Title: Equidistribution of negative statistics and quotients of Coxeter groups of type B and D

Transcription

1 Title: Euidistribution of negative statistics and uotients of Coxeter groups of type B and D Author: Riccardo Biagioli Address: Université de Lyon, Université Lyon 1 Institut Camille Jordan - UMR 5208 du CNRS 43, boulevard du 11 novembre 1918 F Villeurbanne Cedex biagioli@mathuniv lyon1fr Phone: +33 (0) Fax: +33 (0)

2 Abstract: We generalize some identities and -identities previously known for the symmetric group to Coxeter groups of type B and D The extended results include theorems of Foata and Schützenberger, Gessel, and Roselle on various distributions of statistics, like inversion number, major index, and descent number In order to show our results we provide explicit caracterizations of the systems of minimal coset representatives of Coxeter groups of type B and D MSC: 05A19, 05E15 Keywords: Coxeter groups, statistics, major index, descent number, symmetric group, generating functions, -series Date submission Date acceptance

3 Euidistribution of negative statistics and uotients of Coxeter groups of type B and D Riccardo Biagioli Université de Lyon, Université Lyon 1 Institut Camille Jordan - UMR 5208 du CNRS 43, boulevard du 11 novembre 1918 F Villeurbanne Cedex Abstract We generalize some identities and -identities previously known for the symmetric group to Coxeter groups of type B and D The extended results include theorems of Foata and Schützenberger, Gessel, and Roselle on various distributions of statistics, like of inversion number, major index, and descent number In order to show our results we provide explicit caracterizations of the systems of minimal coset representatives of Coxeter groups of type B and D 1 Introduction A well known theorem of MacMahon 17 shows that the length function and the major index are euidistributed over the symmetric group S n We recall that the length of a permutation σ S n is given by the number of inversions, denoted inv(σ) : {(i, j) i < j, σ(i) > σ(j)}, and the major index of σ is the sum of all its descents More precisely, maj(σ) : i, i Des(σ) where Des(σ) : {i n 1 σ(i) > σ(i + 1)} Foata gave a bijective proof of this euidistribution theorem in 9 He studied further his bijection and together with Schützenberger derived the two following results 13 The first one is a refinement of MacMahon s theorem, asserting the euidistribution of major index and number of inversions over descent classes Theorem 11 (Foata-Schützenberger) Let M {m 1,, m t } < {1,, n 1} Then maj(σ) inv(σ) {σ S n Des(σ 1 )M} {σ S n Des(σ 1 )M} Dedicated to the memory of my friend and colleague Giulio Minervini 3

4 The second one concerns the symmetry of the distribution of the major index and the inversion number over the symmetric group Theorem 12 (Foata-Schützenberger) The pairs of statistics (maj, inv) and (inv, maj) have the same distribution on S n, namely S n (t, ) : t maj(σ) inv(σ) t inv(σ) maj(σ) σ S n σ S n Theorem 11 has been extensively studied and generalized in many ways in the last three decades Nevertheless, it still receives a lot of attention as shown by two recent papers of Hivert, Novelli, and Thibon 16, and of Adin, Brenti, and Roichman 3, where a multivariate generalization and an extension to the hyperoctahedral group of it are provided In the latter paper, the problem of finding an analogue of this Foata-Schützenberger theorem for the Coxeter groups of type D is proposed 3, Problem 56 In this paper we answer this uestion Actually, we show that the negative major indices nmaj, introduced in 2 on Coxeter groups of type B, and dmaj, defined in 6 on Coxeter groups of type D, give generalizations of the first and second Foata-Schützenberger identities to B n and D n In our analysis we derive nice relations among uotients, or sets of minimal coset representatives, of B n and D n that are interesting in their own Explicit maps between these uotients are shown, and used to compute some generating functions Finally, we use our results, and the negative descent numbers, to give generalizations to B n and D n of two classical -identities The first one, due to Roselle 19 (see also Rawlings 18, (24)), is the generating function of the inversion number and major index over the symmetric group: for undefined notation see next section Theorem 13 (Roselle) u n S n (t, ) (t; t) n (; ) n n 0 1 (u; t, ),, where S 0 (t, ) 1 The second one is the trivariate distribution of inversion number, major index, and number of descents, due to Gessel 15, Theorem 84, (see also 14) Theorem 14 (Gessel) n 0 u n σ S n t maj(σ) inv(σ) p des(σ) n! (t; ) n+1 2 Preliminaries and notation k 0 p k eu etu et k u In this section we give some definitions, notation and results that will be used in the rest of this work For n N we let n : {1, 2,, n} (where 0 : ) Given n, m Z, n m, 4

