Separation probabilities for products of permutations


 Rafe Robinson
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1 Sepaation pobabilities fo poducts of pemutations Olivie Benadi, Rosena R. X. Du, Alejando H. Moales and Richad P. Stanley Mach 1, 2012 Abstact We study the mixing popeties of pemutations obtained as a poduct of two unifomly andom pemutations of fixed cycle types. Fo instance, we give an exact fomula fo the pobability that elements 1, 2,...,k ae in distinct cycles of the andom pemutation of {1, 2,..., n} obtained as poduct of two unifomly andom ncycles. 1 Intoduction We study cetain sepaation pobabilities fo poducts of pemutations. The achetypal question can be stated as follows: in the symmetic goup S n, what is the pobability that the elements 1,2,...,k ae in distinct cycles of the poduct of two ncycles chosen unifomly andomly? The answe is supisingly elegant: the pobability is 1 k! if n k is odd and 1 k! + 2 k 2!n k+1n+k if n k is even. This esult was oiginally conjectued by Bóna [3] fo k = 2 and n odd. Subsequently, Du and Stanley poved it fo all k and poposed additional conjectues [11]. The goal of this pape is to pove these conjectues, and establish genealizations of the above esult. Ou appoach is diffeent fom the one used in [11]. Let us define a lage class of poblems. Given a tuple A = A 1,...,A k of k disjoint nonempty subsets of {1,...,n}, we say that a pemutation π is Asepaated if no cycle of π contains elements of moe than one of the subsets A i. Now, given two intege patitions λ,µ of n, one can wonde about the pobability P λ,µ A that the poduct of two unifomly andom pemutations of cycle type λ and µ is Asepaated. The example pesented above coesponds to A = {1},..., {k} and λ = µ = n. Clealy, the sepaation pobabilities P λ,µ A only depend on A though the size of the subsets #A 1,...,#A k, and we shall denote σ α λ,µ := P λ,µa, whee α = #A 1,...,#A k is a composition of size m n. Note also that σ α λ,µ = σα λ,µ wheneve the composition α is a pemutation of the composition α. Below, we focus on the case µ = n and we futhe denote σ α λ := σα λ,n. In this pape, we fist expess the sepaation pobabilities σλ α as some coefficients in an explicit geneating function. Using this expession we then pove the following symmety popety: if α = α 1,...,α k and β = β 1,...,β k ae compositions of the same size m n and of the same length k, then σλ α k i=1 α i! = σ β λ k i=1 β i!. 1 O.B. acknowledges suppot fom NSF gant DMS , ANR A3, and ERC ExploeMaps. 1
2 Moeove, fo cetain patitions λ including the cases λ = n and λ = 2 N we obtain explicit expessions fo the pobabilities σλ α fo cetain patitions λ. Fo instance, the sepaation pobability fo the poduct of two ncycles is found to be σ α n σ α n = n m! k i=1 α i! n + kn 1! 1 n m n 1 k 2 + n+m m k m k =0 This includes the case α = 1 k poved by Du and Stanley [11]. 1 m k n++1 m n+k+. 2 Ou geneal expession fo the sepaation pobabilities σλ α is deived using a fomula obtained in [8] about coloed factoizations of the ncycle into two pemutations. This fomula displays a symmety which tuns out to be of cucial impotance fo ou method. Ou appoach can in fact be made mostly bijective as explained in Section 5. Indeed, the fomula obtained in [8] builds on a bijection established in [9]. An altenative bijective poof was given in [2] and in Section 5 we explain how to concatenate this bijective poof with the constuctions of the pesent pape. Outline. In Section 2 we pesent ou stategy fo computing the sepaation pobabilities. This involves counting cetain coloed factoizations of the ncycle. We then gathe ou main esults in Section 3. In paticula we pove the symmety popety 1 and obtain fomulas fo the sepaation pobabilities σλ α fo cetain patitions λ including λ = n o when λ = 2N. In Section 4, we give fomulas elating the sepaation pobabilities σλ α and σα λ when λ is a patition obtained fom anothe patition λ by adding some pats of size 1. In Section 5, we indicate how ou poofs could be made bijective. We gathe a few additional emaks in Section 6. Notation. We denote [n] := {1, 2,..., n}. We denote by #S the cadinality of a set S. A composition of an intege n is a tuple α = α 1,α 2,...,α k of positive intege summing to n. We then say that α has size n and length lα = k. An intege patition is a composition such that the pats α i ae in weakly deceasing ode. We use the notation λ = n esp. λ n to indicate that λ is a composition esp. intege patition of n. We sometime wite intege patitions in multiset notation: witing λ = 1 n 1,2 n 2,...,j n j means that λ has n i pats equal to i. We denote by S n the symmetic goup on [n]. Given a patition λ of n, we denote by C λ the set of pemutations in S n with cycle type λ. It is well known that #C λ = n!