An analogue to the Witt identity

Transcription

1 An analoue to the Witt identity G. A. T. F. da Costa 1 and G. A. Zimmermann 2 Departamento de Matemática Universidade Federal de Santa Catarina Florianópolis-SC-Brasil arxiv: v1 [math.co] 17 Feb 2013 Abstract In this paper we solve combinatorial and alebraic problems associated with a multivariate identity first considered by S. Sherman which he called an analo to the Witt identity. We extend previous results obtained for the univariate case. Keywords: Sherman identity, paths countin, eneralized Witt formula, free Lie alebras Mathematical Subject Classification: 05C30, 05C25, 05C38 1 Introduction In [13] S. Sherman considered the formal identity in the indeterminates z 1,...,z n : 1.1 m 1,...,m R 0 1+z m z m R R N + 1 z m z m R R N R 1+z j 2 where N + and N are the numbers of distinct classes of equivalence of nonperiodic closed paths with positive and neative sins, respectively, which traverse without backtrackin m i times ede i, i 1,...,R, of a raph G R with R > 1 edes formin loops counterclockwisely oriented and hooked to a sinle vertex, m i 1. In [13] Sherman refers to equation 1.1 as an analo to the Witt identity. The reasonwill becomeclear soon. Sherman identity, asweprefer tocall it, forshort, isa special non trivial case of another identity called Feynman identity first conjectured by Richard Feynman. This identity relates the Euler polynomial of a raph to a formal product over the classes of equivalence of closed nonperiodic paths with no backtrackin in the raph and it is an important inredient in a combinatorial j1 1.costa@ufsc.br 2 raciele@ifsc.edu.br 1

2 formulation of the Isin model in two dimensions much studied in physics. In [12] S. Sherman proved Feynman identity for planar and toroidal raphs and recently this identity was proved in reat enerality by M. Loebl in [6] and D. Cimasoni in [4]. Sherman compared equation 1.1 with the multivariante Witt identity [14]: z m z m R R Mm 1,...,m R 1 m 1,...,m R 0 R i1 z i 1.3 Mm 1,...,m R µ m 1,...,m R N! N m 1!...m R! where N m m R > 0, µ is the Möbius function defined by the rules: a µ+1 +1, b µ 0, for p e p eq q, p 1,...,p q primes, and any e i > 1, c µp 1...p q 1 q. The summation runs over all the common divisors of m 1,...,m R. Oriinally, Witt identity appeared associated with Lie alebras. In this context the formula ives the dimensions of the homoeneous subspaces of a finitely enerated free Lie alebra L. If Lm 1,...,m R is the subspace of L enerated by all homoeneous elements of multideree m 1,...,m R, then diml M. However, formula 1.3 has many applications in combinatorics as well [11]. Specially relevant is that M can be interpreted as the number of equivalence classes of closed non periodic paths which traverse counterclockwisely the edes of G R, the same raph associated to Sherman identity 1.1. This property is stated in [13] without a proof but this combinatorial interpretation of Witt formula can be understood reinterpreted as a colorin problem of a necklace with N beads with colors chosen out of a set of R colors such that the coloured beads form a nonperiodic confiuration. In another words, Mm 1,...,m R is the number of nonperiodic coloured necklaces composed of m i ocurrences of the color i, i 1,...,R. In [13] Sherman called attention to this association of both identities 1.1 and 1.2 to paths in the same raph which motivated him to consider the problem of findin a relation of 1.1 to Lie alebras. Interpretin 1.1 in alebraic terms means to relate the exponents N ± to some Lie alebraic data. An investiation of Sherman s problem was initiated in [8] and [9] and a solution obtained for the univariate case of identity 1.1. In the present paper we solve the problem in the multivariate formal case which requires important improvements. The countin method developed in [8] and [9] is based on a sin formula for a path iven in terms of data encoded in the word representation for the path. It played a crucial role in ettin formulas for N ± in the univariate case. However, the countin method based on this sin formula is complicated. In the present paper we make improvements in the countin method in order to apply it to the multivariate case without dependin too much on the sin formula. The formula is used here only to prove a simple Lemma. 2

