SIGNED WORDS AND PERMUTATIONS, III; THE MACMAHON VERFAHREN. Dominique Foata and Guo-Niu Han

Transcription

1 Oct 3, 2005 SIGNED WORDS AND PERMUTATIONS, III; THE MACMAHON VERFAHREN Dominique Foata and Guo-Niu Han Glory to Viennot, Wizard of Bordeaux, A prince in Physics, In Mathematics, Combinatorics, Even in Graphics, Sure, in Viennotics He builds bijections, Top calculations No one can beat him Only verbatim Can we follow him He has admirers, Who have got down here They all celebrate Such a happy fate Sixty years have gone, He still is our don Dedicated to Xavier Lucelle, April 2005 Abstract The MacMahon Verfahren is fully exploited to derive further multivariable generating functions for the hyperoctahedral group B n and for rearrangements of signed words Using the properties of the fundamental transformation the generating polynomial for B n by the flagmajor and inverse flag-major indices can be derived, and also by the length function, associated with the flag-descent number Introduction To paraphrase Leo Carlitz [Ca56], the present paper could have been entitled Expansions of certain products, as we want to expand the product () K (u) := in its infinite version, and (2) K r,s (u) := i 0,j 0 0 i r, 0 j s ( uz ij q i qj 2 ), ( uz ij q i qj 2 ), in its graded version, where { Z, if i and j are both odd; Z ij :=, if i and j are both even; 0, if i and j have different parity

2 DOMINIQUE FOATA AND GUO-NIU HAN The second pair under study, which depends on r variables u,, u r, reads (3) L (u,,u r ) :=, (u i ;q 2 ) (u i qz;q 2 ) (4) L s (u,,u r ) := i r i r (u i ;q 2 ) s/2 + (u i qz;q 2 ) (s+)/2 In those expressions we have used the usual notations for the q- ascending factorial {, if n = 0; (5) (a;q) n := ( a)( aq) ( aq n ), if n ; in its finite form and (6) (a;q) := lim n (a;q) n = ( aq n ); n 0 in its infinite form All those products will be the basic ingredients for deriving the distributions of various statistics attached to signed permutations and words By signed word we understand a word w = x x 2 x m, whose letters are integers, positive or negative If m = (m,m 2,,m r ) is a sequence of nonnegative integers such that m +m 2 + +m r = m, let B m be the set of rearrangements w = x x 2 x m of the sequence m 2 m 2 r m r, with the convention that some letters i ( i r) may be replaced by their opposite values i For typographical reasons we shall use the notation i := i in the sequel When m = m 2 = = m r =, the class B m is simply the hyperoctahedral group B m (see [Bo68], p ) of the signed permutations of order m (m = r) Using the χ-notation that maps each statement A onto the value χ(a) = or 0 depending on whether A is true or not, we recall that the usual inversion number, invw, of each signed word w = x x 2 x n is defined by invw := χ(x i > x j ) j n i<j It also makes sense to introduce invw := j n χ(x i > x j ), i<j and define the flag-inversion number of w by finvw := invw+invw+negw, where negw := χ(x j < 0) As noted in our previous paper j n [FoHa05a], the flag-inversion number coincides with the traditional length function l, when applied to each signed permutation 2

3 SIGNED WORDS AND PERMUTATIONS, III The flag-major index fmaj and the flag descent number fdes, which were introduced by Adin and Roichman [AR0] for signed permutations, are also valid for signed words They read fmajw := 2majw +negw; fdesw := 2desw+χ(x < 0); where majw := j jχ(x j > x j+ ) denotes the usual major index of w and desw the number of descents desw := j χ(x j > x j+ ) Finally, for each signed permutation w let w denote the inverse of w and define ifdesw := fdesw and ifmaj := fmajw Notations In the sequel B n (resp B m ) designates the hyperoctahedral group of order n (resp the set of signed words of multiplicity m = (m,m 2,,m r )), as defined above Each generating polynomial for B n (resp for B m ) by some k-variable statistic will be denoted by B n (t,,t k ) (resp B m (t,,t k )) When the variable t i is missing in the latter expression, it means that the variable t i is given the value The main two results of this paper corresponding to the two pairs of products earlier introduced can be stated as follows Theorem Let (7) B n (t,t 2,q,q 2,Z) := t fdesw t ifdesw 2 q fmajw q ifmajw 2 Z negw w B n bethegeneratingpolynomialfor thegroupb n by thefive-variablestatistic (fdes, ifdes, fmaj, ifmaj, neg) Then, (8) u n (t 2 n 0 ;q2 ) n+(t 2 2 ;q2 2 ) (+t )(+t 2 )B n (t,t 2,q,q 2,Z) n+ = t r ts 2 K r,s(u), where K r,s (u) is defined in (2) r 0,s 0 Theorem 2 For each sequence m = (m,,m r ) let (9) B m (t,q,z) := w B m t fdesw q fmajw Z negw be the generating polynomial for the class B m of signed words by the three-variable statistic (fdes, fmaj, neg) Then (0) (+t)b m (t,q,z) um u m r r = t s L (t 2 ;q 2 s (u,,u r ), ) m + m s 0 where m := m + +m r and L s (u,,u r ) is defined in (4) 3