/z λ whee z λ = i iniλ n i λ! and n i λ is the numbe of pats equal to i in λ. We shall conside symmetic functions in an infinite numbe of vaiables x = {x 1,x 2,...}. Fo any sequence of nonnegative integes, α = α 1,α 2,...,α k we denote x α := x α 1 1 xα xα k k. We denote by [x α ]fx the coefficient of this monomial in a seies fx. Fo an intege patition λ = λ 1,...,λ k we denote by p λ x and m λ x espectively the powe symmetic function and monomial symmetic function indexed by λ see e.g. [10]. That is, p λ x = lλ i=1 p λ i x whee p k x = i 1 xk i, and m λx = α xα whee the sum is ove all the distinct sequences α whose positive pats ae {λ 1,λ 2,...,λ k } in any ode. Recall that the powe symmetic functions fom a basis of the ing of symmetic functions. Fo a symmetic function fx we denote by [p λ x]fx the coefficient of p λ x of the decomposition of fx in this basis. 2 Stategy In this section, we fist tanslate the poblem of detemining the sepaation pobabilities σλ α into the poblem of enumeating cetain sets Sλ α. Then, we intoduce a symmetic function Gα n x,t whose coefficients in one basis ae the cadinalities #Sλ α, while the coefficients in anothe basis 2
3 count cetain coloed sepaated factoizations of the pemutation 1,..., n. Lastly, we give exact counting fomulas fo these coloed sepaated factoizations. Ou main esults will follow as coollaies in Section 3. Fo a composition α = α 1,...,α k of size m n, we denote by A α n the set of tuples A = A 1,...,A k of paiwise disjoint subsets of [n] with #A i = α i fo all i in [k]. Obseve that #A α n = n α 1,α 2,...,α k,n m. Now, ecall fom the intoduction that σλ α is the pobability fo the poduct of a unifomly andom pemutation of cycle type λ with a unifomly andom ncycle to be Asepaated fo a fixed tuple A in A α n. Altenatively, it can be defined as the pobability fo the poduct of a unifomly andom pemutation of cycle type λ with a fixed ncycle to be Asepaated fo a unifomly andom tuple A in A α n since the only popety that mattes is that the elements in A ae andomly distibuted in the ncycle. Definition 1. Fo an intege patition λ of n, and a composition α of m n, we denote by S α λ the set of pais π,a, whee π is a pemutation in C λ and A is a tuple in A α n such that the poduct π 1,2,...,n is Asepaated. Fom the above discussion we obtain fo any composition α = α 1,...,α k of size m, σ α λ = #S λ α n. 3 α 1,α 2,...,α k,n m #Cλ Enumeating the sets Sλ α diectly seems athe challenging. Howeve, we will show below how to enumeate a elated class of coloed sepaated pemutations denoted by Tγ α. We define a cycle coloing of a pemutation π S n in [q] to be a mapping c fom [n] to [q] such that if i,j [n] belong to the same cycle of π then ci = cj. We think of [q] as the set of colos, and c 1 i as set of elements coloed i. Definition 2. Let γ = γ 1,...,γ l be a composition of size n and length l, and let α = α 1,...,α k be a composition of size m n and length k. Fo a nonnegative intege we define Tγ α as the set of quaduples π,a,c 1,c 2, whee π is a pemutation of [n], A = A 1,...,A k is in A α n, and i c 1 is a cycle coloing of π in [l] such that thee ae γ i element coloed i fo all i in [l], ii c 2 is a cycle coloing of the poduct π 1,2,...,n in [k + ] such that evey colo in [k + ] is used and fo all i in [k] the elements in the subset A i ae coloed i. Note that condition ii in Definition 2 and the definition of cycle coloing implies that the poduct π 1,2,...,n is Asepaated. In ode to elate the cadinalities of the sets Sλ α and T γ α, it is convenient to use symmetic functions in the vaiables x = {x 1,x 2,x 3,...}. Namely, given a composition α of m n, we define G α nx,t := p λ x t excessπ,a, λ n π,a S α λ whee the oute sum uns ove all the intege patitions of n, and excessπ,a is the numbe of cycles of the poduct π 1, 2,..., n containing none of the elements in A. Recall that the powe 3