3 In [10] S-J. Kan and M-H. Kim derived dimension formulas for the homoeneousspacesofeneralfreeradedliealebras. Weusesomeoftheir resultstosolve Sherman s problem. At the same time our results ive a combinatorial realization for some of theirs in terms of paths in a raph. The paper is oranized as follows. In section 2, we recall the word representation of a path and some basic definitions. A basic Lemma about the distribution of sins in the set of words of iven lenth is proved. In section 3, we compute formulas for the numbers of equivalence classes of closed nonperiodic paths of iven lenth. The first of these eneralizes Witt formula in the sense that it counts paths that traverse the edes of the raph in all directions and no backtrackins. The other formulas ive the exponents in Sherman s identity 1.1. We also interpret these formulas in terms of a colourin problem. Sherman s problem, that is, to ive an alebraic meanin to the exponents in 1.1 is solved in section 4. 2 Preliminaries A path in G R is an ordered sequence of the edes which does not necessarily respect their orientation. A path is closed and subjected to the constraint that it never oes immediately backwards over a previous ede. Given G r G R, denote by i 1,..., an enumeration of the edes of G r in increasin order. A closed path of lenth N r in G r is best represented by a word of the form 2.1 D e j 1 j 1 D e j 2 j 2...D e j l j l where l r,r +1,...,N, j k {i 1,..., }, j k j k+1, j l j 1, and l e jk N k1 All edes of G r are traversed by a path so that each i k appears at least once in the sequence S l j 1,j 2,...,j l. The order in which the symbols D e j j appear in the word indicates the edes traversed by p and in which order. If the sin of e j is positive neative the path traverses e j times ede j followin the opposite of ede s orientation. A word is called periodic if it equals D e j 1 j 1 D e j 2 j 2...D e jα j α for some > 1 and the word between parenthesis is nonperiodic. Number is called the period of the word. Permutin circularly the symbols D e j j in 2.1 one ets l 3

4 words that represent the same closed path. For example, the word D1 2 D+1 2 D+1 1 D+3 2 is a circular permutation of D 2 +1 D 1 +1 D2 3 D1 2. Circular words are taken to be equivalent because they represent the same closed path. Althouh this is also true for a word and its inversion D e j l j l...d e j 1 j 1 they are not taken equivalent here. This is the reason for the exponent 2 on the riht side of 1.1 as oriinally in [13]. In section 3 we consider sined paths. The sin of a path is iven by the formula 2.2 sinp 1 1+np where np is the number of interal revolutions of the tanent vector of p. From this definition it follows that if p h is a periodic path with odd period, then sinp sinh. If is even, sinp 1. The sin of a path can be computed from its word representation 2.1 usin the formula [9] N+l+T+s+1 where T is the number of subsequences in the decomposition of S l into subsequences see [9] for definition and example of a decomposition and s is the number of neative exponents in 2.1. It follows from the previous sin formulas that periodic words with even period have neative sin. The followin Lemma is important in the proof of several results in section 3. It was assumed in [8] and [9] without a proof. Lemma 2.1 Given G r G R, consider all paths that traverse each ede of G r at least once no backtrackin allowed and the set of all representative words periodic or not, circular permutations and inversions included of fixed lenth N r > 1. Then, half of the words has positive sin and the other half has neative sin. Proof: It suffices to consider the subset of words associated to a fixed sequence S l j 1,j 2,...,j l. For this sequence the numbers N, l and T are fixed. The words with these numbers have sins which depend only ons {0,1,2,...,l}. For N+l+T even, the sin of a word is 1 s+1. If l 2k there are for each odd value of s 2k s words with positive sin. Summin over the odd values of s we et the total number of 2 2k 1 words with positive sin. Summin over the even values of s we et the same number of words with neative sin. If l 2k+1 a similar countin ives 2 2k words with positive neative sins. The case N +l +T odd is analoous. 4