4 DOMINIQUE FOATA AND GUO-NIU HAN It is worth noticing that Reiner [Re93a] has calculated the generating polynomialforb n byanother5-variablestatisticinvolvingeachsignedpermutation and its inverse The bibasic series he has used are normalized by products of the form (t ;q ) n+ (t 2 ;q 2 ) n instead of (t 2 ;q) 2 n+ (t 2 2;q2) 2 n+ Using the properties of the fundamental transformation on signed words described in our first paper [FoHa05a] we obtain the following specialization of Theorem Let () finv B n (t,q) := w B n t fdesw q finvw be the generating polynomial for the group B n by the pair (fdes,finv) Then, the q-factorial generating function for the polynomials finv B n (t,q) (n 0) has the following form: (2) n 0 v n (q 2 ;q 2 ) n finv B n (t,q) = t t 2 +(v( t 2 );q) ( t+(v( t 2 )q;q 2 ) ) From identity(2) we deduce the generating function for the polynomials dess B n (t,q) := w B n t dessw q finvw,where dess isthe traditionalnumber of descents for signed permutations defined by dessw := desw+χ(x < 0) instead of fdesw := 2desw+χ(x < 0) and recover Reiner s identity [Re93a] u n (3) dess (q 2 ;q 2 B n (t,q) = ) n n 0 t te q (u( t)) (v( t);q 2 ), where e q (v( t 2 )) = /(v( t 2 );q) is the traditional q-exponential This is done in section 4 As in our second paper [FoHa05b] we make use of the MacMahon Verfahren technique to prove the two theorems, which consists of transferring the topology of the signed permutations or words measured by the various statistics, fdes, fmaj, to a set of pairs of matrices with integral entries in the case of Theorem and a set of plain words in the case of Theorem 2 for which the calculation of the associated statistic is easy Each time there is then a combinatorial bijection between signed permutations (resp words) and pairs of matrices (resp plain words) that has the adequate properties This is the content of Theorem 3 and Theorem 4 In all our results we have tried to include the variable Z that takes the number neg of negative letters of each signed permutation or word into account This allows us to re-obtain the classical results on the symmetric group and sets of words 4

5 SIGNED WORDS AND PERMUTATIONS, III In the next section we recall the MacMahon Verfahren technique, which was developed in our previous paper [FoHa05b] for signed permutations Notice that Reiner [Re93a, Re93b, Re93c, Re95a, Re95b], extending Stanley s [St72] (P,ω)-partition approach, has successfully developed a (P, ω)-partition theory for the combinatorial study of the hyperoctahedral group, which could have been used in this paper Section 3 contains the proof of Theorem, whose specializations are given in Section 4 We end the paper with the proof of Theorem 2 and its specializations Noticeably, the generating polynomial for the class B m of signed words by the twovariable statistic (fdes, fmaj) is completely explicit, in the sense that we derive the factorial generating function for those polynomials and also a recurrence relation, while only the generating function given by (0) has a simple form when the variable Z is kept 2 The MacMahon Verfahren Let N n (resp NIW(n)) be the set of words (resp nonincreasing words) of length n, whose letters are nonnegative integers As done in [FoHa05b] the MacMahon Verfahren consists of mapping each pair(b, w) NIW(n) B n onto a word c N n as follows Write the signed permutation w as the linear word w = x x 2 x n, where x k is the image of the integer k ( k n) For each k =,2,,n let z k be the number of descents in the right factor x k x k+ x n and ǫ k be equal to 0 or depending on whetherx k ispositiveornegativenext,formthewordsz(w) := z z 2 z n and ǫ(w) := ǫ ǫ 2 ǫ n Now, take a nonincreasing word b = b b 2 b n and define a k := b k +z k, c k := 2a k + ǫ k ( k n), then a(b,w) := a ( a 2 a n and c (b,w) := c c c 2c n Finally, form the two-row matrix c 2 ) c n Its x x 2 x n bottom row is a permutation of 2 n; rearrange the columns in such a way that the bottom row is precisely 2 n Then the word c(b,w) = c c 2 c n which corresponds to the pair (b,w) is defined to be the top row in the resulting matrix Example Start with the pair (b, w) below and calculate all the necessary ingredients: b = w = z(w) = ǫ(w) = a(b,w) = c (b,w) = c(b,w) =