4 symmetic functions p λ x fom a basis of the ing of symmetic functions, so that the contibution of a patition λ to G α n x,t can be ecoveed by extacting the coefficient of p λx in this basis: #Sλ α = [p λx] G α n x,1. 4 As we pove now, the sets T α γ ae elated to the coefficients of G α nx,t in the basis of monomial symmetic functions. Poposition 3. If α is a composition of length k, then G α n x,t + k = γ n m γ x 0 whee the oute sum is ove all intege patitions of n, and t #Tγ α, 5 t := tt 1 t + 1.! Poof. Since both sides of 5 ae polynomial in t and symmetic function in x it suffices to show that fo any nonnegative intege t and any patition γ the coefficient of x γ is the same on both sides of 5. We fist detemine the coefficient [x γ ]G α n x,t + k when t is a nonnegative intege. Let λ be a patition, and π be a pemutation of cycle type λ. Then the symmetic function p λ x can be intepeted as the geneating function of the cycle coloings of π, that is, fo any sequence γ = γ 1,...,γ l of nonnegative integes, the coefficient [x γ ]p λ x is the numbe of cycle coloings of π such that γ i elements ae coloed i, fo all i > 0. Moeove, if π is Asepaated fo a cetain tuple A = A 1,...,A k in A α n, then t + kexcesss,π epesents the numbe of cycle coloings of the pemutation π 1,2,...,n in [k +t] not necessaily using evey colo such that fo all i [k] the elements in the subset A i ae coloed i. Theefoe, fo a patition γ and a nonnegative intege t, the coefficient [x γ ]G α nx,t+k counts the numbe of quaduples π,a,c 1,c 2, whee π,a,c 1,c 2 ae as in the definition of Tγ α t except that c 2 might actually use only a subset of the colos [k + t]. Note howeve that all the colos in [k] will necessaily be used by c 2, and that we can patition the quaduples accoding to the subset of colos used by c 2. This gives [x γ ]G α nx,t + k = 0 t #T α γ. Now extacting the coefficient of x γ in the ighthand side of 5 gives the same esult. This completes the poof. In ode to obtain an explicit expession fo the seies G α nx,t it emains to enumeate the sets Tγ α which is done below. Poposition 4. Let be a nonnegative intege, let α be a composition of size m and length k, and let γ be a patition of size n m and length l. Then the set Tγ α specified by Definition 2 has cadinality #Tγ α nn l!n k! n + k 1 =, 6 n k l + 1! n m if n k l + 1 0, and 0 othewise. The est of this section is devoted to the poof of Poposition 4. In ode to count the quaduples π,a,c 1,c 2 satisfying Definition 2, we shall stat by choosing π,c 1,c 2 befoe choosing the tuple A. Fo compositions γ = γ 1,...,γ l, δ = δ 1,...,δ l of n we denote by B γ,δ the set of 4
5 tiples π,c 1,c 2, whee π is a pemutation of [n], c 1 is a cycle coloing of π such that γ i elements ae coloed i fo all i [l], and c 2 is a cycle coloing of the pemutation π 1,2,...,n such that δ i elements ae coloed i fo all i [l ]. The poblem of counting such sets was fist consideed by Jackson [5] who actually enumeated the union Bi,j n := B γ,δ using epesentation theoy. It was late poved in [8] that γ,δ =n, lγ=i, lδ=j #B γ,δ = nn l!n l! n l l + 1!, 7 if n l l +1 0, and 0 othewise. The poof of 7 in [8] uses a efinement of a bijection designed in [9] in ode to pove Jackson s fomula fo #Bi,j n. Anothe bijective poof of 7 is given in [2], and we shall discuss it futhe in Section 5 a poof of 7 using epesentation theoy can be found in [12]. One of the stiking featues of the counting fomula 7 is that it depends on the compositions γ, δ only though thei lengths l, l. This symmety will pove paticulaly handy fo enumeating Tγ α. Let, α, γ be as in Poposition 4, and let δ = δ 1,...,δ k+ be a composition of n of length k +. We denote by Tγ,δ α the set of quaduples π,a,c 1,c 2 in Tγ α such that the cycle coloing c 2 has δ i elements coloed i fo all i in [k + ] equivalently, π,c 1,c 2 B γ,δ. We also denote d α δ := k δi i=1 i. It is easily seen that fo any tiple π,c1,c 2 B γ,δ, the numbe d α δ counts the tuples A A α n such that π,a,c 1,c 2 Tγ,δ α. Theefoe, #T α γ = δ =n, lδ=k+ #T α γ,δ = δ =n, lδ=k+ d α δ #B γ,δ, whee the sum is ove all the compositions of n of length k +. Using 7 then gives #Tγ α nn l!n k! = d α δ n k l + 1! δ =n, lδ=k+ if n k l + 1 0, and 0 othewise. In ode to complete the poof of Poposition 4, it only emains to pove the following lemma. Lemma 5. If α has size m and length k, then n + k 1 d α δ =. n m δ =n, lδ=k+ Poof. We give a bijective poof illustated in Figue 1. One can epesent a composition δ = δ 1,...,δ k+ as a sequence of ows of boxes the ith ow has δ i boxes. With this epesentation, d α δ := k δi i=1 α i is the numbe of ways of choosing αi boxes in the ith ow of δ fo i = 1,...,k. Hence δ =n, lδ=k+ dα δ counts αmaked compositions of size n and length k+, that is, sequences of k+ nonempty ows of boxes with some maked boxes in the fist k ows, with a total of n boxes, and α i maks in the ith ow fo i = 1,...,k; see Figue 1. Now αmaked compositions of size n and length k+ ae clealy in bijection by adding a maked box to each of the ows 1,...,k, and making the last box of each of the ows k + 1,...,k + with α maked compositions of size n + k and length k + such that the last box of each ow is maked, whee α = α 1 +1,α 2 +1,...,α k +1,1,1,...,1 is a composition of length k +. Lastly, these objects ae clealy in bijection by concatenating all the ows with sequences of n + k boxes with m + k + maks, one of which is on the last box. Thee ae n+k 1 n m such sequences, which concludes the poof of Lemma 5 and Poposition 4. 5
