A BIJECTION BETWEEN WORDS AND MULTISETS OF NECKLACES

Transcription

1 A BIJECTION BETWEEN WORDS AND MULTISETS OF NECKLACES I. GESSEL, A. RESTIVO, AND C. REUTENAUER 1. Introuction In [4] has been prove the bijectivity of a natural mapping Φ from wors on a totally orere alphabet onto multisets of primitive necklaces (circular wors) on this alphabet. This mapping has many enumerative applications; among them, the fact that the number of permutations in a given conjugation class an with a given escent set is equal to the scalar prouct of two representations naturally associate to the class an the set. The irect mapping is efine by associating to each wor its stanar permutation, an then replacing in the cycles of the latter each element by the corresponing letter of the wor. The inverse mapping φ 1 is constructe using the lexicographic orer of infinite wors. In the present article, we replace the stanar permutation by the costanar permutation. That is, the permutation obtaine by numbering the positions in the wor from right to left (instea from left to right as it is one for the stanar permutation). This a priori useless generalization has however striking properties. Inee, it inuces a bijection Ξ between wors an multisets of necklaces, which are intimately relate to the Poincaré- Birkhoff-Witt theorem applie to the free Lie superalgebra (instea of the free Lie algebra as it is the case for Φ). See Section 3 for the exact escription of these multisets. Another striking property of the bijection Ξ is hat the inverse bijection uses, instea of the lexicographic orer, the alternate lexicographic orer of infinite wors; this means that one compares the first letters of the infinite wors for the given orer of the alphabet, then if equality, the secon letters for the opposite orer, an so on. The alternate lexicographic orer is very natural. Inee it correspons to the orer of real numbers given by their expansion into continue fractions. This is well-known an use implicitly very often, see for example the book [3] on the the Markoff an Lagrange spectra. The proof of the bijectivity of Ξ that we give has as a byprouct a new proof of the bijectivity of Φ, that we give in Section 2. In Section 5 we recall the symmetric functions that are inuce by Φ, an then show that the symmetric functions inuce by Ξ are relate to the free Lie superalgebra an equal to the image of the formers uner the funamental involution of symmetric functions. Date: March 1, To our frien Toni Machí. 1

2 2 I. GESSEL, A. RESTIVO, AND C. REUTENAUER 2. Reminer: the lexicographic bijection Recall that the stanar permutation of a wor w = a 1 a n of length n on a totally orere alphabet A is the permutation σ, enote st(w), such that for any i,j {1,...,n}, the conition σ(i) < σ(j) is equivalent to a i < a j or a i = a j an i < j. The permutation st(w) may be obtaine by numbering from left to right the letters of w, starting witht the smallest letter, then the secon smallest, an so on. Equivalently, st(w) is the unique shortest permutation σ S n such that w.σ 1 is an increasing wor. Here, the length l(σ) of a permutation σ is as usually the number of its inversions an w.τ enotes the right action of the permutation τ S n on the wor w, efine by: w.τ = a τ(1) a τ(n). As an example, we have st(baacbcab) = , = an baacbcab = aaabbbcc. Two wors are conjugate if they may be written uv an vu for some wors u,v. A conjugation class is calle a necklace, or a circular wor. It is primitive if the wors in the class are not a nontrivial power of another wor; equivalently, the circular wor, viewe as regular n-gon with labels in A, is not fixe by any nontrivial rotation. If w is a wor, we enote the corresponing circular wor by (w), imitating the wor an cycle notation of permutations. We escribe now the bijection Φ of [4] between the set A of wors on A an the set M of multisets of primitive necklaces on A. Actually, we present a slight variant of it (replacing