6 DOMINIQUE FOATA AND GUO-NIU HAN For each c = c c n N n and let totc := c + + c n, maxc := max{c,,c n } and let oddc denote the number of odd letters in c The proof of the following theorem can be found in [FoHa05b, Theorem 4] Theorem 2 For each nonnegative integer s the above mapping is a bijection (b,w) c(b,w) of the set of pairs (b,w) = (b b 2 b n, x x 2 x n ) NIW(n) B n such that 2b +fdesw = s onto the set of words c = c c 2 c n N n such that maxc = s Moreover, (2) 2b +fdesw = maxc(b,w); 2totb+fmajw = totc(b,w); negw = oddc(b,w) We end this section by quoting the following classical identity, where b is the first letter of b (22) = u s q totb (u;q) n+ s 0 b NIW(n),b s 3 Proof of Theorem We apply the MacMahon Verfahren just described to the two pairs (b,w) and (β,w ), where b and β are two nonincreasing words The pair (b,w) (resp (β,w )) is mapped onto a word c := c(b,w) = c c 2 c n (resp C := c(β,w ) = C C 2 C n ) of length n, with the properties (3) 2b +fdesw = maxc; 2totb+fmajw = totc; (32) 2β +ifdesw = maxc; 2totβ +ifmajw = totc There remains to investigate the relation between the two words c and C before pursuing the calculation Form the two-row matrix ( ) ( ) c c = c 2 c n, C C C 2 C n where c = c c 2c n designates the nonincreasing rearrangement of the word c The following two properties can be readily verified: (i) for each i =,2,,n the letters c i and C i are of the same parity; (ii)if c i = c i+ is even (resp is odd), then C i C i+ (resp C i C i+ ) Conversely, the following property holds: (iii) if ( c C) is a two-row matrix of length n having properties (i) and (ii), there exists one and only one signed permutation w and two nonincreasing words b, β satisfying (3) and (32) 6

7 SIGNED WORDS AND PERMUTATIONS, III Example We take the same example for w as in the previous section, but we also calculate C = c(β,w ) b = w = z(w) = ǫ(w) = a(b,w) = c := c (b,w) = c := c(b,w) = ( ) c = C β = w = z(w ) = ǫ(w ) = a(β,w ) = C := c (β,w ) = C := c(β,w ) = ( Let (i < i < < i r ) (resp (j < j 2 < < j s )) be the increasing sequence of the integers i (resp the integers j) such that c i is even (resp is odd) Define ( e for even and o for odd ) c j ( ) 2f 2d e e = 2g ( e 2f 2d o o ) + + = 2g o + ) := ( c i c i 2 c i r C i C i2 C ir := ( c j c j 2 c j s C j C j2 C js so that the two two-row matrices ( ) ( f d e e c = g e := i /2 c i 2 /2 c i r /2 C i /2 C i2 /2 C ir /2 ( ) f d o o = := g o ), ) ; ) ; ( (c j )/2 (c j 2 )/2 (c j s )/2 (C j )/2 (C j2 )/2 (C js )/2 may be regarded as another expression for the two-row matrix ( c C) Define the integers i max and j max by: { (maxc )/2, if maxc is odd; i max := maxc/2, if maxc is even; { (maxc )/2, if maxc is odd; j max := maxc/2, if maxc is even Then, to the pair d e, d o there corresponds a pair of unique finite matrices D e = (d e ij ), Do = (d o ij ) (0 i i max,0 j j max ) (and no other pair of smaller dimensions), where d e ij (resp do ij ) is equal to the number of the two-letters in d e (resp in d o ) that are equal to ( i j) 7 ),

8 DOMINIQUE FOATA AND GUO-NIU HAN On the other hand, with f designating the length of the word f, totc = tot2f e +tot(2f o +) = 2totf e +2totf o + f o = 2 id e ij +2 id o ij + i,j i,j i,j d o ij; totc = tot2g e +tot(2g o +) = 2totg e +2totg o + g o = 2 jd e ij +2 jd o ij + d o ij i,j i,j i,j The following proposition follows from Theorem 2 Proposition 3 For each pair of nonnegative integers (r, s) the map (b,β,w) (D e,d o ) is a bijection of the triples (b,β,w) such that 2b +fdesw r, 2β +ifdesw s onto the pairs of matrices D e = (d e i,j ), D o = (d o i,j ) (0 i r,0 j s) Moreover, (33) 2totb+fmajw = 2 id e ij +2 id o ij + d o ij ; i,j i,j i,j (34) 2totβ +ifmajw = 2 jd e ij +2 jd o ij + d o ij ; i,j i,j i,j (35) negw = d o ij i,j (36) d e ij + d o ij = w i,j i,j Again work with the same example as above To the two-row matrix ( ) ( ) c = C there corresponds the pair ( ) ( ) d e =, d o = As maxc = 2 is even (resp maxc = is odd), we have i max = 6, j max = 5 and D e = ; D o = Also verify that 2totb+fmajw = 35 = 2 (2+2+6)+2 (3+3)+3; 2totβ +ifmajw = 27 = 2 (2+4)+2 (+5)+3 and negw = 3 8