5 3 Countin paths in G r Fix a subraph G r G R. Call θ ± m i1,...m ir the number of equivalence classes of closed nonperiodic paths of lenth N r and ± sins that traverse m i1 times ede i 1,..., m ir times ede of G r, m ij > 0, j 1,...,r, with no backtracks, m i m ir N and zero times the edes in G R \G r. In this section we derive formulas for θ : θ + + θ and θ ±. Notice that θ ± is just another name for the exponents N ± in 1.1 showin only the nonzero entries in N ±. Firstly, we compute θ. In the case r 1, a path with m i > 1 is periodic. The non periodic ones are two, the path with lenth N 1 and its inversion so that θm i 0 if m i > 1 and θm i 2, if m i 1. In the other cases, θ is iven next. Theorem 3.1 For r 2, define 3.1 F, m i 2 M a1 2 2a a where M min{m i1,m i2 } and, if r 3, 3.2 F,..., m N ar 1 a 1 2 a a r {S a} c1 mi2 1 a 1 mic 1 t ic 1 where {S a } is the set of sequences j 1,...,j a such that j k {i 1,..., } and j k j k+1, j a j 1. Number t ic counts how many times ede i c occurs in a sequence S a. Use is made of the convention that the combination symbol in 3.2 is zero whenever t ic > m ic. Then, 3.3 θm i1,...,m ir F,..., m m i1,...m ir µ The summation is over all the common divisors of m i1,...,m ir, and µ is the Möbius function. Proof: The number Kl,m i1,...m ir of words with the same values of m i1,...,m ir and l {r,r +1,...,N} is iven by Kl,m i1,...,m ir 2 l {Sl} r c1 mic 1 n ic 1 Let s explain this formula a bit. Number n ic counts the number of ocurrences of ede i c in a sequence S l j 1,...,j l. The combination symbol counts the number of 5

6 unrestricted partitions of m ic into n ic nonzero positive parts [1] so that the product times 2 l there are 2 l ways of assinin + and sins to the exponents in 2.1 ives the total number of words representin paths which traverse m i1 times ede i 1,..., m ir times ede of G r G R in all possible ways. Then, one sums over all sequences S l with the convention that a combination symbol equals zero whenever m < n. In the set of Kl,m i1,...,m ir words there is the subset of nonperiodic words plus their circular permutations and inversions and the subset of periodic words if any whose periods are the common divisors of l, and m i1,...,m ir plus their circular permutations and inversions. Denote by Kl,m i1,...,m ir the number of elements in the former set. The words with period are of the form D e k 1 k 1 D e k 2 k 2...D e kα k α where α l/, and D e k 1 k 1 D e k 2 k 2...D e kα k α is nonperiodic so that the number of periodic words with period plus their circular permutations and inversions is iven by Kl/,m i1 /,...,m ir /. Therefore, Kl,m i1,...,m ir l,k,m i1,...,m ir K l, m i 1,..., m The summation is over all the common divisors of l,m i1,...,m ir. Applyin Möbius inversion formula [2] it follows that l 3.4 Kl,m i1,...,m ir µk, m i 1,..., m l,m i1,...,m ir where µ is the Möbius function. To eliminate circular permutations divide 3.4 by l. Summin over all possible values of l one ets a formula for the number θm i1,...m ir : 3.5 θm i1,...,m ir N lr Kl,m i1,...,m ir l Upon substitution of 3.4 into 3.5 one ets, for the case r 3, θm i1,...,m ir N lr 1 l l,m i1,...,m ir µ2 l r {S l } c1 mic 1 n ic 1 Proceed now as follows. For a iven common divisor of m i1,...,m ir, sum over all valuesofl whicharemultipleof. Then, sumoverallpossibledivisorsofm i1,...,m ir. Write l a, and n t. In the case r 3 one has r/ a N/ but unless 6

