d) in terms of Entringer numbers.Definition 16.A path x in B(u, v) is said to be valid if there exists some cd-monomial M such that s M (x) = 1.A sequence s = ((i 1 , j 1 ), (i 2 , j 2 ), . . ., (i n , j n )) with i k , j k 0 and n) is said to be valid if it corresponds to a valid path in B(u, v).Equivalently, the sequence s is said to be valid if s satisfies: (i)  1.
It is easy to verify that E n (n) = 0 and it suffices to show that T n (k) satisfies the relation (5).We analyze when a valid path in B(u k , v) can be extended to a valid path in B(u, v). For . By Proposition 9, we can label the edges u k → u i k (1 i n − 1) and u i k → u ij k (1 j n − 2) by (n − i − 1, i − 1) and (n − j − 2, j − 1) respectively.By induction, T n−1 (t) is the number of valid paths in B(u k , v) beginning with u k → u t k for 1 t n − 1.Moreover, T n−1 (n − 1) = 0 and for 1 r n − 2, Now we extend the path x to a path x in B(u, v).Let To compute s 1,a (x) or s 1,b (x), we need to flip the path x at u k .Let , we deduce that x is a valid path in B(u, v) if and only if If (6) holds, then we have k i n − 1. Hence by induction, the number of valid paths in B(u, v) extended from the first edges u If (7) holds, then we get 1 i α and i j n − i − 1, where α = min{n − k, k − 1}.Thus by induction, the number of valid paths in B(u k , v) which begin with u k → u i k (1 i α) and can be extended to a valid path in B(u, v) is the electronic journal of combinatorics 22(2) (2015), #P2.45 Combining (8), (9) and by induction, we derive that Since α = n − k or k − 1, and it is easy to check that It follows from ( 10) and (11) that as desired.This completes the proof.
Corollary 18.The number of valid paths in B(u, v) or the number of valid sequences of length n is equal to the Euler number E n .In other words, ψ u,v (1, 1) = ψ Bn (1, 1) = E n .
Recall that the numbers t n appearing in the Taylor expansion P n (k).
Definition 19.A valid path x in B(u, v) of length 2n + 1 is said to be alternating if w(x) = baba • • • ba.The label sequence s = ((i 1 , j 1 ), (i 2 , j 2 ), . . ., (i 2n+1 , j 2n+1 )) of an alternating path is called an alternating sequence.Equivalently, the sequence s is said to be alternating if s is valid and j 2r−1 > j 2r and i 2r > i 2r+1 for 1 r n.Proof.Assume that F n (k) is the number of alternating paths in B(u, v) with first edge u → u k labeled by (2n − k + 1, k − 1) for 1 k 2n + 1.It is easy to check that F 1 (1) = 0, F 1 (2) = 1, F 1 (3) = 0. We claim that for 1 k n + 1, and for n + 1 < k 2n + 1, We prove (13) first.Let be a path in B(u, v).By Proposition 9, we can assume that the edges u → u k , u k → u i k and u i k → u ij k are labeled by (2n − k + 1, k − 1), (2n − i, i − 1) and (2n − j − 1, j − 1) respectively.Since the path x is valid and w(x) = baba • • • ba, by Theorem 15, we see that Suppose that x = (u i k → u ij k → • • • → v) is an alternating path in B(u i k , v).By induction, the number of alternating paths in B(u i k , v) with first edge u i k → u ij k is F n−1 (j).If 1 k n + 1, then we have 2n − k + 1 k − 1.By (15), we see that i k − 1.By (16), we have i j 2n − i.That is to say, only the paths beginning with u k → u i k (1 i k − 1) and u i k → u ij k (i j 2n − i) will contribute to F n (k).Therefore, the equation ( 13) holds.
the electronic journal of combinatorics 22(2) (2015), #P2.45 is the polynomial obtained by summing the ascent-descent sequences of all the Bruhat paths from u to v:φ u,v (a, b) = x∈B(u,v) w(x).The complete cd-index ψ u,v (c, d) of the interval [u, v] is obtained by a change of variable in the ab-index φ u,v (a, b) of [u, v].Let c = a + b and d = ab + ba.Billera and Brenti

Corollary 8 .
If the flip condition holds for the interval [u, v] and monomial M , then the coefficient of M in ψ u,v (c, d) is equal to |T M (u, v)| and hence is nonnegative.
and so s m,b (x) = 0 again, which is a contradiction.This means s m,b (x) = −1 for any m.Thus the flip F satisfies the flip condition.Now we proceed to show that the flip F is compatible with the reflection order O. Let M (c, d) be a cd-monomial with M (a, ba) = γ 1 • • • γ n−1 .It suffices to show that for any integer m ∈ [n − 1] and any path p in B(u, v), we have s m,γm (p) = s m,γm (F(p)).

) Theorem 17 .
Suppose that the atoms of [u, v] are u 1 , u 2 , . . ., u n , and the edge u → u k has label (n − k, k − 1) for 1 k n.Then the number of valid paths in B(u, v) with first edge u → u k is the Entringer number E n (k).In other words, the number of valid sequences beginning with (n − k, k − 1) is E n (k).Proof.Let T n (k) denote the number of valid paths in B(u, v) with first edge u → u k labeled by (n − k, k − 1) for 1 k n.It is easy to check that T 2 (1) = 1, T 2 (2) = 0. Then the electronic journal of combinatorics 22(2) (2015), #P2.45

Theorem 20 .
Let [u, v]  be a Bruhat interval such that BG(u, v) is isomorphic to the Boolean lattice B 2n+1 .Suppose that the atoms of [u, v] are u 1 , u 2 , . . ., u 2n+1 , and the edgeu → u k (1 k 2n+1) has label (2n−k +1, k −1).Then the number of alternating paths in B(u, v) beginning with the edge u → u k is the Poupard number P n (k).In other words, the number of alternating sequences of length 2n + 1 beginning with (2n− k + 1, k − 1) is P n (k).the electronic journal of combinatorics 22(2) (2015), #P2.45 Let E n denote the Euler number, i.e., the number of up-down (or alternating) permutations on [n].Denote by E n (k) the number of up-down permutations of length n beginning with k (1 k n).Clearly, we have the electronic journal of combinatorics 22(2) (2015), #P2.45
[6] we enumerate the valid paths beginning with the same first edge.Let E n be the Euler number, i.e., the number of up-down permutations on[n].It is well known that tan u + sec u = It is clear that E n = n k=1 E n (k).The numbers E n (k) are called Euler and Bernoulli numbers, or Entringer numbers, see[6].The Entringer numbers E n (k) for n, k 7 are displayed in Table are called the reduced tangent numbers.It is easy to see that t n = E 2n+1 /2 n , where E 2n+1 are the Euler numbers.By [8, Theorem 1.1], we have