stanar permutation by its inverse) in orer to make it compatible with the Burrows-Wheeler transform. This is no essential change. Let w = a 1 a n be a wor an τ the inverse of its stanar permutation. With each cycle (j 1,...,j i ) of τ, associate the necklace (a j1,...,a ji ). It turns out that this necklace is primitive. The mapping Φ which associates with w the multiset of necklaces obtaine by taking all cycles of τ is bijective. Consier the example above. The cycles of τ are (1,2,3,7,4), (5) an (6,8). The corresponing multiset of necklaces is therefore Φ(w) = {(baaac),(b),(cb)}. In orer to escribe the inverse bijection, we follow [8]. Given a multiset of primitive necklaces, consier the corresponing multiset of wors (there are i wors for each necklace of length i); note that these wors are all primitive. Put them in orer, from top to bottom, in a right justifie tableau, where the orer is as follows: u < v if an only if uuuu... < vvvv... (lexicographic orer for infinite wors). Note that if there are nontrivial multiplicities in the multiset, then there are repeate rows.then the inverse image of the multiset is the last column, rea from top to bottom. As an example, take the previous multiset {(baaac),(b),(cb)}. The multiset of wors is {baaac, aaacb, aacba, acbaa, cbaaa, b, cb, bc}. The orer on the corresponing infinite wors is aaacb... < aacba... < acbaa... < baaac... < bb... < bc... < cbaaa... < cbcb... Thus the tableau is

3 A BIJECTION BETWEEN WORDS AND MULTISETS OF NECKLACES 3 a a a c b a a c b a a c b a a b a a a c b b c c b a a a c b Its last column is baacbcab as esire. We give now a new proof of the bijectivity of Φ. It is simpler than the proof in [4]. Let w, τ be as above. 1. If p < q, then a τ(p) a τ(q) : inee, this follows from the efinition of st(w) by numbering from left to right an from the fact that τ is the inverse of st(w). 2. If p < q an if a τ(p) = a τ(q), then τ(p) < τ(q): the efinition of st(w) by numbering from left to right shows that if i < j an a i = a j, then st(i) < st(j). Suppose now that p < q an a τ(p) = a τ(q). Put i = τ(p) an j = τ(q); then a i = a j ; if we ha τ(p) > τ(q), that is, i > j, we woul have st(i) > st(j), that is, p > q, a contraiction. Thus τ(p) < τ(q). 3. For p = 1,...,n, let w p = a τ(p) a τ 2 (p)...a τ k (p) where k is the length of the cycle of τ containing p. Note that τ k (p) = p an therefore the last letter of w p is a p. 4. The p-th row of the tableau associate to the multiset Φ(w) is w p : this is a consequence of the fact that if p < q, then w p < w q (lexicographical orer), which follows from 1. an 2., observing that w p = a τ(p)a τ 2 (p)a τ 3 (p) It follows that the last column of this tableau is w, which proves the injectivity of Φ. 6. If p < q an w p = w q, then τ i (p) < τ i (q): inee, since w p = w q, we have w τ i (p) = w τ i (q). Since moreover p < q, by 2., τ(p) < τ(q) (since w p = w q implies that their first letters coincie, that is, a τ(p) = a τ(q) ). This proves that τ i (p) < τ i () by inuction on i. 7. Suppose that some w p is not primitive. Then w p = u l, l 2. Let q = τ h (p), where h is the length of u. Then w p = w q. Moreover p q since h < w p = length of the cycle of τ at p. Suppose that p < q (the case p > q is similar). Then by 6., p < q, τ h (p) < τ h (q), τ 2h (p) < τ 2h (q),...,τ (l 1)h (p) < τ (l 1)h (q). By efinition of q, this means that p < τ h (p), τ h (p) < τ 2h (p), τ 2h (p) < τ 3h (p),...,τ (l 1)h (p) < τ lh (p) = p, a contraiction. 8. Let P be a set of representatives of the orbits of τ. Then Φ(w) is the multiset {(w p ) p P }. Therefore, by 7., the image of Φ is containe in M. Now, there are well-known length-preserving bijections between A an M (for example using factorization into Lynon wors), so the surjectivity of Φ follows. 3. The alternating lexicographic bijection The costanar permutation of a wor w = a 1 a n of length n on a totally orere alphabet A is the permutation σ, enote cost(w), such that