7 1 it is not admissible to have a < r because all r edes of the raph should be traversed. For this reason, r a N/. Result 3.2 follows. In the case r 2, l is even and, for each l, only sequences of the form i 1,i 2,...,i 1,i 2 with n i1 n i2 l 2 are possible. Put l 2a, a 1,2,...,M min{m 1,m 2 } to et 3.1. Example 1. From 3.1, F1,1 F1,2 F2,1 F1,3 F3,1 4, F2,2 12, F1,4 F4,1 F1,5 F5,1 4, F2,3 F3,2 20, F2,4 F4,2 28, F3, From 3.3, θ1,1 θ1,2 θ2,1 3 θ1,3 θ3,1 θ1,4 θ4,1 θ1,5 θ5,1 4, θ2,2 10, θ2,3 θ3,2 20, θ3,3 56. Example 2. From 3.2, F1,1,1 16, F1,1,2 F1,2,1 F2,1,1 32, F1,2,2 F2,1,2 F2,2,1 112, F1,1,3 F1,3,1 F3,1,1 48, F1,1,4 F1,4,1 F4,1,1 64, F1,2,3 F3,1,2 F2,3,1 F3,2,1 F1,3,2 F2,1,3 256, F2,2, From 3.3, θ1,1,1 16, θ1,1,2 θ2,1,1 θ1,2,1 32, θ1,2,2 θ2,1,2 θ2,2,1 112, θ1,1,3 θ3,1,1 θ1,3,1 48, θ1,1,4 θ4,1,1 θ1,4,1 64, θ1,2,3 θ3,1,2 θ2,3,1 θ3,2,1 θ1,3,2 θ2,1,3 256, θ2,2, Remarks. a Notice that θ, likewise Witt formula, is iven in terms of Möbius function. However, formula 3.3 counts closed nonperiodic paths traversin the edes of G R in all directions and no backtrackin and in that sense eneralizes Witt formula. Also, our formula has an alebraic meanin of a dimension. See section 4. b If m i1,...,m ir are coprime, F θ. Otherwise, F can be rational. For instance, F3,3 172/3. But F : NF, N m i m ir, is always a positive inteer which counts the number of all words of lenth N. For example, in the case N 4, m 1 m 2 2, F 48. ThewordsareD 1 ±2 D 2 ±2, D1 1 D 2 +1 D 1 +1 D 2 +1, D 1 +1 D2 1 D 1 +1 D 2 +1, D1 1 D 1 2 D+1 1 D+1 2, D 1 1 D+1 2 D+1 1 D 1 2, D 1 1 D 1 2 D 1 1 D+1 2, and D 1 1 D 1 2 D+1 1 D 1 2, plus four circular permutations for each of them, and the four periodic words D ±1 1 D ±1 2 2 plus two circular permutations for each. In terms of F, θm i1,...,m ir 1 N m i1,...m ir µf m i1,..., m Althouh the Möbius function is neative for some divisors, nevertheless the riht hand side is always a positive number because F m i 1,..., m ir counts words in a subset of the words counted by F m i1,...,m ir. c Given a circular necklace with N beads consider the problem of countin inequivalent nonperiodic colourins of these beads with 2r colors {c i,c i }, i 1,...,r, 7

8 with m i occurrences of the index i, N m i, with the restriction that no two colors c i and c i same index occur adjacent in a colourin. Now, consider an oriented raph with r loops hooked to a sinle vertex. Each loop ede corresponds to a color c i. A nonperiodic closed nobacktrackin path of lenth N in the raph corresponds to a colourin and a color c i corresponds to an ede bein traversed in the opposite orientation. The presence of a sinle vertex in the raph reflects the fact that adjacent to a bead with, say, color c i any other with distinct index may follow. The number of inequivalent colourins is iven by θ. As a basic test of our countin ideas, we prove Sherman s statement in [13] relatin Witt formula to paths in G R : Proposition 3.2 Relative to raph G R, formula 1.2 ives the number M of equivalence classes of closed non periodic paths of lenth N > 0 which traverse counterclockwisely m i 0 times ede i, i 1,2,...,R, m m R N. Proof: Denote by m i1,...,m ir, r R, the non zero entries in Mm 1,...,m R which we call M r m i1,...,m ir. Words representin counterclockwise paths have positive exponents so that the factors 2 2a and 2 a in formulas 3.1 and 3.2 are not needed, hence, 3.6 M r m i1,...,m ir where 3.7 F c, m i 2 µ F c m i1,...,m ir M 1 1 a a 1 a1 with M min{m i1,m i2 }, if r 2; and 3.8 F c,...m N 1 a ar r {S a} c1,..., m mi2 1 a 1 mic 1 t ic 1 if r 3. In the case r 2 suppose m i1 m i2. Usin formula 5.3 with l 2, section 5, it follows that m i1 a1 1 1 a a 1 mi2 1 a 1 m i2 N! + mi 2 m i1 N m i 1!m i 2! 1 8