6 Figue 1: A 2,1,2maked composition of size n = 12 and length 5 and its bijective tansfomation into a sequence n + k = 15 boxes with m + k + = = 10 maks, one of which is on the last box. 3 Main esults In this section, we exploit Popositions 3 and 4 in ode to deive ou main esults. All the esults in this section will be consequences of the following theoem. Theoem 6. Fo any composition α of m n of length k, the geneating function G α n x,t + k in the vaiables t and x = {x 1,x 2,...} has the following explicit expession in the bases m λ x and t : G α nx,t + k = n m =0 t n + k 1 n m λ n, lλ n k +1 nn lλ!n k! n k lλ + 1! m λ x. 8 Moeove, fo any patition λ of n, one has #S α λ = [p λx]g α nx,1 and σ α λ = #S λ α n. α 1,α 2,...,α k,n m #Cλ Theoem 6 is the diect consequence of Popositions 3 and 4. One of the stiking featues of 8 is that the expession of G α n x,t + k depends on α only though its size and length. This symmety popety then obviously also holds fo #Sλ α = [p λx]g α nx,1, and tanslates into the fomula 1 fo sepaation pobabilities as stated below. Coollay 7. Let λ be a patition of n, and let α = α 1,...,α k and β = β 1,...,β k be compositions of the same size m and length k. Then, #S α λ = #Sβ λ, 9 o equivalently, in tems of sepaation pobabilities, σ α λ k i=1 α i! = σ β λ k i=1 β i!. We now deive explicit fomulas fo the sepaation pobabilities fo the poduct of a unifomly andom pemutation π, with paticula constaints on its cycle type, with a unifomly andom ncycle. We focus on two constaints: the case whee π is equied to have p cycles, and the case whee π is a fixedpointfee involution fo n even. 3.1 Case when π has exactly p cycles Let Cn,p denote the set of pemutations of [n] having p cycles. Recall that the numbes cn,p = #Cn,p = [x p ]xx + 1x + 2 x + n 1 ae called the signless Stiling numbes of the fist kind. We denote by σ α n,p the pobability that the poduct of a unifomly andom pemutation in Cn,p with a unifomly andom ncycle is Asepaated fo a given set A in A α n. By a easoning simila to the one used in the poof of 3, one gets σ α n,p = 1 n α 1,α 2,...,α k,n m We now compute the pobabilities σ α n,p explicitly. cn,p λ n,lλ=p #S α λ. 10 6
7 Theoem 8. Let α be a composition of m with k pats. Then, σ α n,p = n m! k i=1 α n m i! 1 k n + k 1 cn k + 1,p, 11 cn, p n m n k + 1! =0 whee cn,p ae signless Stiling numbes of the fist kind. Fo instance, Theoem 8 in the case m = n gives the pobability that the cycles of the poduct of a unifomly andom pemutation in Cn, p with a unifomly andom ncycle efine a given set patition of [n] having blocks of sizes α 1,α 2,...,α k. This pobability is found to be σ α n,p = k i=1 α i! cn, p cn k + 1,p. n k + 1! We now pove Theoem 8. Via 10, this amounts to enumeating S α n,p := λ n,lλ=p Sα λ, and using Theoem 6 one gets #S α n,p = [p λ x]g α n x,1 whee An,p,l := = λ n,lλ=p n m =0 µ n, lµ=p 1 k n k +1 n + k 1 n m l=1 [p µ x] λ n, lλ=l Lemma 9. Fo any positive integes p,l n µ n, lµ=p [p µ x] λ n, lλ=l nn l!n k! n k l + 1! An,p,l, 12 m λ x. The next lemma gives a fomula fo An,p,l. m λ x = whee ca,b ae the signless Stiling numbes of the fist kind. n 1 1 l p cl,p, 13 l 1 l! Poof. Fo this poof we use the pincipal specialization of symmetic functions, that is, thei evaluation at x = 1 a := {1,1,...,1,0,0...} a ones. Since p γ 1 a = a lγ fo any positive intege a, one gets λ n, lλ=l m λ 1 a = n p=1 a p µ n, lµ=p [p µ x] λ n, lλ=l m λ x. The ighthand side of the pevious equation is a polynomial in a, and by extacting the coefficient of a p one gets [p µ x] m λ x = [a p ] m λ 1 a. µ n, lµ=p λ n, lλ=l λ n, lλ=l Now, fo any patition λ, m λ 1 a counts the atuples of nonnegative integes such that the positive ones ae the same as the pats of λ in some ode. Hence m λ 1 a counts the atuples λ n, lλ=l λ n, lλ=l of nonnegative integes with l positive ones summing to n. This gives, n 1 a m λ 1 a =. l 1 l a Extacting the coefficient of a p gives 13 since [a p ] l 7 = 1l p cl,p. l!