4 4 I. GESSEL, A. RESTIVO, AND C. REUTENAUER for any i,j {1,...,n}, the conition σ(i) < σ(j) is equivalent to a i < a j or a i = a j an i > j. The permutation cost(w) may be obtaine by numbering from right to left the letters of w, starting witht the smallest letter, then the secon smallest, an so on. For example, b a a c b c a b so that cost(baacbcab) = Let ω be the longest permutation in S n. Then it is easy to see that cost(w) = st(w.ω) ω. It follows from the similar property for st(w) that cost(w) is the unique longest permutation σ S n such that w.σ 1 is an increasing wor. We consier on infinite wors the alternating lexicographic orer: given two infinite wors a = a 0 a 1 a 2... an b = b 0 b 1 b 2..., we write a < alt b if either a 0 < b 0, or a 0 = b 0 an a 1 > b 1, or a 0 = b 0,a 1 = b 1 an a 2 < b 2, an so on. More formally, if for some i, one has a 0 = b 0,...,a i 1 = b i 1 an a i < b i if i is even an a i > b i if i is o. We enote by M the set of multisets of necklaces on A escribe as follows: M M if an only if the necklaces in M are of the form (w) for some primitive wor w, or of the form (ww) for some primitive wor of o length w; moreover, the necklaces of o length appear at most once in M. We efine now a mapping Ξ, similar to the mapping of Section 2, but with the stanar permutation replace by the costanar permutation. It will turn out that the image of this mapping will be M (instea of M), an that the inverse bijection may be compute by using the same kin of tableau as before, except that the lexicographic orer must be replace by the alternating lexicographic orer. Let w = a 1 a n be a wor an τ the inverse of its costanar permutation. With each cycle (j 1,...,j i ) of τ, associate the necklace (a j1,...,a ji ). The mapping Ξ associates with w the multiset of necklaces obtaine by taking all cycles of τ. Consier the example w = baacbcab above. The cycles of τ = = are (1,7,6), (2,3), (4,8) an (5). The corresponing multiset of necklaces is therefore Ξ(w) = {(bac),(aa),(cb),(b)}. In orer to the efine the inverse, we efine the alternating tableau of a multiset of necklaces M: associate with M the multiset of wors which is the union with multiplicities of the wors in the conjugation classes appearing in M (there are i wors, possibly repeate, for each circular wor of length i, so that the carinality of this multiset of wors is equal to the total length of M). Now put in orer, from top to bottom, these wors in a right justifie tableau, where the orer is the alternating orer of the infinite powers of these wors; that is: u is above v in the tableau if uuu... < alt vvv... If the infinite powers are equal, the corresponing wors are put in any orer. For example, consier the multiset {(bac),(aa),(cb),(b)}; its associate multiset of wors is {bac,acb,cba,aa,aa,cb,bc,b}; these are orere by acb... < alt aa... = aa... < alt bc... < alt bb... < alt bac... < alt cba... < alt cbcb... Therefore,

5 A BIJECTION BETWEEN WORDS AND MULTISETS OF NECKLACES 5 the alternating tableau is a c b a a a a b c b b a c c b a c b Theorem 3.1. The previous mapping is a bijection between A an M. The inverse mapping associates with a multiset the last column of its alternating tableau rea from top to bottom. For later use, we consier the set I of inepenent sets of necklaces: a set E of necklaces is calle inepenent if for any two circular wors (u),(v) in E, the wors u an v have no conjugate power. Lemma 3.1. There are canonical bijections between M, M an I. Proof. Note that necklaces are canonically in bijection with powers of Lynon wors. Using this ientification, M is the set of finite multisets of Lynon wors; moreover, an element of I is a finite set of powers of istinct Lynon wors; we then obtain a bijection from I to M by replacing each power w l of a Lynon wor w by the Lynon wor w with multiplicity l. Moreover, we obtain a bijection from M into M by replacing each Lynon wor of o length, having multiplicity