9 Similarly, if m i2 m i1. In the case r 3 define I, 3.9 I m i > 0 m i m ir N F c,..., m Upon substitution of 3.8 into 3.9 and exchanin the summation symbols, we et N 1 r mic 1 I a t ic 1 ar {S a} c1 m i > 0 m i m ir N Applyin Lemma 5.2, section 5, I N 1 a ar {S a} N 1 a 1 N 1 N 1 a a 1 ar rw r a where rwa r r 1 r+j j j1 is the number of sequences in {S a } [9]. Usin that and we et N 1 N 1 a a 1 ar N 1 N 1 a a 1 ar 3.10 I N j 1 a + 1 a+r j 1 a N j N 1 1 a+r 1 r+1 N r r 1 r+j j j1 j N Stirlin numbers S N,r of second kind are iven by the formula [3] 3.11 S N,r 1 r! r r 1 k k k0 r k N 1 r! r r 1 r+j j j0 j N 9

10 so that 3.12 I r! N S N,r Stirlin numbers have the property that 3.13 m i > 0 m i m ir N N! r!s!...m ir! m i 1 Comparin relations 3.12, 3.13 and 3.9, 3.14 F c,..., m N N! m 1!...mr! Upon substitution of 3.14 into 3.6 the result follows. In the sequel we compute formulas for θ + and θ. N,r Theorem 3.3 Suppose any of the followin conditions is satisfied: a N m i m ir < 2r; b m i1,...,m ir are coprime; c m i1,...,m ir are not all odd nor even; d m i1,...,m ir are all odd. Then, 3.15 θ m i1,...,m ir θ + m i1,...,m ir Proof: Similar to Theorem 1 in [8] usin Lemma 1.. The case where m i1,...,m ir are all even numbers is iven in the next theorem. Theorem 3.4 The number θ + m i1,...m ir is iven by µ 3.16 θ + m i1,...,m ir G,..., m odd m i1,...,m ir where the summation is over all the common odd divisors of m i1,...,m ir, and G F 2 with F as in 3.1 and 3.2. Suppose m i1,...,m ir are all even numbers. Then, 3.17 θ m i1,...,m ir θ + m i1,...,m ir θ + 2,..., m 2, Proof: First, suppose that all common divisors of m i1,...,m ir are odd numbers. In this case, µ θm i1,...,m ir F,..., m odd m i1,...,m ir 10

11 Since θ θ + +θ and θ + θ Theorem?? it follows that θ 2θ +, hence, 3.18 θ + 1 µ 2 F,..., m odd m i1,...,m ir If the numbers m i1,...,m ir are all even then aain θ + is iven by 3.18 for in this casethem i shave commondivisorswhich areeven numbers butsinceperiodicwords with even period have neative sin, hence, only the odd divisors are relevant to et θ +. The reason why one should have the factor 1/2 is that by Lemma 1 when one considers the set of all possible words representin paths of a iven lenth which traverse m i1,...,m ir times the edes of G r, half of them have positive sin and the other half have neative sin. To account for the positive half one needs the factor 1/2. Let s now compute θ in the even case. Write θ 1 2 µ F + µ F odd m i1,...,m ir even m i1,...,m ir + µ F odd m i1,...,m ir even m i1,...,m ir µ F µ 2θ + + F even m i1,...,m ir Usin that θ θ + +θ, it follows that µ θ θ + + F even m i1,...,m ir µ F odd m i1,...,m ir Now, the relevant even divisors are {2n} where n are the odd common divisors of {m i }. For the other possible divisors if any use that µ2 j n 0, j 2. Usin that µ2n µn the summation over the even divisors is equal to provin the result. θ + m i 1 2,..., m 2 Remark. Likewise θ, the numbers θ ± can be interpreted as the number of inequivalent nonperiodic colourins of a circular necklace with N beads. However, now these colourins are classified as positive or neative accordin to formula 2.3. It is positive neative if the number N+l+T +s is odd even. In this case, s is the number of c colors present in a colourin. Interpret T in terms of the color indices. 11

12 Definition. Let s 1,...,s r be arbitrary positive inteers. Let the number P be defined as follows. If s 1,...,s r are all even numbers, µ 3.19 Ps 1,...,s r G s1,..., s r even s 1,...,s r Otherwise, Ps 1,...,s r 0. Also, define { Gs1,...,s r if s 1,...,s r not all even 3.20 H Gs 1,...,s r 1 k s 1,...,s r k Ps 1 k,..., sr otherwise k Lemma P µ s 1,...,s r G H Proof: From the above definition, G H if s 1,...,s r not all even. Otherwise, G H 1 P s1,..., s r s 1,...,s r Now, apply Lemma 5.1, section 5, to et the result. Theorem θ + m i1,...,m ir µ H,..., m m i1,...,m ir Proof: When m i1,...,m ir are not all even, their odd divisors are the only possible common divisors. In this case, P 0 and µ θ + H odd m i1,...,m ir with H G. In the case m i1,...,m ir are all even the sum over odd divisors of 12