9 SIGNED WORDS AND PERMUTATIONS, III In the following summations b and β run over the set of nonincreasing words of length n By using identity (22) we have u n (t 2 ;q2 ) n+(t 2 2 ;q2 2 ) (+t )(+t 2 )B n (t,t 2,q,q 2,Z) n+ n 0 = n 0 u n (+t )(+t 2 )B n (t,t 2,q,q 2,Z) = u n (t 2r n,r,s u n t r n,r,s = +t 2r + )(t 2s 2 +t 2s + 2 ) w B n b,β,b r,β s ts 2 = n,r,su n t r t s 2 w B n b,β,2b r,2β s r 0,s 0 b,β,b r,β s t 2r t 2s 2 q 2totb q 2totβ 2 t fdesw t ifdesw 2 q fmajw+2totb q ifmajw+2totβ 2 Z negw t fdesw t ifdesw 2 q fmajw+2totb q ifmajw+2totβ 2 Z negw w B n,b,β 2b +fdesw r,2β +ifdesw s q fmajw+2totb q ifmajw+2totβ 2 Z negw We can continue the calculation by using (33), (34), (35) the last summation being over matrices D e, D o of dimensions (r + ) (s + ), that is, = n,r,su n t r ts 2 = r,s = r,s = r,s t r ts 2 = t r ts 2 t r ts 2 r 0,s 0 q 2Σid e ij +2Σido ij +Σdo ij q 2Σjde ij +2Σjdo ij +Σdo ij 2 Z Σdo ij D e,d o u Σd e Σ(2i)d ij q e ij q Σ(2j)de ij 2 u Σd D e D o (uq 2i q2j 2 )de ij (uzq 2i+ q 2j+ D e ij D o ij ( uq 2iq2j 2 ) ( uzq 2i+ ij t r t s 2 0 i r,0 j s ij ( uz ij q i qj 2 ) q 2j+ 2 ) o ij q Σ(2i+)d o ij q Σ(2j+)do ij 2 Z Σdo ij 2 ) do ij (0 i r,0 j s) (0 i r,0 j s) 4 Specializations and Flag-inversion number First, when Z := 0 and t, t 2, q, q 2 are replaced by their square roots, identity (8) becomes (4) n 0 u n A n (t,t 2,q,q 2 )= (t ;q ) n+ (t 2 ;q 2 ) n+ 9 r 0,s 0 t r t s 2 (u;q,q 2 ) r+,s+,

10 DOMINIQUE FOATA AND GUO-NIU HAN where A n (t,t 2,q,q 2 ) is the generating polynomial for the symmetric group S n by the quadruple (des,ides,maj,imaj), an identity first derived by Garsia and Gessel [GaGe78] Other approaches can be found in [Ra79], [DeFo85] Remember that when a variable is missing in B n (t,t 2,q,q 2,Z) it means that the variable has been given the value Multiply both sides of (8) by ( t 2 ) and let t 2 = We get: (42) u n (t 2 n 0 ;q2 ) n+(q2 2;q2 2 ) (+t )B n (t,q,q 2,Z) n = t r r 0 0 i r,j 0 Again multiply both sides by ( t ) and let t = : (43) u n (q 2;q2 ) n(q2 2;q2 2 ) B n (q,q 2,Z) = n n 0 i 0,j 0 ( uz ij q i qj 2 ) ( uz ij q i qj 2 ) Also the (classical) generating function for the polynomials A n (q,q 2 ) can be derived from identity (43) and reads: n 0 u n (q ;q ) n (q 2 ;q 2 ) n A n (q,q 2 ) = (u;q,q 2 ), Withq = the denominator ofthefraction ontheright side ofidentity (42) becomes { (u;q 2 2 ) (r/2)+ (uq 2 Z;q 2 2) r/2, if r is even; Hence, n 0 (u;q 2 2 )(r+)/2 (uq 2 Z;q2 2)(r+)/2, if r is odd u n ( t 2 )n+ (q2 2;q2 2 ) (+t )B n (t,q 2,Z) n t 2s+ = ((u;q 2 s 0 2 ) (uq 2 Z;q2 2) ) + s+ s 0 ( ) = t 2 +(u;q2 2 ) (uq 2 Z;q2 2) t +(uq 2 Z;q2) 2 t 2s ((u;q2 2) (uq 2 Z;q2 2) ) s (u;q2 2) Now replace u by v( t 2 ) This implies the following result stated as a theorem 0