8 Using Lemma 9 in 12 gives #S α n,p = n! n m 0 which we simplify using the following lemma. a Lemma 10. Fo any nonnegative intege a, n k +1 1 k n + k 1 n k 1 l p cl,p, 14 n m l 1 l! q=0 l=1 a 1 q+1 p cq + 1,p = q q + 1! ca + 1,p. a + 1! Poof. The lefthand side equals [x p ] a a x q=0 q q+1. Using the ChuVandemonde identity this equals [x p ] x+a a+1 which is pecisely the ighthand side. Using Lemma 10 in 14 gives #S α n,p = n! n m =0 1 k n + k 1 cn k + 1,p, 15 n m n k + 1! which is equivalent to 11 via 3. This completes the poof of Theoem 8. In the case p = 1, the expession 11 fo the pobability σ α 1 = σn α of m k tems instead. We state this below. can be witten as a sum Coollay 11. Let α be a composition of m with k pats. Then the sepaation pobabilities σn α sepaation fo the poduct of two unifomly andom ncycles ae σn α = n m! k i=1 α i! 1 n m n 1 m k k 2 1 m k n++1 m +. n + kn 1! n+m m k =0 n+k+ The equation in Coollay 11, aleady stated in the intoduction, is paticulaly simple when m k is small. Fo α = 1 k i.e. m = k one gets the esult stated at the beginning of this pape: σ 1k n = { 1 k! if n k odd, 1 k! + 2 k 2!n k+1n+k if n k even. 16 In ode to pove Coollay 11 we stat with the expession obtained by setting p = 1 in 11: σ α n = n m! k i=1 α i! n 1! = n m! k i=1 α i! n 1! n m =0 We now use the following polynomial identity. i=0 1 k 1 n k + 1 n+k 1 [x n m ]1 + x 1 k a+b+1 b =0 n + k 1 n m x + m k + 1 i=0 n + k Lemma 12. Fo nonnegative integes a, b, one has the following identity between polynomials in x: a x i a 1 1 b b = i + b + 1 i a + 1 i x + 1 a+i+1 x b+1 a+i i x i+1 8
9 1 Poof. It is easy to see that the lefthand side of 18 is equal to ta t b dt. Now this integal can be computed via integation by pats. By a simple induction on b, this gives the ighthand side of 18. x b+1 x Now using 18 in 17, with a = n + k 1 and b = m k, gives σn α = n m! k i=1 α i! [x n m 1 + x 1 k m k ] n + kn 1! n+m m k x m k+1 =0 = n m! k i=1 α i! 1 n m n 1 m k k 2 1 m k + n + kn 1! n+m m k This completes the poof of Coollay 11. =0 n+k+ m k n+k+ n++1 m 1 + x n++1 x Case when π is a fixedpointfee involution Given a composition α of m 2N with k pats, we define HN α t := π,a S α 2 N t excessπ,a, whee excessπ,a is the numbe of cycles of the poduct π 1,2,...,2N containing none of the elements of A and whee π is a fixedpointfee involution of [2N]. Note that HN αt = [p 2 Nx]G α 2N x,t. We now give an explicit expession fo this seies. Theoem 13. Fo any composition α of m 2N of length k, the geneating seies HN α t + k is given by H α Nt + k = N min2n m,n k+1 =0 t 2N + k 1 k+ N 2N k! 2 2N m N k + 1!. 19 Consequently the sepaation pobabilities fo the poduct of a fixedpointfee involution with a 2Ncycle ae given by σ α 2 N = k i=1 α i! 2N 1!2N 1!! min2n m,n k+1 =0 1 k 2N + k 1 k+ N 1 2N k! 2 2N m N k + 1!. 20 Remak 14. It is possible to pove Theoem 13 diectly using ideas simila to the ones used to pove Theoem 6 in Section 2. This will be explained in moe detail in Section 5. In the poof given below, we instead obtain Theoem 13 as a consequence of Theoem 6. The est of this section is devoted to the poof of Theoem 13. Since H α N t = [p 2 Nx]Gα 2N x,t, Theoem 6 gives HNt α + k = 21 2N m N k +1 t 2N + k 1 2NN s!2n k! [p 2N m N k s + 1! 2 Nx] m λ x. =0 We then use the following esult. s=0 λ 2N, lλ=n+s 9
10 Lemma 15. Fo any nonnegative intege s N, [p 2 Nx] m λ x = λ 2N, lλ=n+s 1 s 2 s s!n s!. Poof. Fo patitions λ,µ of n, we denote S λ,µ = [p λ x]m µ x and R λ,µ = [m λ x]p µ x. The matices S = S λ,µ λ,µ n and R = R λ,µ λ,µ n ae the tansition matices between the bases {p λ } λ, n and {m λ } λ n of symmetic functions of degee n, hence S = R 1. Moeove the matix R is easily seen to be lowe tiangula in the dominance ode of patitions, that is, R λ,µ = 0 unless λ 1 + λ λ i µ 1 + µ µ i fo all i 1 [10, Pop ]. Thus the matix S = R 1 is also lowe tiangula in the dominance ode. Since the only patition of 2N of length N + s that is not lage than the patition 2 N in the dominance ode is 1 2s 2 N s, one gets [p 2 Nx] λ 2N, lλ=n+s m λ x = [p 2 Nx]m 1 2s 2N sx. 22 To compute this coefficient we use the standad scala poduct, on symmetic functions see e.g. [10, Sec. 7] defined by p λ,p µ = z λ if λ = µ and 0 othewise, whee z λ was defined at the end of Section 1. Fom this definition one immediately gets [p 2 N]m 1 2s 2 N s = 1 p z 2 N,m 1 2s 2 N s = 1 2 N N!2 N p 2 N,m 1 2s 2N s. 23 Let {h λ } denote the basis of the complete symmetic functions. It is well known that h λ,m µ = 1 if λ = µ and 0 othewise, theefoe p 2 N,m 1 2s 2 N s = [h 1 2s 2 N s]p 2 N. Lastly, since p 2 N = p 2 N and p 2 = 2h 2 h 2 1 one gets N p 2 N,m 1 2s 2 N s = [h 1 2s 2 N s]p 2 N = [h2s 1 hn s 2 ]2h 2 h 2 1 N = 2 N s 1 s. 24 s Putting togethe 22, 23 and 24 completes the poof. By Lemma 15, Equation 21 becomes 2N m HN α t + k = =0 = 2N = 2N t 2N m =0 N k +1 2N + k 1 2N m s=0 t 2N + k 1 2N k! 2N m N k + 1! min2n m,n k+1 =0 t 2NN s!2n k! N k s + 1! N k +1 s=0 2N + k 1 2N k! 2N m N k + 1! 1 s 2 s s!n s! N k s 1 s 2 N k +1, whee the last equality uses the binomial theoem. This completes the poof of Equation 19. Equation 20 then immediately follows fom the case t = 1 k of 19 via 3. This completes the poof of Theoem Adding fixed points to the pemutation π In this section we obtain a elation between the sepaation pobabilities σ α λ and σα λ, when the patition λ is obtained fom λ by adding some pats of size 1. Ou main esult is given below. 2 s 10