13 m i1,...,m ir can be expressed as θ + µ G odd m i1,...,m ir µ G µ G m i1,...,m ir even m i1,...,m ir µ G P m i1,...,m ir µ G m i1,...,m ir m i1,...,m ir µ H µ m i1,...,m ir G H Example 3. θ ± 1,1 θ ± 1,2 θ ± 2,1 θ ± 1,3 θ ± 3,1 θ ± 1,4 θ ± 4,1 θ ± 1,5 θ ± 5,1 2, θ + 2,2 6, θ 2,2 4, θ ± 2,3 θ ± 3,2 10, θ + 2,4 14, θ 2,4 12, θ + 4,2 14, θ 4,2 12, θ ± 3,3 28. Example 4. θ ± 1,1,1 8, θ ± 1,1,2 θ ± 2,1,1 θ ± 1,2,1 16, θ ± 1,2,2 θ ± 2,1,2 θ ± 2,2,1 56, θ ± 1,1,3 θ ± 3,1,1 θ ± 1,3,1 24,θ ± 1,1,4 θ ± 4,1,1 θ ± 1,4,1 32, θ ± 1,2,3 θ ± 3,1,2 θ ± 2,3,1 θ ± 3,2,1 θ ± 1,3,2 θ ± 2,1,3 128, θ + 2,2,2 524, θ 2,2, Sherman identity and Lie alebras In this section we relate our previous results with Lie alebras and solve Sherman s problem. The solution is provided by the followin proposition by S. -J. Kan and M. -H. Kim in [10]. Proposition 4.1 Let V k 1,...,k r Z>0V r k1,...,k r be a Z r >0-raded vector space over C with dimv k1,...,k r dk 1,...,k r <, for all k 1,...,k r Z r >0, and let L k 1,...,k r Z>0L r k1,...,k r be the free Lie alebra enerated by V. Then, the dimensions of the subspaces L k1,...,k r are iven by 4.1 diml k1,...,k r k 1,...,k r µ W k1,..., k r 13

14 where summation is over all common divisors of k 1,...,k r and W is iven by 4.2 Wk 1,...,k r s Tk 1,...,k r s 1! s! i 1,...,1 The exponents s i1,..., are the components of s T, di 1,..., s i 1,...,ir 4.3 Tk 1,...,k r {s s i1,..., s i1,..., Z 0, and 4.4 s i 1,...,1 i 1,...,1 s i1,...,, s! Moreover, the numbers diml k1,...,k r satisfy s i1,..., i 1,..., k 1,...,k r }, i 1,...,1 s i1,...,! 4.5 k 1,...,k r1 1 z k zr kr diml k 1,...,kr 1 fz 1,...,z r where 4.6 fz 1,...,z r : k 1,...,k r1 dk 1,...,k r z k zkr r This function is associated with the eneratin function of the W s, 4.7 z 1,...,z r : by the relation k 1,...,k r1 4.8 e 1 f Wk 1,...,k r z k z kr r Identity 4.5 is a consequence of the famous Poincaré-Birkhoff-Witt theorem for the free Lie alebra. Computation of the formal loarithm of the left hand side of 4.5 and its expansion ives that the infinite product equals the exponential in 4.8. Raise both members of 4.5 to the power 1, compute the formal loarithm of both members and expand them. Identification of the coefficients of the same order, definition 4.2 and application of Möbius inversion ives 4.1. See [10] for 14