5 we let n, m : {n, n + 1,, m} We let P : {1, 2, 3, } The cardinality of a set A will be denoted by A and we let ( ) n 2 : {S n S 2} Given a set A, we denote A < : {a 1, a 2, } where a 1 < a 2 < For our study we need notation for -analogs of the factorial, binomial coefficient, and multinomial coefficient These are defined by the following expressions n : n 1 ; n! : n n ; n n! : m m!n m! ; n n! : m 1, m 2,, m t m 1!m 2! m t! As usual we let (a; ) 0 : 1 (a; ) n : (1 a)(1 a) (1 a n 1 ) (a; ) : n 1(1 a n 1 ) Moreover, for r, s N we let (a; t, ) r,s : 1 if r or s are zero (1 at i 1 j 1 ) if r, s 1, 1 i r 1 j s and Finally, (a; t, ), : i 1 (1 at i 1 j 1 ) j 1 eu : n 0 u n n!, is the -analogue of the exponential function The following -binomial theorem is well known (see eg 4) Theorem 21 ( x; ) n n n m m0 (m+1 2 ) x m 21 Coxeter groups of type B and D We denote by B n the group of all bijections β of the set n, n \ {0} onto itself such that β( i) β(i) for all i n, n \ {0}, with composition as the group operation This group is usually known as the group of signed permutations on n, or as the hyperoctahedral group of rank 5

6 n If β B n then we write β β(1),, β(n) and we call this the window notation of β As set of generators for B n we take S B : {s B 1,, sb n 1, sb 0 } where for i n 1 s B i : 1,, i 1, i + 1, i, i + 2,, n and s B 0 : 1, 2,, n It is well known that (B n, S B ) is a Coxeter system of type B (see eg, 8, 81) 4 s0 s1 s2 s3 sn 2 sn 1 Figure 1: The Dynkin diagram of B n To give an explicit combinatorial description of the length function l B of B n with respect to S B, we need the following statistics For β B n we let Note that, if β B n, inv(β) : {(i, j) i < j, β(i) > β(j)}, N 1 (β) : {i n β(i) < 0}, and { ( ) N 2 (β) : n {i, j} β(i) + β(j) < 0} 2 N 1 (β) + N 2 (β) {i n β(i)<0} β(i) (1) For example if β 3, 1, 6, 2, 4, 5 B 6 then inv(β) 9, N 1 (β) 4, and N 2 (β) 14 The following characterizations of the length function, and of the right descent set of β B n are well known 8 Proposition 22 Let β B n Then where β(0) : 0 l B (β) inv(β) + N 1 (β) + N 2 (β), and Des B (β) {i 0, n 1 β(i) > β(i + 1)}, We denote by D n the subgroup of B n consisting of all the signed permutations having an even number of negative entries in their window notation, more precisely D n : {γ B n N 1 (γ) 0 (mod 2)} It is usually called the even-signed permutation group As a set of generators for D n we take S D : {s D 0, sd 1,, sd n 1 } where for i n 1 s D i : s B i and s D 0 : 2, 1, 3,, n There is a well known direct combinatorial way to compute the length, and the right descent set of γ D n, (see, eg, 8, 82) 6