11 Theoem 16. Let λ be a patition of n with pats of size at least 2 and let λ be the patition obtained fom λ by adding pats of size 1. Then fo any composition α = α 1,...,α k of m n+ of length k, m k #Sλ α = p=0 n + p n + m + p n n + m + m p n Equivalently, in tems of sepaation pobabilities, σ α λ = n! n+ α 1,...,α k,n+ m n+ m k p=0 n+p n n+m+ p n+m n + m + p 1 m k n + m p + m p n n+m+ p 1 n+m n m + p!m k p + 1! Fo instance, when α = 1 k Theoem 16 gives n+ k σλ 1k = n + + k 2 + k n + + k 1 n + k n n + k n+ m k p σ 1k λ. #S m k p+1,1k 1 λ. σ m k p+1,1k 1 λ. The est of the section is devoted to poving Theoem 16. Obseve fist that 26 is a simple estatement of 25 via 3 using the fact that #C λ = n+ n #Cλ. Thus it only emains to pove 25, which amounts to enumeating Sλ α. Fo this pupose, we will fist define a mapping Ψ fom Sλ α to Ŝα λ, whee Ŝα λ is a set closely elated to Sα λ. We shall then count the numbe of peimages of each element in Ŝα λ unde the mapping Ψ. Roughly speaking, if π,a is in Sλ α and the tuple A = A 1,...,A k is thought as making some elements in the cycles of the pemutation ω = π 1,2,...,n +, then the mapping Ψ simply consists in emoving all the fixed points of π fom the cycle stuctue of ω and tansfeing thei maks to the element peceding them in the cycle stuctue of ω. We intoduce some notation. A multisubset of [n] is a function M which associates to each intege i [n] its multiplicity Mi which is a nonnegative intege. The intege i is said to be in the multisubset M if Mi > 0. The size of M is the sum of multiplicities n i=1 Mi. Fo a composition α = α 1,...,α k, we denote by A ˆα n the set of tuples M 1,...,M k of disjoint multisubsets of [n] i.e., no element i [n] is in moe than one multisubset such that the multisubset M j has size α j fo all j [k]. Fo M = M 1,...,M k in Âα n we say that a pemutation π of [n] is Msepaated if no cycle of π contains elements of moe than one of the multisubsets M j. Lastly, fo a patition λ of n we denote by Ŝα λ the set of pais π,m whee π is a pemutation in C λ, and M is a tuple in Â α n such that the poduct π 1,2,...,n is Msepaated. We now set λ,λ,α,k,m,n, to be as in Theoem 16, and define a mapping Ψ fom Sλ α to Ŝα λ. Let π be a pemutation of [n + ] of cycle type λ, and let e 1 < e 2 < < e n [n + ] be the elements not fixed by π. We denote ϕπ the pemutation π defined by setting πi = πj if π e i = e j. Obseve that π has cycle type λ. Remak 17. If e 1 < e 2 < < e n [n + ] ae the elements not fixed by π and π = ϕπ, then the cycle stuctue of the pemutation π 1,2,...,n + is obtained fom the cycle stuctue of π 1,2,...,n by eplacing each element i [n 1] by the sequence of elements F i = e i,e i + 1,e i +2,...,e i+1 1, and eplacing the element n by the sequence of elements F n = e n,e n +1,e n + 2,...,n+,1,2,...,e 1 1. In paticula, the pemutations π 1,2,...,n and π 1,2,...,n+ have the same numbe of cycles
12 Now given a pai π,a in S α λ, whee A = A 1,...,A k, we conside the pai Ψπ,A = π,m, whee π = ϕπ and M = M 1,...,M k is a tuple of multisubsets of [n] defined as follows: fo all j [k] and all i [n] the multiplicity M j i is the numbe of elements in the sequence F i belonging to the subset A j whee the sequence F i is defined as in Remak 17. It is easy to see that Ψ is a mapping fom S α λ to Ŝα λ. We ae now going to evaluate #S α λ by counting the numbe of peimages of each element in Ŝα λ unde the mapping Ψ. As we will see now, the numbe of peimages of a pai π,m in Ŝα λ only depends on M. Lemma 18. Let π,m Ŝα λ, whee M = M 1,...,M k. Let s be the numbe of distinct elements appeaing in the multisets M 1,...,M k, and let x = k j=1 M jn be the multiplicity of the intege n. Then the numbe of peimages of the pai π,m unde the mapping Ψ is n + + s if x = 0, #Ψ 1 n + m π,m = 27 n + + s n + + s 1 x + othewise. n + m n + m Poof. We adopt the notation of Remak 17, and fo all i [n] we denote M i = k j=1 M ji the multiplicity of the intege i. In ode to constuct a peimage π,a of π,m, whee A = A 1,...,A k, one has to i choose fo all i [n] the length f i > 0 of the sequence F i with n