11 SIGNED WORDS AND PERMUTATIONS, III Theorem 4 Let B n (t,q 2,Z) be the generating polynomial for the group B n by the triple (fdes,ifmaj,neg) Then, v n (44) (q 2 n 0 2 ;q2 2 ) B n (t,q 2,Z) n t ( ) = t 2 +(v( t2 );q2 2 ) (v( t 2 )q 2Z;q2 2) t +(v( t 2 )q 2 Z;q2) 2 Several consequences are drawn from Theorem 4 First, when Z = 0 and when t, q 2 are replaced by their square roots, we get v n t (45) A n (t,q 2 ) =, (q 2 ;q 2 ) n t +(v( t );q 2 ) n 0 where A n (t,q 2 ) is the generating polynomial for the group S n by the pair (des, imaj), but also by the pair (des, inv) (see [St76], [FoHa97]) Let iw := w denote the inverse of the signed permutation w At this stage we have to remember that the bijection Ψ of B n onto itself that we have constructed in our first paper [FoHa05a] preserves the inverse ligne of route [FoHa05a, Theorem 2], so that the chain i Ψ i w w w 2 w 3 ( ) ( ) ( ) ( ) fdes ifdes ifdes fdes ifmaj fmaj finv finv shows that the four pairs (fdes, ifmaj), (ifdes, fmaj), (ifdes, finv) and (fdes,finv) are all equidistributed over B n Therefore, t fdesw q ifmajw = t ifdesw q fmajw = t ifdesw q finvw = t fdesw q finvw, w w w w where w runs over B n The rightmost generating polynomial was designated by finv B n (t,q) in () Therefore we can derive a formula for finv B n (t,q) by using its (fdes,ifmaj) interpretation We have then finv B n (t,q) = B n (t,q 2,Z) with t = t, q 2 = q and Z = Let Z := in (44); as (v( t 2 );q2 2 ) (v( t 2 )q 2;q2 2) = (v( t 2 );q 2), identity (44) implies identity (2) To recover Reiner s identity (3) we make use of (2) by sorting the signed permutations according to the parity of their flag descent numbers: B n (t,q) =: B n (t2,q) + tb n (t2,q), so that dess B n (t 2,q) = B n (t2,q) + t 2 B n(t 2,q) Hence v n dess B (q 2 ;q 2 n (t 2,q)= (v( t2 )q;q 2 ) t 2 +t 2 (v( t2 )q;q 2 ) ) n t 2 +(v( t 2 );q) t 2 +(v( t 2 );q) n 0 = t 2 t 2 +(v( t 2 );q) (v( t 2 )q;q 2 )

12 DOMINIQUE FOATA AND GUO-NIU HAN As/(v( t 2 );q) canalsobeexpressedastheq-exponentiale q (v( t 2 )), we then get identity (3) 5 The MacMahon Verfahren for signed words Let w = x x 2 x m be a signed word belonging to the class B m, where m = (m,m 2,,m r ) is a sequence of nonnegative integers such that m + m m r = m Remember that this means that w is a rearrangement of m 2 m 2 r m r, with the convention that some letters i ( i r) may be replaced by their opposite values i Again, let ǫ := ǫ(w) := ǫ ǫ 2 ǫ m be the binary word defined by ǫ i = 0 or, depending on whether x i is positive or negative The MacMahon Verfahren bijection for signed words is constructed as follows First, compute the word z = z z 2 z m, where z k is equal to the number of descents in the right factor x k x k+ x m, as well as the word ǫ = ǫ ǫ 2 ǫ m mentioned above, so that, as in the case of signed permutations, (5) fmajw = 2totz +totǫ Next, define a k := b k + z k, c k := 2a k + ǫ k ( k m), then ( a := a a) 2 a ( m and c := c c 2 c m Finally, form the two-row matrix c c = c 2 ) c m Its bottom row is a rearrangement of absw x x 2 x m the word m 2 m 2 r m r Make the convention that two biletters ( c ) ( k x k and c l x l ) commute if and only if x k and x l are different and rearrange the biletters of that biword in such a way that the bottom row is precisely m 2 m 2 r m r The top row in the resulting two-row matrix is then the juxtaposition product of r nonincreasing words b () = b () b () m, b (2) = b (2) b (2) m 2,, b (r) = b (r) b (r) m r, of length m, m 2,, m r, respectively Moreover, (52) totb () +totb (2) + +totb (r) = totc = 2tota+totǫ = 2totb+2totz +totǫ = 2totb+fmajw On the other hand, (53) 2b +fdesw = 2b +2z +ǫ = c = max i b (i) ( i r) As in the case of signed permutations, we can easily see that for each nonnegative integer s the map (b,w) (b (),b (2),,b (r) ) is a bijection of 2

13 SIGNED WORDS AND PERMUTATIONS, III the set of pairs (b,w) NIW(m) R m such that 2b +fdesw=s onto the set of juxtaposition products b () b (2) b (r) such that max i b (i) = s The reverse bijection is constructed in the same way as in the case of signed permutations Example Start with the pair (b,w), where w belongs to B m with m = (2,2,2,2,2,2),r = 6, m = 2, the negative elements being overlined Id = b = w = z = ǫ = a = c = absw = b () b (r) = m r m r = Weverifythat2totb+fmajw = 2totb+2totz+totǫ = = 03 = totb () + + totb (6) and 2b + fdesw = 2b + 2z + ǫ = = 4 = c = max i b (i) The combinatorial theorem for signed words that corresponds to Theorem 2 is now stated Theorem 5 For each nonnegative integer s the above mapping is a bijection of the set of pairs (b,w) = (b b 2 b m, x x 2 x m ) NIW(m) B m such that 2b + fdesw = s onto the set of juxtaposition products b () b (r) NIW(m ) NIW(m r )such thatmax i b (i) = s Moreover, (52) and (53) hold, together with (54) negw = odd(b () b (r) ) Relation (54) is obvious, as the negative letters of w are in bijection with the odd letters of the juxtaposition product Now consider the generating polynomial B m (t,q,z), as defined in (9) Making use of (22) and of the usual q-identities = (u;q) N n 0 [ ] N +n = n q [ ] N +n n b NIW(N),b n 3 q u n ; q totb ;