7 s 0 s s s s s s n-2 n-1 Figure 2: The Dynkin diagram of D n Proposition 23 Let γ D n Then where γ(0) : γ(2) 22 Negative statistics l D (γ) inv(γ) + N 2 (γ), and Des D (γ) {i 0, n 1 γ(i) > γ(i + 1)}, In 2, Adin, Brenti and Roichman introduced the following statistics on B n For β B n let NDes(β) : Des(β) { β(i) β(i) < 0}, where denotes disjoint union, and define nmaj(β) : i, and ndes(β) : NDes(β) i NDes(β) Note that NDes(β) is a multiset It follows from (1) that nmaj(β) maj(β) + N 1 (β) + N 2 (β), and (2) ndes(β) des(β) + N 1 (β) (3) For the element β 3, 1, 6, 2, 4, 5 B 6, nmaj(β) 29, and ndes(β) 7 In 6, a notion of descent multiset for γ D n is introduced DDes(γ) : Des(γ) { γ(i) 1 γ(i) < 0} \ {0}, and the following statistics are defined dmaj(γ) : i, and ddes(γ) : DDes(γ) It easily follows that i DDes(γ) dmaj(γ) maj(γ) + N 2 (γ), and (4) ddes(γ) des(γ) + N 1 (γ) + ɛ(γ), (5) 7

8 where ɛ(γ) : { 1 if 1 γ(n) 0 if 1 γ(n) (6) For example if γ 4, 1, 3, 5, 2, 6 D 6 then dmaj(γ) 21, and ddes(γ) 5 The statistics nmaj and dmaj are usually called negative major indices; ndes and ddes negative descent numbers for B n and D n, respectively The negative major indices are Mahonian statistics, namely they are euidistributed with the length over the group, nmaj(β) lb(γ), and dmaj(γ) ld(γ) β B n β B n γ D n γ D n The pairs (ndes, nmaj) and (ddes, dmaj) give generalizations to B n and D n of a famous identity of Carlitz, see 2, Theorem 32, and 6, Theorem Quotients of Coxeter groups To show some of the next results we will need of the following decomposition that comes from the general theory of Coxeter group We refer the reader to 8 for any undefined notation Let (W, S) be a Coxeter system, for J S we let W J be the parabolic subgroup of W generated by J, and W J : {w W l(ws) > l(w) for all s J}, the set of minimal left coset representatives of W J, or the (right) uotient The uotient W J is a poset according to the Bruhat order The following is well known (see 8, 24) Proposition 24 Let (W, S) be a Coxeter system, and let J S Then: i) Every w W has a uniue factorization w w J w J such that w J W J and w J W J ii) For this factorization l(w) l(w J ) + l(w J ) As a first application of this decomposition to the groups B n (and D n ), let us consider the parabolic subgroup generated by J : S B \{s B 0 } In this case, by looking at the Dynkin diagram in Figure 1, we obtain that B J S n Moreover it is not hard to see that Hence from Proposition 24 we get B J : B J n {u B n u(1) < u(2) < < u(n)} (7) B n σ S n {uσ u B J } (8) 8

9 Note that in the case D n, for J : S D \ {s D 0 }, a similar decomposition holds, D n {uσ u D J }, σ S n where once again D J S n, and D J {u D n u(1) < u(2) < < u(n)} Remark 25 The construction or right uotient can be mirrored, by considering left descents Let J S A left uotient of W is defined by J W : {w W l(sw) > l(w) for all s J} Proposition 24 holds for left uotients too, but the factorization in i) becomes w w J Jw, with J w J W Left and right uotients are isomorphic posets, by means of the inversion map In the next section, we will work with subsets of B n and D n that are left uotients They are called descent classes for reasons that will be immediately clear 3 Combinatorial description of descent classes Let us fix a subset of descents M : {m 1, m 2,, m t } < 0, n 1 The set B(M) : {β B n Des B (β 1 ) M}, (9) is usually called a B-descent class Note that this set is nothing but a left uotient of B n More precisely, it is the one corresponding to the subset J S \ M, where M : {s i i M} The following result can be found in 3, Lemma 41 Lemma 31 Let β B n, and M {m 1,, m t } < 0, n 1 Let m t+1 : n Then Des B (β 1 ) M if and only if there exist (uniue) integers r 1,, r t satisfying m i r i m i+1 for all i, and such that β is a shuffle of the following increasing seuences: 1, 2,, m 1, r 1, r 1 + 1,, (m 1 + 1), r 1 + 1, r 1 + 2,, m 2, r t, r t + 1,, (m t + 1), r t + 1, r t + 2,, n (10) Some of these seuences may be empty, if r i m i or r i m i+1 for some i, or if m i 0 The following one is an explicit description of D-descent classes D(M) : {γ D n Des D (γ 1 ) M} (11) 9