l, by w 2 with multiplicity l/2 times if l is even; an by w 2 with multiplicity (l 1)/2 times together with w with multiplicity 1, if l is o. 4. Proof of the main result Recall from Section 3 that we enote by τ the inverse of the costanar permutation of w = a 1 a n. Lemma 4.1. If p < q, then a τ(p) a τ(q). Proof. The efinition of cost(w) by numbering from right to left implies that the numbere positions are successively τ(1), τ(2),... an so on. This implies the lemma. Lemma 4.2. If p < q an if a τ(p) = a τ(q), then τ(p) > τ(q). Proof. The efinition of cost(w) by numbering from right to left shows that if i < j an a i = a j, then cost(i) > cost(j). Suppose now that p < q an a τ(p) = a τ(q). Put i = τ(p) an j = τ(q); then a i = a j ; if we ha τ(p) < τ(q), that is, i < j, we woul have cost(i) > cost(j), that is, p > q, a contraiction. Thus τ(p) > τ(q). For p = 1,...,n, let w p = a τ(p) a τ 2 (p)... a τ k (p) where k is the length of the cycle of τ containing p. Note that therefore τ k (p) = p an the last letter of w p is a p. Let P be a set of representatives of the orbits of τ. Then observe that Ξ(w) is the multiset {(w p ) p P }.

6 6 I. GESSEL, A. RESTIVO, AND C. REUTENAUER Corollary 4.1. If p < q then w p alt w q. Proof. Since wp = a τ(p) a τ 2 (p)a τ 3 (p)..., the corollary immeiately follows from the two lemmas. We may euce the injectivity of the mapping Ξ from the corollary. By efinition of Ξ, the multiset of wors which is the union with multiplicities of the wors in the conjugation classes appearing in Ξ is exactly the multiset of wors w p, p = 1,...,n. Thus, the alternating tableau (as efine in Section 3) of Ξ(w) is by the corollary equal to the right justifie tableau whose p-th row is the wor w p. Since the last letter of that wor is a p, the injectivity follows, together with the construction of the inverse through the alternating tableau. Lemma 4.3. If p < q an w p = w q, then τ i (p) < τ i (q) if i is even an τ i (p) > τ i (q) if i is o. Proof. Note that if w p = w q, then w τ i (p) = w τ i (q). If moreover p < q, then by Lemma 4.2, τ(p) > τ(q) (since w p = w q implies that their first letters coincie, that is, a τ(p) = a τ(q) ). The lemma follows by inuction on i. Corollary 4.2. Let w p = u l for some wor u an some integer l 1. (i) if the length of u an l are both o, then l = 1; (ii) if the length of u is even, then l = 1. Proof. Let h be the length of u an q = τ h (p). We have therefore w p = w q. Note that hl is the length of w p, that is the length of the cycle of τ at p. We assume by contraiction that l > 1, so that p an q are not equal. We assume below that p < q, since the arguments are quite similar for p > q. (i) Suppose that h an l are both o. Then, since hl is o, we have by Lemma 4.3 τ hl (p) > τ hl (q). Now, τ hl (p) = p. Moreover, τ hl (q) = τ hl (τ h (p)) = τ h (p) = q. Thus, p > q, a contraiction. (ii) Suppose that h is even. Then by Lemma 4.3, p < q, τ h (p) < τ h (q), τ 2h (p) < τ 2h (q),...,τ (l 1)h (p) < τ (l 1)h (q). That is: p < τ h (p), τ h (p) < τ 2h (p), τ 2h (p) < τ 3h (p),...,τ (l 1)h (p) < τ lh (p) = p, a contraiction. We may show now that the image of Ξ is containe in the set M. It is enough to show that each wor w p is either primitive or the square of a primitive wor of o length; an that if w p = w q is of o length, then p = q. The corollary shows that if w p is not primitive, then w p = u l with u of o length an l even; if l 4, then l = 2m, m 2 an w p = v m, where v = u 2 is of even length, which contraicts the corollary; hence l = 2. Suppose that w p = w q is of o length k (which is the length of the cycle of τ at p an q); if p < q, then by Lemma 4.3, we have τ k (p) > τ k (q), that is, p > q, a contraiction. Thus we must have p = q. All this shows that the image of Ξ is containe in M. Now, there exist length-preserving bijections between A an M : for example using factorization into Lynon wors an using Lemma 3.1. Thus the surjectivity of Ξ follows.