14 DOMINIQUE FOATA AND GUO-NIU HAN we have +t (t 2 ;q 2 B m (t,q,z) = [ ] m+s (t 2s +t 2s + ) ) m+ s B m (t,q) s 0 q 2 = [ m+ s t r ] /2 B s m (t,q,z) /2 s 0 q 2 = t s q 2totb t fdesw q fmajw Z negw s 0 w B m = s 0 t s b NIW(m), 2b s b NIW(m),w B m 2b +fdesw s q 2totb+fmajw Z negw But using (52), (53), (54) we can write +t (t 2 ;q 2 ) m+ B m (t,q,z) = s 0 t s b (),,b (r), max i b (i) s and (+t)b m (t,q,z) um u m r r = t s (t 2 ;q 2 ) m + m s 0 q totb() + +totb (r) Z oddb() + +oddb (r) i r m i 0 u mi i q totb(i) Z oddb(i), b (i) using the notation: m := m + +m r There remains to evaluate m b um q totb Z oddb, where the second sum is over all nonincreasing words b = b b m of length m such that b s Let i < < i k m (resp j < < j l m) be the sequence of the integers i (resp j) such that b i is even (resp b j is odd) Then, b is completely characterized by the pair (b e,b o ), where b e := (b i /2)(b ik /2) and b o := ((b j )/2)((b jl )/2) Moreover, totb = 2totb e +2totb o + b o Hence, m 0b, b =m,b s u m q totb Z oddb = b,b s = = b e,2b e s u b u b q totb Z oddb e q 2totbe b o,2b o s (uqz) bo q 2totbo (u;q 2 ) s/2 + (uqz;q 2 ) (s+)/2 This achieves the proof of Theorem 3 4

15 SIGNED WORDS AND PERMUTATIONS, III 6 Specializations As has been seen in this paper a graded form such as (0) has an infinite version (again obtained by multiplying the formula by ( t) and letting t := ) given by (6) m B m (q,z) um u m r r = (q 2 ;q 2 ) m i r = i r (u i ;q 2 ) (u i qz;q 2 ) e q 2(u i )e q 2(u i qz), where e q 2(u) denotes the usual q-exponential with basis q 2 ([GaRa90], p 9) The polynomial B m (q,z) is the generating polynomial for the class B m by the pair (fmaj, neg), namely (62) B m (q,z) = w q fmajw Z negw (w B m ) (63) On the other hand, Let finv B m (q,z) := w B m q finvw Z negw bethegeneratingpolynomialfortheclassb m bythepair(finv,neg)using a different approach (the derivation is not reproduced in the paper), we can prove the identity (64) (q;q) m + +m r finv B m (q,z) = ( Zq;q) m + +m r (q;q) m (q;q) mr [ ] m + +m r = ( Zq;q) m + +m r m,,m r, q using the traditional notation for the q-multinomial coefficient In general, B m (q,z) finv B m (q,z) This can be shown by means of a combinatorial argument Let Z := in (6) and make use of the q-binomial theorem (see [An76], p 7, or [GaRa90], chap ), on the one hand, and let Z := in (64), on the other hand We get the evaluations [ ] m + +m r (65) B m (q) = ( q;q) m + +m r = finv B m (q) m,,m r q 5

16 DOMINIQUE FOATA AND GUO-NIU HAN This shows that fmaj and finv are equidistributed over each class B m, a property proved bijectively in our first paper [FoHa05a] Next, let q := in (64) We obtain ( ) (66) finv B m (Z) = (+Z) m m + +m r + +m r (= Bm (Z) ), m,,m r an identity which is equivalent to (67) m B m (Z) um u m r r m! = i r exp(u i ) exp(u i Z) Thus, the q-analog of (66) yields (64) with a combinatorial interpretation in terms of the flag-inversion number finv, while(6) may be interpreted as q 2 -analog of (67) with an interpretation in terms of the flag-maj index number fmaj Finally, for Z = 0, formula (0) yields the identity m A m (t,q) um u m r r = t s, (t;q) + m (u i ;q) s+ s 0 i r where A m (t,q) is the generating polynomial for the class of the rearrangements of the word m 2 m 2 r m r by (des,maj) As done by Rawlings [Ra79], [Ra80], the polynomials A m (t,q) can also be defined by a recurrence relation involving either the polynomials themselves, or their coefficients 7 The Signed-Word-Euler-Mahonian polynomials We end the paper by showing that the polynomials B m (t,q) = B m (t,q,z) Z= can be calculated not only by their factorial generating function given by (0) for Z :=, but also by a recurrence formula ( Definition A sequence B m (t,q) = k 0t ) k B m,k (q) (m = (m,,m r ); m 0,,m r 0)) of polynomials in two variables t, q, is said to be signed-word-euler-mahonian, if one of the following four equivalent conditions holds: () The (t 2,q 2 )-factorial generating function for the polynomials (7) C m (t,q) := (+t)b m (t,q) is given by identity (0) when Z =, that is, (72) m C m (t,q) um u m r r (t 2 ;q 2 = t s ) + m (u i ;q) s+ s 0 i r 6