10 Lemma 32 Let γ D n, and M {m 1,, m t } < 0, n 1 Let m t+1 : n Then Des D (γ 1 ) M if and only if there exist (uniue) integers r 1,, r t satisfying m i r i m i+1 for all i, and such that γ is a shuffle of the following increasing seuences There are three cases, and six possible blocks of seuences 1) If 0 M: (m 1 0) with t (r i m i ) 0 (mod 2) i1 2) If 0, 1 M: (note m 1 2) r 1, r 1 + 1,, 2, 1, r 1 + 1, r 1 + 2,, m 2, r t, r t + 1,, (m t + 1), r t + 1, r t + 2,, n, 1, 2,, m 1, 1, 2,, m 1 r 1, r 1 + 1,, (m 1 + 1), r 1, r 1 + 1,, (m 1 + 1) r 1 + 1, r 1 + 2,, m 2, r 1 + 1, r 1 + 2,, m 2 r t, r t + 1,, (m t + 1), r t, r t + 1,, (m t + 1) r t + 1, r t + 2,, n r t + 1, r t + 2,, n (12) (13) with t (r i m i ) 0 (mod 2); i1 t (r i m i ) 1 (mod 2) i1 3) If 0 M and 1 M: (note m 2 2, and r 1 2) 1 r 1,, 2, 1 2, 3,, m 2, 1, 2, 3,, m 2 r 1 + 1, r 1 + 2,, m 2 r 2, r 2 + 1,, (m 2 + 1), r 2, r 2 + 1,, (m 2 + 1), r 2, r 2 + 1,, (m 2 + 1) r 2 + 1, r 2 + 2,, m 3, r 2 + 1, r 2 + 2,, m 3, r 2 + 1, r 2 + 2,, m 3 r t, r t + 1,, (m t + 1), r t, r t + 1,, (m t + 1), r t, r t + 1,, (m t + 1) r t + 1, r t + 2,, n r t + 1, r t + 2,, n r t + 1, r t + 2,, n (14) with t (r i m i ) 0 (mod 2); i2 t (r i m i ) 1 (mod 2); i2 t (r i m i ) 0 (mod 2) Some of these seuences may be empty, if r i m i or r i m i+1 for some i, or if m i 0 i1 10