7 A BIJECTION BETWEEN WORDS AND MULTISETS OF NECKLACES 7 5. Applications to symmetric functions an representations of the symmetric group 5.1. With the lexicographical bijection. We recall first the construction an the properties of the symmetric functions relate to the bijection Φ of [4]. Given a multiset M of necklaces (or circular wors) of total length n, we associate with it a partition of n, enote λ(m) an calle the cycle type of M: the parts of λ(m) are the lengths of the necklaces forming M, with multiplicities. The same construction applies to permutations in S n, viewe as multisets (actually sets) of their cycles. Let A be an infinite alphabet, which will be consiere as a set of noncommuting an of commuting variables, epening on the context. Given a wor a 1...a n an the corresponing necklace (a 1...a n ) on A, the evaluation of both of them is the commutative monomial a 1...a n in Z[A]. The evaluation of a multiset of primitive necklaces M is the monomial in Z[A] which is the prouct of the evaluations of all necklaces that form M. Fix a partition λ. In [4] (see also [11] Th. 9.41) is constructe the following element of Z[[A]]: P λ = ev(m), λ(m)=λ where the sum is over all multisets of primitive necklaces on A of cycle type λ. Since bijections from A into A preserve primitive necklaces, it is easy to see that P λ is a symmetric function. Note that for a partition with a single part n, this symmetric function P n is simply the sum of the evaluations of all primitive necklaces of length n; equivalently, the sum of all the evaluations of the Lynon wors of length n. It follows immeiately from the bijectivity of Φ that one has also P λ = ev(w), λ(st(w))=λ where the sum is over all wors on A whose stanar permutation has cyle type λ. This symmetric function may be escribe as follows. Let λ = 1 n 1...k n k: this means that the part i has multiplicy n i in λ. Then by the efinition of P λ one sees that P λ = P i n i. 1 i k Now, by the combinatorial efinition of plethysm (see [5] Section I.8), one sees that P i n i = h ni [P i ], where h n enotes the n-th homogeneous symmetric functions, with the notations of [5]. Thus we have P λ = h ni [P i ]. 1 i k These symmetric functions correspon to representations an characters of the symmetric groups as is escribe in [4, 11]. We briefly review the construction. Consier the algebra of noncommutative polynomials Q A

8 8 I. GESSEL, A. RESTIVO, AND C. REUTENAUER on the infinite set of noncommuting variables a A. The free Lie algebra is the smallest subspace L of Q A containing the variables in A an close uner the Lie bracket [f,g] = fg gf. Denote by L n the n-th homogeneous part of L. Denote (f 1,...,f l ) = 1 f l! σ(1) f σ(l). σ S l Now for any partition λ, enote U λ the subspace of Q A spanne by the elements (f 1,...,f l ), for all P i L λi. Then it is known (an follows from the Poincaré-Birkhoff-Witt theorem) that Q A = λ U λ, where the sum is over all partitions λ. Then P λ is the (Frobenius) character (or characteristic map as in [5]) of the representation of the symmetric group S n acting on the multilinear part of egree n of U λ ; equivalently, P λ is the multivariate generating function of U λ. In particular P n is the character of the n-th Lie representation. For later use, we recall the formula giving P n : P n = 1 µ()p n n, n where µ is the Möbius function an p the power sum symmetric function With the alternating bijection. We mimick now the previous contruction by replacing the multisets an the bijection. Recall that we enote by M the set of multisets of necklaces on A escribe as follows: M M if an only if the necklaces in M are of the form (w) for some primitive wor w, or of the form (ww) for some primitive wor of o length w; moreover, the necklaces of o length appear at most once in M. Fix a partition λ. Then efine Q λ = ev(m) Z[[A]]. M M, λ(m)=λ For the same reason as before, this is a symmetric function. By our bijection between M an wors, we have also Q λ = ev(w). w A, λ(cost(w))=λ We may as before escribe the symmetric function as follows. Denote by e n the n-th elementary symmetric function. Then the efinition of M implies that for λ = 1 n 1...k n k, we have (1) Q λ = 1 i k, i even h ni [Q i ] 1 i k, i o e ni [Q i ]. Recall from [5] that there is a funamental involution on the ring of symmetric functions, enote by ω.