17 (73) SIGNED WORDS AND PERMUTATIONS, III (2) For each multiplicity m we have: C m (t,q) (t 2 ;q 2 ) + m = s 0 Let m+ r := (m,,m r,m r +) (3) The recurrence relation (74) ( q m r+ )B m+r (t,q) holds with B (0,,0) (t,q) = [ ] [ t s m +s mr +s s q s = ( t 2 q 2+2 m )B m (t,q) q m r+ ( t)(+tq)b m (tq,q), (4) The following recurrence relation holds for the coefficients B m,k (q) (75) (+q + +q m r )B m+r,k(q) = (+q + +q m r+k )B m,k (q) +q m r+k B m,k (q)+(q m r+k +q m r+k+ + +q 2 m + )B m,k 2 (q), while B (0,,0),0 (q) = and B (0,,0),k (q) = 0 for every k 0 Theorem 7 The conditions (), (2), (3) and (4) in the previous definition are equivalent Proof The proofs of the equivalences [() (2)] and [(3) (4)] are easy and therefore omitted For proving the equivalence [() (3)] proceed as follows Let C(t,q;u,,u r ) denote the right side of (72) and form the q-difference C(t,q;u,,u r ) C(t,q;u,,u r,u r q) applied to the sole variable u r We get ] q (76) C(t,q;u,,u r ) C(t,q;u,,u r,u r q) t s t s = (u ;q) s+ (u r ;q) s+ (u ;q) s+ (u r q;q) s+ s 0 s 0 = t s [ u ] r (u ;q) s+ (u r ;q) s+ u r q s+ s 0 t s [ = u r q s+ u ] r (u ;q) s+ (u r ;q) s+ u r q s+ s 0 = u r ( C(t,q;u,,u r ) qc(tq,q;u,,u r,u r q) ) Now, let C(t,q;u,,u r ) := m C m(t,q)u m u m r r /(t 2 ;q 2 ) + m and express each term C() occurring in identity (76) as a factorial series in 7

18 the u i s We obtain ( q mr+ )C m+r (t,q) um m = m DOMINIQUE FOATA AND GUO-NIU HAN u m r+ r (t 2 ;q 2 ) 2+ m ( t 2 q 2+2 m )C m (t,q) um u m r+ (t 2 ;q 2 ) 2+ m m q m r+ ( t 2 )C m (tq,q) um u m r+ (t 2 ;q 2 ) 2+ m Taking the coefficients of u m u m r r um r+ r yields ( q m r+ )C m+r (t,q) = ( t 2 q 2+2 m )C m (t,q) q m r+ ( t 2 )C m (tq,q), which in its turn is equivalent to (76) in view of (7) All the steps of the argument are reversible Remark The fact that B m (t,q) is the generating polynomial for the class B m by the pair (fdes,fmaj) can also be proved by the insertion technique using (75) The argument has been already developed in [ClFo95a, 6] for ordinary words Again let m := m = m + +m r With each word from B m+ r associate (m r + ) new words obtained by marking one and only one letter equal to r or r Let Bm+ r denote the class of all those marked signed words If w = x x i x m+ is such a word, where the i-th letter is marked (accordingly, equal to either r or r), let mark i be the number of letters equal to r or r in the right factor x i+ x i+2 x m+ and define: On the other hand, let Clearly, fmaj w := fmajw+mark i w B m,k (q) := w B m+ r,fdesw=k q fmaj w B m,fdesw=k q fmajw w = (+q + +q m r )B m+r,k(q) Now each word w from the class B m gives rise to 2(m+) distinct marked signed words of length (m+), when the marked letter r or r is inserted between letters of w, as well as in the beginning of and at the end of the word As in the case of the signed permutations, we can verify that for each j = 0,,,2m+ there is one and only one marked signed word w of length (m+) derived by insertion such that fmaj w = fmajw+j On the other hand, fdes is not modified if r is inserted to the right of w, or if r or r is inserted into a descent x i > x i+ Furthermore, fdes increases by one, if x > 0 (resp x < 0) and r (resp r) is inserted to the left of w For all the other insertions fdes increases by 2 8