11 Proof The only difference with respect to the B n case is for the 0, 1 descents They depend on the relative positions of ±1 and ±2 in the window notation of γ The following are the D-descent classes of all elements of B 2 We have that D( ) {1, 2, 1, 2} D({0}) {2, 1, 2, 1} D({1}) { 2, 1, 2, 1} D({0, 1}) {1, 2, 1, 2} From this, the parity conditions t i1 (r i m i ) 0 or 1 (mod 2), and Lemma 31 the result follows Remark 33 Let us fix a subset of descents M : {m 1, m 2,, m t } Consider the decompositions of B n and D n given by Proposition 24 by using left uotients Recall that B n 2 n n! and that D n 2 n 1 n! By looking at the Dynkin diagrams in Figure 1 and Figure 2, it is easy to derive the following eualities If 0 M, then B(M) 2 D(M) ; If 0, 1 M, then B(M) D(M) ; If 0 M, and 1 M, then B(M) m 2 D(M) Now we make explicit these eualities by showing relations between D and B left uotients Proposition 34 Let M 0, n 1, such that 0 M Then i) B(M) splits into the disjoint union B(M) D(M) D(M), where D(M) : { γ γ(1), γ(2),, γ(n) γ D(M)} {γ s B 0 γ D(M)} ii) Moreover ld(β) 2 ld(γ) β B(M) γ D(M) Proof Let γ D(M) By Lemma 32 γ is a shuffle of the seuences in (12), and so it can also be obtained as a shuffle of the seuences in (10) Hence γ B(M) Now, let us change the sign to the first entry of γ, by getting γ We are changing the sign of r i, or of r i + 1 for i t, in one of the seuences in (12) Note that this operation does not create a new B-descent for γ Hence γ B(M) \ D(M) More precisely, γ can be obtained by shuffling the same seuences that give γ where the twos involving r i are replaced either by r i + 1,, (m i + 1) and r i, r i + 1,, m i, or by r i 1,, (m i + 1) 11

12 and r i + 2,, m i, depending if it is the sign of r i, or of r i + 1, that changes All those seuences belong to (10) So i) follows by Remark 33 Now, it is easy to see that for all γ D(M), one has l D (γ) l D ( γ) To see that, suppose γ(1) > 0 Then inv( γ) inv(γ) (γ(1) 1) and N 2 ( γ) N 2 (γ) + (γ(1) 1), and so the length l D is stable If γ(1) < 0 a similar computations holds, hence ii) follows Note that the two subsets D(M) and D(M) are not isomorphic as posets, when they are considered as sub-posets of (B(M), < B ), where < B denotes the B-Bruhat order An example is given for n 3 and M {0, 2} Proposition 35 Let M 0, n 1, such that 0, 1 M Then i) The map ϕ : B(M) D(M) defined by { β, if β D n ; β s B 0 β, otherwise, is a bijection ii) Moreover ld(β) ld(γ) β B(M) γ D(M) Proof Let β B(M), it is a shuffle of the seuences in (10) If β D n, then it is also a shuffle of the seuences in the first block of (13) Hence β D(M) Now suppose that β D n Since 0 Des B (β 1 ), then 1 βn By multiplying on the left by s B 0, we change the sign of 1, and so the parity of β Hence s B 0 β D n Actually, we obtain an element which is a shuffle of the seuences in second block of (13) From Remark 33 i) follows Since N 2 (γ) N 2 (ϕ(γ)) and inv(γ) inv(ϕ(γ)), one has l D (γ) l D (ϕ(γ)), and so ii) follows The map ϕ is not a poset isomorphism between (B(M), < B ) and (D(M), < D ), where < B and < D denote the corresponding Bruhat orders When n 3, and M {2}, B(M) is a chain, while in D(M) there are two not comparable elements Proposition 36 Let M 0, n 1, such that 0 M, and 1 M Then i) B(M) splits as the disjoint union of the following m 2 subsets B(M) D 1 (M) D 12 (M) D 12m2 (M) Each D 1i (M) is in bijection with D(M), and it is recursively defined as follows: 12

13 1) D 1 (M) is obtained by shuffling the seuences defining D(M) where 1 (if present) is replaced with 1 2) For each i 2, D 12i (M) is obtained by shuffling the seuences defining D 12i 1 (M) where: 1 and ±i are swiched if they are in the same seuence; i is replaced by 1, and 1 is replaced by i, otherwise This case happens when 1 is at the beginning of a seuence of type 1, (i 1),, 2, and i is the initial value of the seuence i, i + 1,, m 2 ii) Moreover ld(β) m 2 ld(γ) β B(M) γ D(M) Before writing down the proof let us consider an example Example 37 Consider n 4 and M {1, 3} Then D(M) is given by the shuffles of the following blocks of increasing seuences (written in column) 1 1, 2, 3 2, 1 3, 2, 1 D(M) 2, Then B(M) splits as disjoint union of the following three subsets: 1 1, 2, 3 2, 1 3, 2, 1 D 1 (M) 2, D 12 (M) D 123 (M) 2 2, 1, 3 1, 2 3, 1, 2 1, , 3, 1 3, 2 1, 3, 2 3, Proof The transformations defining D 1i (M) involve only the first two seuences of the three blocks of (14) It is easy to see that D 1i (M) B(M) for all i m 2, and that D 1i (M) and D 1j (M) are disjoint if i j Hence the decomposition in i) follows from Remark 33 13