9 A BIJECTION BETWEEN WORDS AND MULTISETS OF NECKLACES 9 Lemma 5.1. If n is twice an o number, then ω(p n ) is equal to the sum of the evaluations of the Lynon wors of length n together with the square of Lynon wors of length n/2. Proof. The sum of the evaluations of the Lynon wors of length n is equal to P n as note above. By efinition of plethysm, the sum of the evaluations of the squares of the Lynon wors of length m is equal to P m [p 2 ]; inee this sum is equal to the sum of the evaluations of the wors obtaine from Lynon wors of length m by oubling each letter in it. Thus, putting n = 2m, we have that the sum of the lemma is equal to P n + P m [p 2 ] = 1 µ()p n n + 1 m n m µ()p m [p 2 ]. Now, suppose that m is o. Since p [p 2 ] = p 2, the last sum is equal to 1 µ()p n n + 1 µ()p n n + 1 µ()p m m [p 2 ] n, o n, even = 1 µ()p n n + 1 n ( µ(2)p m µ()p m 2 ). n,o m m Since for o, µ(2) = µ(), this is equal to 1 n ( µ()p n + µ()p m 2 ). n,o m m On the other han, since ω(p ) = ( 1) 1 p, we have ω(p n ) = 1 µ()( 1) ( 1) n p n n. n Now ( 1) n = n n an n is even, so that the last sum is equal to 1 n ( µ()p n µ()p n ) n, o n, even = 1 n ( µ()p n µ(2)p m 2 ). n, o m This conclues the proof. Lemma 5.2. ([4] p. 201) If n is not twice an o number, then ω(p n ) = P n. We euce the following result. Corollary 5.1. Q λ = ω(p λ ). Proof. We first show that ω(p n ) = Q n. If n is not twice an o number, in view of Lemma 5.2, this amounts to show that P n = Q n. Now P n is the sum of the evaluations of the primitive necklaces of length n; an Q n is the sum of the evaluations of the necklaces in M of length n; these necklaces are primitive because of the efinition of M an the hypothesis on n. Hence P n an Q n are equal. Now suppose that n is twice an o number. Then in view of Lemma 5.1, we have to show that Q n is the sum of the evaluations of the Lynon wors

10 10 I. GESSEL, A. RESTIVO, AND C. REUTENAUER of length n an of the square of Lynon wors of length n/2. Now this follows from the efinition of M, since Q n is the sum of the evaluations of the necklaces of length n in M. By the formula above giving P λ an Q λ, an since ω is an automorphism, in orer to prove the lemma, it is enough to show that ω(h n [P i ]) = h n [Q i ] if i is even, an = e n [Q i ] is i is o. Now, for homogeneous symmetric functions f,g with g of egree i, it is well-known that ω(f[g]) = f[ω(g)] if i is even, an = ω(f)[ω(g)] if i is o. Hence the corollary follows from the equality ω(h n ) = e n an ω(p i ) = Q i. Remark There is an alternative proof of the corollary that works on the other sie of the bijections. Since it is of some interest, we sketch it now. It rests on the fact that if a multiset of monomials is etermine by a family of weak an strict inequalities on the variables, an if the sum of this multiset is a symmetric function f, then ω(f) is obtaine by interchanging weak an strict inequalities, see [6] Th.3.1. We show on an example that this result implies the corollary. Inee, consier the permutation σ = S n an a wor w of length 7, that we may write as w = a 5 a 3 a 7 a 1 a 6 