19 SIGNED WORDS AND PERMUTATIONS, III Hence,allthemarkedsignedwordsw fromb m+ r,suchthatfdesw = k are derived by insertion from three sources: (i) the set {w B m : fdesw = k} and the contribution is: (+q + +q m r+k )B m,k (q); (ii) the set {w B m : fdesw = k } and the contribution is: q m r+k B m,k (q); (iii) the set {w B m : fdesw = k 2} and the contribution is: (q m r+k + +q 2 m + )B m,k 2 (q) Remark 2 Let m := n and B n (t,q) = B m (t,q), so that B n (t,q) is now the generating polynomial for the set of signed permutations of order n Then (73), (74) and (75) become (77) (+t)b n (t,q) (t 2 ;q 2 = t s (+q + +q s ) n ) n+ s 0 (78) ( q)b n (t,q) = ( t 2 q 2n )B n (t,q) q( t)(+tq)b n (tq,q) (79) B n,k (q) = (+q + +q k )B n,k (q) +q k B n,k (q)+(q k +q k+ + +q 2n )B n,k 2 (q) The last three relations have been derived by Brenti et al [ABR0], Chow and Gessel [ChGe04], Haglund et al [HLR04] Concluding remarks The statistical study of the hyperoctahedral group B n was initiated by Reiner ([Re93a], [Re93b], [Re93c], [Re95a], [Re95b]) It had been rejuvenated by Adin and Roichman [AR0] with their introduction of the flag-major index, which was shown [ABR0] to be equidistributed with the length function See also their recent papers on the subject[abr05],[rero05] Another approach to Theorems and 2 would be to make use of the Cauchy identity for the Schur functions, as was done in [ClFo95b] References [AR0] Ron M Adin and Yuval Roichman, The flag major index and group actions on polynomial rings, Europ J Combin, vol 22, 200, p [ABR0] Ron M Adin, Francesco Brenti and Yuval Roichman, Descent Numbers and Major Indices for the Hyperoctahedral Group, Adv in Appl Math, vol 27, 200, p [ABR05] Ron M Adin, Francesco Brenti and Yuval Roichman, Equi-distribution over Descent Classes of the Hyperoctahedral Group, to appear in J Comb Theory, Ser A, 2005 [An76] George E Andrews, The Theory of Partitions Addison-Wesley, Reading MA, 976 (Encyclopedia of Mathand its Appl, 2) [Bo68] N Bourbaki, Groupes et algèbres de Lie, chap 4, 5, 6 Hermann, Paris, 968 9

20 DOMINIQUE FOATA AND GUO-NIU HAN [Ca56] L Carlitz, The Expansion of certain Products, Proc Amer Math Soc, vol 7, 956, p [ChGe04] Chak-On Chow and Ira M Gessel, On the Descent Numbers and Major Indices for the Hyperoctahedral Group, Manuscript, 8 p, 2004 [ClFo95a] Ron J Clarke and Dominique Foata, Eulerian Calculus, II: An Extension of Han s Fundamental Transformation, Europ J Combinatorics, vol 6, 995, p [ClFo95b] R J Clarke and D Foata, Eulerian Calculus, III: The Ubiquitous Cauchy Formula, Europ J Combinatorics, vol 6, 995, p [DeFo85] Jacques Désarménien and Dominique Foata, Fonctions symétriques et séries hypergéométriques basiques multivariées, Bull Soc Math France, vol 3, 985, p 3-22 [FoHa97] Dominique Foata and Guo-Niu Han, Calcul basique des permutations signées, I : longueur et nombre d inversions, Adv in Appl Math, vol 8, 997, p [FoHa05a] Dominique Foata and Guo-Niu Han, Signed Words and Permutations, I; a Fundamental Transformation, to appear in Proc Amer Math Soc, 2006 [FoHa05b] Dominique Foata and Guo-Niu Han, Signed Words and Permutations, II; The Euler-Mahonian Polynomials, Electronic J Combinatorics, vol (2) (The Stanley Festschrift), , R22 [GaGe78] Adriano M Garsia and Ira Gessel, Permutations Statistics and Partitions, Adv in Math, vol 3, 979, p [GaRa90] George Gasper and Mizan Rahman, Basic Hypergeometric Series London, Cambridge Univ Press, 990 (Encyclopedia of Math and Its Appl, 35) [HLR04] J Haglund, N Loehr and J B Remmel, Statistics on Wreath Products, Perfect Matchings and Signed Words, Europ J Combin, vol 26, 2005, p [Ra79] Don P Rawlings, Permutation and Multipermutation Statistics, Ph D thesis, Univ Calif San Diego Publ IRMA Strasbourg, 49/P-23, 979 [Ra80] Don P Rawlings, Generalized Worpitzky identities with applications to permutation enumeration, Europ J Comb, vol 2, 98, p [Re93a] V Reiner, Signed permutation statistics, Europ J Combinatorics, vol 4, 993, p [Re93b] V Reiner, Signed permutation statistics and cycle type, Europ J Combinatorics, vol 4, 993, p [Re93c] V Reiner, Upper binomial posets and signed permutation statistics, Europ J Combinatorics, vol 4, 993, p [Re95a] V Reiner, Descents and one-dimensional characters for classical Weyl groups, Discrete Math, vol 40, 995, p [Re95b] V Reiner, The distribution of descents and length in a Coxeter group, Electronic J Combinatorics, vol 2, 995, # R25 [ReRo05] Amitai Regev, Yuval Roichman, Statistics on Wreath Products and Generalized Binomial-Stirling Numbers, to appear in Israel J Math, 2005 [St76] Richard P Stanley, Binomial posets, Möbius inversion, and permutation enumeration, J Combinatorial Theory Ser A, vol 20, 976, p Dominique Foata Institut Lothaire, rue Murner F Strasbourg, France foata@mathu-strasbgfr Guo-Niu Han IRMA UMR 750 Université Louis Pasteur et CNRS 7, rue René-Descartes F Strasbourg, France guoniu@mathu-strasbgfr 20