14 Since changing 1 into 1 in a signed permutation γ affects neither inv(γ) nor N 2 (γ), it follows that ld(γ) ld(γ) γ D(M) Now let us show that for all i 2 γ D 1i (M) l D(γ) γ D 1 (M) γ D 1i 1 (M) l D(γ) Let γ D 1i 1 (M) Consider the block in (14) whose a particular shuffle gives γ If 1 and ±i are in the same seuence, it can be either of the form, 1, i,, m 2, or of the form r 1,, i, 1, 2 Now consider the shuffle giving γ, where 1 has been switched with ±i We get a new element γ D 1i (M) It is clear that γ has one more inversion with respect to γ, and so the D-length go up by 1 In fact, all other seuences in the block (whose shuffle gives γ) are made by elements that are either all bigger or all smaller of both 1 and ±i Hence the difference between inv(γ) and inv( γ) depends only on the relative positions of 1 and ±i within the same seuence Suppose that 1 and i are not in the same seuence This means that 1 is at the beginning of the seuence 1, (i 1),, 2 and i is at the beginning of the seuence i, i + 1,, m 2 So γ D 1i (M), the element corresponding to γ after the switch, is obtained by shuffling a block that contains the following two seuences i, (i 1),, 2 and 1, i + 1,, m 2 Once again all other seuences of the block are made by elements that are either all smaller or bigger of both 1 and i The difference between the values of inv( γ) and inv(γ) depends only on the relative positions of 1 and i Hence γ loses i 2 inversions with respect to γ (the ones given by the 1 at the beginning of the seuence), and N 2 ( γ) N 2 (γ) + (i 1) thanks to i So l D ( γ) l D (γ) Euidistribution over descent classes In this section we show generalizations of Theorem 11 to Coxeter groups of type B and D We need the following classical result; see 14, Theorem 31, and 20, Example 225 for a proof Theorem 41 Let n P and M {m 1, m 2,, m t } < n 1 Then maj(σ) inv(σ) n m 1, m 2 m 1,, n m t {σ S n Des(σ 1 ) M} {σ S n Des(σ 1 ) M} 14

15 Theorem 42 Let n P and M {m 1, m 2,, m t } < 0, n 1 Then nmaj(β) lb(β) {β B n Des B (β 1 ) M} {β B n Des B (β 1 ) M} n m 1, m 2 m 1,, n m t {β B n Des B (β 1 ) M} n im 1 +1 fmaj(β) (1 + i ) (15) Proof Let us denote by Sh(r 1,, r t ) the set of signed permutations obtained as shuffles of the seuences in (10), with prescribed r 1,, r t From Theorem 41 it follows that maj(β) inv(β) n (16) m 1, r 1 m 1,, r t m t, n r t β Sh(r 1,,r t) β Sh(r 1,,r t) In fact inversion number and major index of a shuffle depend only on the order of the elements in the shuffled seuences From this, and the definitions of nmaj(β) maj(β) + N 1 (β) + N 2 (β) and of l B (β) inv(β) + N 1 (β) + N 2 (β), the first euality in (15) follows The second euality and the sum have been computed in 3 The symbol fmaj denote the flag-major index introduced by Adin and Roichman in 1 By the Principle of Inclusion-Exclusion we obtain Corollary 43 {β B n Des B (β 1 )M} nmaj(β) {β B n Des B (β 1 )M} l B(β) {β B n Des B (β 1 )M} fmaj B (β) The following lemma will be useful in the computation of our main result Theorem 45 Lemma 44 Let n P and M {m 1, m 2,, m t } < 0, n 1 Then n 1 ld(β) n (1 + i ) m 1, m 2 m 1,, n m t im 1 {β B n Des B (β 1 ) M} Proof Let β B(M) Recall that l D (β) l B (β) N 1 (β), and that l B (β) inv(β) + β(i)<0 β(i) Note that β(i) < 0 if and only if there exists a j such that m j + 1 β(i) r j Therefore β(i) β(i)<0 t (m i + 1) + + r i i1 t i1 t i1 (r i m i )m i + (r i m i )(r i m i + 1) (r i m i )(r i + m i + 1) 15