i=1 f i = n +, ii choose the position b [f n ] coesponding to the intege n + in the sequence F n, iii if M j i > 0 fo some i [n] and j [k], then choose which M j i elements in the sequence F i ae in the subset A j. Indeed, the choices i, ii detemine the pemutation π C λ since they detemine the fixedpoints of π, which is enough to ecove π fom π, while by Remak 17 the choice iii detemines the tuple of subsets A = A 1,...,A k. We will now count the numbe ways of making the choices i, ii, iii by encoding such choices as ows of maked and unmaked boxes as illustated in Figue 2. We teat sepaately the cases x = 0 and x 0. Suppose fist x = 0. To each i [n] we associate a ow of boxes R i encoding the choices i, ii, iii as follows: 1 if i n and M i = 0, then the ow R i is made of f i boxes, the fist of which is maked, 2 if i n and M i > 0, then the ow R i is made of f i + 1 boxes, with the fist box being maked and M i othe boxes being maked the maks epesent the choice iii, 3 the ow R n is made of f n + 1 boxes, with the fist box being maked and an additional box being maked and called special maked box this box epesents the choice ii. Thee is no loss of infomation in concatenating the ows R 1,R 2,...,R n given that M is known indeed the ow R i stats at the i + N i th maked box, whee N i = h<i M h. This concatenation esults in a ow of n + + s + 1 boxes with n + m + 1 maks such that the fist box is maked and the last mak is special ; see Figue 2. Moeove thee ae n++s n+m such ows of boxes and any of them can be obtained fo some choices of i, ii, iii. This poves the case x = 0 of Lemma 18. We now suppose x > 0. We eason similaly as above but thee ae now two possibilities fo the ow R n, depending on whethe o not the intege n+ belongs to one of the subsets A 1,...,A k. In ode to encode a peimage such that n + belong to one of the subsets A 1,...,A k the condition 3 above must be changed to 12
13 R 1 f 1 = 4 R 2 f 2 = 2 R 3 f 3 = 2 R 4 f 4 = 5 R 5 f 5 = 1 R 6 f 6 = 3,x = 0 Figue 2: Example of choices 1,2,3 encoded by a sequence of boxes, some of which being maked indicated in gay, with one mak being special indicated with a coss. Hee n = 6, k = 2, = 11, x = 0 and the multisubsets M 1,M 2 ae defined by M 1 1 = 1, M 2 3 = 1, M 1 4 = 3, and M j i = 0 fo the othe values of i,j. 3 the ow R n is made of f n + 1 boxes, with the fist box being maked and x othe boxes being maked, one of which being called special maked box. In this case, concatenating the ows R 1,R 2,...,R n gives a ow of n + + s boxes with n + m maks, with the fist box being maked and one of the x last maked boxes being special. Thee ae x n++s 1 n+m 1 such ows and each of them comes fom a unique choice of i, ii and iii. Lastly, in ode to encode a peimage such that n + does not belong to one of the subsets A 1,...,A k the condition 3 above must be changed to 3 the ow R n is made of f n + 1 boxes, with the fist box being maked and x + 1 othe boxes being maked, one of which being called special maked box. In this case, concatenating the ows R 1,R 2,...,R n gives a ow of n + + s boxes with n + m + 1 maks, with the fist box being maked and one of the x+1 last maked boxes being special. Thee ae x + 1 n++s 1 n+m such ows and each of them comes fom a unique choice of i, ii and iii. Thus, in the case x > 0 one has n + + s 1 n + + s 1 #Ψ 1 π,m = x + x + 1 n + m 1 n + m This completes the poof of Lemma 18. = x n + + s + n + m n + + s 1 n + m We now complete the poof of Theoem 16. Fo any composition γ = γ 1,...,γ k, we denote by Ŝ α,γ λ the set of pais π,m in Ŝα λ, whee the tuple M = M 1,...,M k is such that fo all j [k] the multisubset M j which is of size α j contains exactly γ j distinct elements. Summing 27 gives n + + γ n + + γ 1 #Ψ 1 π,m = EX + PX = 0 + PX > 0 n + m n + m π,m Ŝ α,γ λ. #Ŝα,γ λ, 28 whee X is the andom vaiable defined as X = k j=1 M jn fo a pai π,m chosen unifomly andomly in Ŝα,γ λ, EX is the expectation of this andom vaiable, and PX > 0 = 1 PX = 0 is the pobability that X is positive. Lemma 19. With the above notation, EX = m n, and PX > 0 = γ n. Poof. The poof is simply based on a cyclic symmety. Fo i [n] we conside the andom vaiable X i = k j=1 M ji fo a pai π,m chosen unifomly andomly in Ŝα,γ λ. It is easy to see that all the vaiables X 1,...,X n = X ae identically distibuted since the set Ŝα,γ λ is unchanged by cyclically shifting the value of the integes 1,2,...,n in pais π,m Ŝα,γ λ. Theefoe, n n n EX = EX i = E X i = Em = m, i=1 i=1 13