a 2 a 4. Then st(w) = σ if an only if a 1 a 2 < a 3 a 4 < a 5 a 6 < a 7. Moreover cost(w) = σ if an only if a 1 < a 2 a 3 < a 4 a 5 < a 6 a 7. We recall now some facts on Lie superalgebras which will allow us to escribe the representation which has the character Q λ. Consier again Q A. The free oly generate Lie superalgebra is the smallest subspace L of Q A containing the variables in A an close uner the following operation (f,g are homogeneous polynomials): [f,g] := fg ( 1) eg(f)eg(g) gf. Denote by L n the n-th homogeneous part of L. Denote also, for homogeneous polynomials f 1,...,f l, (f 1,...,f l ) := 1 P ( 1) i<j,σ(i)>σ(j) eg(f i)eg(f j ) f l! σ(1) f σ(l). σ S l Then it is known (an follows from the super version of the Poincaré- Birkhoff-Witt theorem, see [10]) that Lemma 5.3. Q A = U λ, λ where the sum is over all partitions λ an where, for λ = λ 1 λ l, U λ is spanne by the elements (f 1,...,f l ), for all f i L λ i. We give a proof for the reaer s convenience. Proof. We claim that for any homogeneous polynomials f 1,...,f l an any permutation σ, one has f 1 f l ( 1) P i<j,σ(i)>σ(j) eg(f i)eg(f j ) f σ(1) f σ(l) mo.(l ) l 1 where the last symbol enotes the space spanne by proucts of no more than n 1 elements of L. The claim will be prove below. It implies that f 1...f l (f 1,...,f l ) mo.(l ) l 1.

11 A BIJECTION BETWEEN WORDS AND MULTISETS OF NECKLACES 11 Now let g 1,g 2,... enote a homogeneous basis of n even L n an h 1,h 2,... enote a homogeneous basis of n o L n. Orer the g i s an the h j s naturally, with the former before the latter. Then by [10] Th.3.1 (a version of the Poincaré-Birkhoff-Witt theorem), the set of all proucts in weakly increasing orer of the g i an h j s, with the restriction that a h j can appear at most once, forms a basis of Q < A >. The previous equation implies therefore that if each prouct f 1...f l is replace by (f 1,...,f l ), then this new set is also a basis. Therefore, since the latter operator is l-multilinear, the set of these elements with λ equal to the multiset {eg(f 1 ),...,eg(f l )} (consiering partitions as multiset of integers) is a basis of U λ. Thus the lemma follows. It remains to prove the claim. We o it by inuction on the length (number of inversions) of σ. If it is 0, there is nothing to prove. Otherwise we may write σ = α τ, with τ the ajacent transposition (k,k + 1) an α having one less inversion than σ. Write i(σ) = i<j,σ(i)>σ(j) eg(f i)eg(f j ). Then i(σ) = i(α)+eg(f α(k) )eg(f α(k+1) ) since σ(1)... σ(l) = α(1) α(k 1)α(k + 1)α(k)α(k + 2) α(l) an α(k) < α(k + 1) by assumption on the inversions. We have f α(k) f α(k+1) = [f α(k),f α(k+1) ] + ( 1) eg(f α(k))eg(f α(k+1) ) f α(k+1) f α(k). Thus, by multiplying appropriately on the left an on the right, we obtain f α(1)...f α(l) ( 1) eg(f α(k))eg(f α(k+1) ) f σ(1)...f σ(l). Now by inuction, f 1...f l ( 1) i(α) f α(1)...f α(l). Thus we obtain that f 1...f l ( 1) i(σ) f σ(1)...f σ(l), as esire. Theorem 5.1. Let λ be a partition of n. Then Q λ is the character of the symmetric group S n acting on the multilinear part of U λ. Equivalently it