16 Moreover N 1 (β) t i1 (r i m i ), and so l D (β) inv(β) + inv(β) + t 1 2 (r i m i )(r i + m i + 1) (r i m i ) t ( ) ri m i (r i m i )(m i 1) 2 i1 i1 Hence by (16) β B(M) l D(β) r 1,,r t β Sh(r 1,,r t) inv(β) P t i1 ( r i m i +1 2 )+(r i m i ) n P t i1 ( r i m i +1 2 )+(r i m i )(m i 1) m r 1,,r 1, r 1 m 1,, n r t t m n t i+1 m i+1 m i (r i m i +1 2 )+(r i m i )(m i 1) m 1, m 2 m 1,, n m t r i1 r i m i m i i m n t i+1 1 (1 + j ) (17) m 1, m 2 m 1,, n m t i1 jm i n 1 n (1 + j ) m 1, m 2 m 1,, n m t jm 1 where the sum runs over m i r i m i+1, and (17) is obtained by applying the -binomial Theorem 21 with x (m i 1) Theorem 45 Let n P and M {m 1, m 2,, m t } < 0, n 1 Then dmaj(γ) l D(γ) γ D(M) γ D(M) n m 1, m 2 m 1,, n m t n m 1, m 2 m 1,, n m t n m 1, m 2 m 1,, n m t (1 + i ) if 0 M; n 1 i1 n 1 (1 + i ) if 0, 1 M; im 1 n 1 i1 (1 + i ) if 0 M, and 1 M m 2 Proof Once again the first euality follows from (16) and the definitions of dmaj and l D The computation of the sum is now an easy application of Lemma 44, together with Propositions 34, 35, and 36 As corollary we obtain the desired generalization 16

17 Corollary 46 dmaj(γ) ld(γ) {γ D n Des D (γ 1 )M} {γ D n Des D (γ 1 )M} Remark 47 If we replace Des B with the usual descent set Des, Corollary 43 is still valid It easily follows from Theorem 42 since Des(β 1 ) M if and only of Des B (β 1 ) M {0} Analogously, by replacing Des D with Des, Corollary 46 holds for the Coxeter group of type D The two corollaries are not true if as descent set one chooses NDes for B n and DDes for D n 5 Symmetry of the joint distribution In this section we find generalizations of Foata-Schützenberger Theorem 12, Roselle Theorem 13, and Gessel Theorem 14 The following is an easy computation Lemma 51 Let n P Then Moreover u B J p N1(u) N1(u)+N2(u) u D J p N1(u)+ɛ(u) N2(u) S n S n 1 p S P i S i n (1 + p i ) ( p; ) n i1 p S P n 1 i S i (1 + p i ) ( p; ) n 1 Proposition 52 The distribution of (nmaj, l B ) over B n is symmetric, namely B n (t, ) : t nmaj(β) lb(β) t lb(β) nmaj(β) β B n β B n Proof Let consider the decomposition (8) of B n Let u B J (or D J ) and σ S n Then the following eualities hold i1 maj(uσ) maj(σ) and inv(uσ) inv(u) Moreover N 1 (uσ) N 1 (u) and N 2 (uσ) N 2 (u) 17