15 details. In [10], 4.1 is called the eneralized Witt formula, W is called the Witt partition function and 4.5 the eneralized Witt identity. Formulas 3.3 and 3.22 have exactly the form of 4.1 with correspondin Witt partition functions iven by F, H, respectively, so we will interpret θ and θ + as ivin the dimensions of the homoeneous spaces of raded Lie alebras. In each case, the alebra is enerated by a raded vector space whose dimensions can be computed recursively from 4.2 as a function of the Witt partition function. However, a eneral formula can be obtained from 4.8 usin 4.6 as the formal Taylor expansion of 1 e. This ives dk 1,...,k r k 1!...k r! with 4.10 z 1,...,z r : k z k zkr r k 1,...,k r1 1 e z1...z r0 Wk 1,...,k r z k 1...z kr and W F,H iven by 3.1, 3.2, Furthemore, diml k1,...,k r θ,θ + iven by 3.3, 3.22 satisfy the eneralized Witt identity 4.5 with the correspondin dimensions iven by 4.9. In fact, an explicit formula for 4.9 can be derived as follows: Theorem 4.2 A formula for the numbers dk 1,...,k r is k 4.11 dk 1,...,k r 1 λ+1 λ1 pλ,k i1 q [Wl i1,...,l ir ] a i where k k k r, q 1+ r i1 k i+1, p λ,k is the set of all a i {0,1,2,...} such that q i1 a i λ, q i1 a il ij k j, and the vectors l i l i1,...,l ir, l ij satisfyin 0 l ij k j, j 1,...,r, i 1,...,q and r j1 l ij > 0. Set Wl i 0 if l ij 0 for some j; otherwise, W is the Witt partition function. Proof: A eneralization of Faà di Bruno s relation due to Constantine and Savits in [5] and [7] ives a formula for the k -th derivative of the exponential of a function z 1,...,z r. From this formula and 4.9, 4.11 follows.. Example 5. We compute d2,2, explicitly. In this case, k 1 k 2 2, k 4, q 8. The possible vectors l 2,2 are l 1 0,1, l 2 1,0, l 3 1,1, l 4 0,2, l 5 2,0, l 6 2,1, l 7 1,2 and l 8 2,2. Next we ive the values of a 1,...,a 8 0 satisfyin a i! 8 a i λ, i1 8 a i l i 2,2 i1 15

16 Define the vector a a 1,...,a 8. The possible a s for each λ are as follows. For λ 1, a 0,...,0,1. For λ 2, 0,1,0,0,0,0,1,0, 0,0,2,0,0,0,0,0, 0,0,0,1,1,0,0,0,1,0,0,0,0,1,0,0. Forλ 3,0,2,0,1,0,0,0,0,2,0,0,0,1,0,0,0, 1,1,1,0,0,0,0,0. For λ 4, 2,2,0,0,0,0,0,0. We et The dimensions up to d3,3 are: d2,2 W2,2 1 2 W1,12 N 2 N 3 N 4 d1,1 W1,1 d1,2 W1,2, d2,1 W2,1 d1,3 W1,3, d3,1 W3,1 d2,2 W2,2 1 2 W1,12 N 5 N 6 d1,4 W1,4, d4,1 W4,1 d2,3 W2,3 W1,1W1,2 d3,2 W3,2 W1,1W2,1 d1,5 W1,5, d5,1 W5,1 d2,4 W2,4 W1,1W1,3 1 2 W1,22 d4,2 W4,2 W1,1W3,1 1 2 W2,12 d3,3 W3,3 W1,1W2,2 W1,2W2, W1,13 For r 3, the dimensions up to d2,2,2 are: N 3 N 4 N 5 N 6 d1,1,1 W1,1,1 d1,1,2 W1,1,2, d1,2,1 W1,2,1, d2,1,1 W2,1,1 d1,2,2 W1,2,2, d2,1,2 W2,1,2, d2,2,1 W2,2,1 d1,1,3 W1,1,3, d1,3,1 W1,3,1, d3,1,1 W3,1,1 d1,1,4 W1,1,4, d1,4,1 W1,4,1, d4,1,1 W4,1,1 d1,2,3 W1,2,3, d3,1,2 W3,1,2, d2,3,1 W2,3,1 d3,2,1 W3,2,1, d1,3,2 W1,3,2, d2,1,3 W2,1,3 d2,2,2 W2,2,2 1 2 W2 1,1,1 Example 6. Relative to θ with W G and applyin data from previous examples for the case r 2 we find the dimensions d1,1 d1,2 d2,1 d1,3 d3,1 d1,4 d4,1 d2,3 d3,2 d1,5 d5,1 d2,2 d2,4 d4,2 d3,3 4. In the case r 3, the dimensions are d1,1,1 16