is the multivariate generating function of U λ. Proof. It suffices to prove the secon assertion. We consier first the case of a partition λ with only one part n. We have to fin a finely homogeneous (that is, homogeneous with respect to each variable) basis of U λ = L n whose multivariate generating function is Q λ = Q n. This will follow from the theory of Lynon bases, or more generally of Hall bases, aapte to the Lie superalgebra case, see [9], [11] Inee, a basis of L n is obtaine as follows. Take a set of Hall trees an evaluate it in L by interpreting the binary operation as the bracketing [,]. Then L has as basis the set of all polynomials f obtaine in this way, together the polynomials [f,f] with f of o egree. Since the multivariate generating function of the Hall trees (or equivalently Lynon wors) of egree n is classically equal to P n (see e.g. [11] Th. 7.2), we see by Lemma 5.1 an 5.2 that L n has a multihomogeneous basis whose multivariate generating function is ω(p n ), equal to Q n by Cor.5.1. Consier now a general partition λ. The proof of Lemma 5.3 shows that U λ has the following basis: the set B λ of (f 1,...,f l ), where f 1,...,f l is a weakly increasing sequence of elements in the set B = {g 1,g 2,...,h 1,h 2,...}. Here the f i s an the g j s are as in the proof of this lemma, an we may even assume that they are finely homogeneous. Then the basis B λ that we

12 12 I. GESSEL, A. RESTIVO, AND C. REUTENAUER obtain is also finely homogeneous. It follows, by Eq. (1) an the efinition of plethysm, that its multivariate generating function is Q λ. References [1] M. Burrows, D.J. Wheeler, A block sorting ata compression algorithm, Technical report, DIGITAL System Research Center, [2] M. Crochemore, J. Désarménien, D. Perrin, A note on the Burrows-Wheeler transformation, Theoretical Computer Science 332, 2005, [3] T.W. Cusick an M.E. Flahive, The Markoff an Lagrange spectra, Amer. Math. Soc [4] I. Gessel, C. Reutenauer, Counting permutations with given cycle structure an escent set, J. Combin. Theory. A 64, 1993, [5] I. Maconal, Symmetric functions an Hall polynomials, Cambrige University Press, [6] C. Malvenuto, C.Reutenauer, Plethysm an conjugation of quasi-symmetric functions, Discrete Mathematics 193 (1998) [7] S. Mantaci, A. Restivo, M. Sciortino, Burrows-Wheeler transform an Sturmian wors, Information an Processing Letters 86, 2003, [8] S. Mantaci, A. Restivo, G. Rosone, M. Sciortino, An extension of the Burrows- Wheeler Transform, Theoretical Computer Science 387, 2007, [9] G. Mélançon, Réécritures ans le groupe libre, l algèbre libre et l algèbre e Lie libre, PhD, Université u Québec à Montréal, [10] R. Ree, Generalize Lie elements, Canaian Journal of Mathematics 12, 1960, [11] C. Reutenauer, Free Lie algebras, Oxfor University Press, Ira Gessel, Department of Mathematics, Braneis University aress: gessel@braneis.eu Antonio Restivo, Dipartimento i matematica, Università i Palermo aress: restivo@math.unipa.it Christophe Reutenauer: Département e mathématiques, Université u Québec à Montréal aress: reutenauer.christophe@uqam.ca