18 Then from Theorem 12 it follows t lb(β) nmaj(γ) t inv(uσ)+n1(uσ)+n2(uσ) maj(uσ)+n1(uσ)+n2(uσ) β B n u B J σ S n t N1(u)+N2(u) N1(u)+N2(u) t inv(σ) maj(σ) u B J σ S n t N1(u)+N2(u) N1(u)+N2(u) t maj(σ) inv(σ) u B J σ S n t maj(uσ)+n1(uσ)+n2(uσ) inv(uσ)+n1(uσ)+n2(uσ) u B J σ S n t nmaj(β) lb(β) β B n The analogous result holds for D n The proof is very similar to that of B n and is left to the reader Proposition 53 The pair of statistics (dmaj, l D ) is symmetric, namely D n (t, ) : t dmaj(γ) ld(γ) t ld(γ) dmaj(γ) γ D n γ D n Note that, the flag-major index and the D-major index 7 do not share with nmaj and dmaj this symmetric distribution property The following identities are generalizations of Theorem 13 of Roselle to B n and D n They easily follow from the proof of Proposition 52, Lemma 51, and from Theorem 13 Proposition 54 (Roselle Identities for B n and D n ) u n B n (t, ) (t; t) n (; ) n ( t; t) n n 0 u n 1 + D n (t, ) (t; t) n (; ) n ( t; t) n 1 n 1 1 (u; t, ),, (B 0 (t, ) : 0); 1 (u; t, ), Similarly the following identities, which generalize Gessel formula, follow from the proof of Proposition 52, Lemma 51, and Theorem 14 Proposition 55 (Gessel Identities for B n and D n ) 1 n 0 1 t + n 1 u n β B n t nmaj(σ) lb(β) p ndes(β) p k eu etu et k u ; n! ( tp; t) n (t; ) n+1 k 0 u n γ D n t dmaj(γ) ld(γ) p ddes(γ) p k eu etu et k u n! ( tp; t) n 1 (t; ) n+1 k 0 18

19 6 Concluding remarks As we mentioned along the paper, there exists another family of statistics, the flag-statistics, defined on Coxeter groups of type B, D (see 1 and 7), and more generally on complex reflection groups 5 Several generating functions involving flag-statistics have already been computed In particular, we refer to the series of papers of Foata and Han 10, 11, 12, for a complete overview on the argument We remark that among the series computed, none involve a combination of flag-statistics and length This is why we conclude the paper with the following interesting proposal Problem 61 What kind of identities, generalizing the ones of Roselle and Gessel, might be obtained by using flag-statistics? References 1 R M Adin and Y Roichman, The Flag Major Index and Group Actions on Polynomial Rings, Europ J Combinatorics, 22 (2001), R M Adin, F Brenti and Y Roichman, Descent Numbers and Major Indices for the Hyperoctahedral Group, Adv in Appl Math, 27 (2001), R M Adin, F Brenti and Y Roichman, Eui-distribution over descent classes of the Hyperoctahedral Group, J Combin Theory Ser A, 113 (2006), GE Andrews, The Theory of Partitions, The Encyclopedia of Mathematics and Its Applications Series, Addison-Wesley Pub Co, NY, 300 pp (1976) 5 E Bagno and R Biagioli, Colored-Descent Representations of Complex Reflection Groups G(r, p, n), Israel J Math, 160 (2007), R Biagioli, Major and descent statistics for the even-signed permutation group, Adv in Appl Math, 31 (2003), R Biagioli and F Caselli, Invariant algebras and major indices for classical Weyl groups, Proc London Math Soc, 88 (2004), A Björner and F Brenti, Combinatorics of Coxeter Groups, Grad Texts in Math 231, Springer-Verlag, Berlin, D Foata, On the Netto Inversion Number of a Seuence, Proc Amer Math Soc, 19 (1968), D Foata and GN Han, Signed words and permutations, II: the Euler-Mahonian polynomials, Electronic J Combinatorics, 11(2), R22, (2005) 19

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