17 8, d1,1,2 d2,1,1 d1,2,1 16, d1,2,2 d2,1,2 d2,2,1 56, d1,1,3 d3,1,1 d1,3,1 24, d1,1,4 d4,1,1 d1,4,1 32, d1,2,3 d3,1,2 d2,3,1 d3,2,1 d1,3,2 d2,1,3 128, d2,2, Example 7. Relative to θ + with W H we find the dimensions d1,1 d1,2 d2,1 d1,3 d3,1 d1,4 d4,1 d2,3 d3,2 d1,5 d5,1 2,d2,2 5,d2,4 d4,2 9,d3,3 28 for the case r 2. Also, d1,1,1 8, d1,1,2 d2,1,1 d1,2,1 16, d1,2,2 d2,1,2 d2,2,1 56, d1,1,3 d3,1,1 d1,3,1 24, d1,1,4 d4,1,1 d1,4,1 32,d1,2,3 d3,1,2 d2,3,1 d3,2,1 d1,3,2 d2,1,3 128, d2,2,2 504 for r 3. Remark. In spite of the neative terms in the formulas for the dimensions they ive positive results. To understand why, consider, for example, the case d2,2 W2,2 1 2 W1,12 with Wa,b F a+bf, hence, d2,2 is four times the result in example 6. In the set of words counted by F 2,2 48 there is a subset whose elements are wordsthat areobtainedluin toetherthe wordsintheset counted byw1,1 8. The luin produces an overcountin which is corrected by the one half factor. So, d2,2 is positive. The same arument can be used to et positivity for the other formulas. Theorem 4.3 For each G r G R, z m i 1...z m ir θ + e z2 i,...,z 2 1 ir +z i 1,...,z ir m i1,...,m ir 1 i m i1,...,m ir 1 1 z m i 1 i 1...z m ir θ e +z2 i 1,...,z 2 ir z i 1,...,z ir Proof: To prove 4.12 multiply and divide its left hand side by 1 z m i 1...z m ir θ + m i1,...,m ir 1 i 1 and use 4.8. To et 4.13 write 1 z m i 1...z m ir θ m i1,...,m ir 1 i 1 Nr m i > 0 m i m ir N 17 1 z m i 1 i 1...z m ir θ

18 Decompose the product over N into three products, namely, one over all N < 2r, one over all even N 2r and another one over all odd N > 2r. Then, apply Theorem 2 and Theorem 3, formula Theorem m i1,...,m ir 1 1+z m i 1...z m ir θ + 1 z m i 1...z m ir θ 1 i 1 i 1 Proof: Multiply 4.12 and The left hand side of 1.1 equals R R 1+z j 2 j1 r2 G r m i1,...,m ir >0 From 4.14 Sherman identity follows. 1+z m i 1...z m ir θ + 1 z m i 1...z m ir θ i 1 i 1 5 Two Lemmas Lemma 5.1 If 5.1 n 1,...,n k d n 1,...,n k µd d f n1 d,..., n k d then 5.2 fn 1,...,n k 1 d n1 d,..., n 1 d d n 1,...,n k Proof: Set Gn 1,...,n k : n n k n 1,...,n k and n1 F d,..., n k n1 : d d n k n1 f d d,..., n k d Then 5.1 can be expressed in the form Gn 1,...,n k n1 d,..., n 1 d d n 1,...,n k µdf 18

19 Möbius inversion ives Therefore, Fn 1,...,n k n n k fn 1,...,n k d n 1,...,n k G d n 1,...,n k and the result follows. The converse is also true. n1 d,..., n k d n1 d n k n1 d d,..., n k d Lemma 5.2 Let N α n n l, n 1,...,n l, n i > 0, a partition of α. Then, 5.3 l i1 k i N l i1 ki 1 n i 1 N 1 α 1 with the convention that a bracket in the left side is zero whenever k i < n i. Proof: Usin it follows that q α 1 q α q α 1 q α l i1 Nα q n i Nα 1 q n i N 1 α 1 l k i n i l i1 k i N i1 k i n i l i1 q N ki 1 n i 1 ki 1 n i 1 q k i q N Comparison with previous expression and the convention ives the result. Acknowledements. We kindly thank Prof. Peter Moree Max Planck Institute for Mathematics, Bonn and Prof. Thomas Ward University of East Anlia, UK for correspondence reardin the positivity of Möbius inversion formula. References [1] G. E. Andrews, The theory of Partitions, Addison-Wesley, Publishin Co.,

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