14 and n PX > 0 = n n PX i > 0 = E i=1 i=1 1 Xi >0 = E γ = γ. We now enumeate the set Ŝα,γ λ. Obseve that any pai π,m in Ŝα,γ λ can be obtained in a unique way fom a pai π,a in S γ λ by tansfoming A = A 1,...,A k into M = M 1,...,M k as follows: fo each j [k] one has to assign a positive multiplicity M j i fo all i A j so as to get a multisubset M j of size α j. Thee ae α j γ j 1 ways of pefoming the latte task, hence #Ŝα,γ λ = Using this esult and Lemma 19 in 28 gives π,m Ŝα,γ λ k i=1 αi 1 #S γ γ i 1 λ. m + n γ n + + γ #Ψ 1 π,m = + γ k n + + γ 1 n n + m n n + m i=1 αi 1 #S γ γ i 1 λ. Obseve that the above expession is 0 unless γ is less o equal to α componentwise. Finally, one gets #S α λ = γ α, lγ=k m + n γ n + + γ + γ k n + + γ 1 αi 1 #S γ n n + m n n + m γ i 1 λ, 29 whee the sum is ove compositions γ with k pats, which ae less o equal to α componentwise. Lastly, by Coollay 7, the cadinality #S γ λ only depends on the composition α though the length and size of α. Theefoe, one can use 29 with α = m k+1,1 k 1, in which case the compositions γ appeaing in the sum ae of the fom γ = m k p + 1,1 k 1 fo some p m k. This gives 25 and completes the poof of Theoem Bijective poofs and intepetation in tems of maps In this section we explain how cetain esults of this pape can be intepeted in tems of maps, and can be poved bijectively. In paticula, we shall intepet the sets Tγ,δ α of sepaated coloed factoizations defined in Section 2 in tems of maps. We can then extend a bijection fom [1] in ode to pove bijectively the symmety popety stated in Coollay Intepetations of sepaated coloed factoizations in tems of maps We fist ecall some definitions about maps. Ou gaphs ae undiected, and they can have multiple edges and loops. Ou sufaces ae twodimensional, compact, boundayless, oientable, and consideed up to homeomophism; such a suface is chaacteized by its genus. A connected gaph is cellulaly embedded in a suface if its edges ae not cossing and its faces connected components of the complement of the gaph ae simply connected. A map is a cellula embedding of a connected gaph in an oientable suface consideed up to homeomophism. A map is epesented in Figue 3. By cutting an edge in its midpoint one gets two halfedges. A map is ooted if one of its halfedges is distinguished as the oot. In what follows we shall conside ooted bipatite maps, and conside i=1 14
15 a b Figue 3: a A ooted bipatite oneface map. b A ooted bipatite teeooted map the spanning tee is indicated by thick lines. The oot halfedge is indicated by an aow. the unique pope coloing of the vetices in black and white such that the oot halfedge is incident to a black vetex. By a classical encoding see e.g. [6], fo any patitions λ,µ of n, the solutions π 1,π 2 C λ C µ of the equation π 1 π 2 = 1,2,...,n ae in bijection with the ooted oneface bipatite maps such that black and white vetices have degees given by the pemutations λ and µ espectively. That is, the numbe of black esp. white vetices of degee i is equal to the numbe of pats of the patition λ esp. µ equal to i. Let γ = γ 1,...,γ l, δ = δ 1,...,δ l be compositions of n and let α = α 1,...,α k be a composition of m n. A ooted bipatite map is γ,δcoloed if its black vetices ae coloed in [l] that is, evey vetex is assigned a colo in [l] in such a way that γ i edges ae incident to black vetices of colo i, and its white vetices ae coloed in [l ] in such a way that δ i edges ae incident to white vetices of colo i. Though the above mentioned encoding, the set B γ,δ of coloed factoizations of the ncycles defined in Section 2 coesponds to the set of γ,δcoloed ooted bipatite oneface maps. Similaly, the sets Tγ,δ α of sepaated coloed factoizations coesponds to the set of γ,δcoloed ooted bipatite oneface maps with some maked edges, such that fo all i [k] exactly α i maked edges ae incident to white vetices coloed i. The esults in this pape can then be intepeted in tems of maps. Fo instance, one can intepet 8 in the case m = k = 0 no maked edges as follows: n p λ xt #white vetices = G n x,t = t nn lλ!n! n 1 m λ x, n lλ + 1! n M B λ =1 λ n λ n, lλ n +1 whee B λ is the set of ooted bipatite oneface maps such that black vetices have degees given by the patition λ. The esults in Subsection 3.2 can also be intepeted in tems of geneal i.e., nonnecessaily bipatite maps. Indeed, the set M N = B 2 N can be intepeted as the set of geneal ooted oneface maps with N edges because a bipatite map in which evey black vetex has degee two can be intepeted as a geneal map upon contacting the black vetices. Theefoe one can intepet 19 in the case m = k = 0 no maked edges as follows: M M N t #vetices = H N t = N N+1 =1 t N 2N! 2 N + 1! This equation is exactly the celebated HaeZagie fomula [4]. 5.2 Bijection fo sepaated coloed factoizations, and symmety 2N N In this section, we explain how some of ou poofs could be made bijective. In paticula we will use bijective esults obtained in [1] in ode to pove the symmety esult stated in Coollay 7. 15
16 We fist ecall the bijection obtained in [1] about the sets B γ,δ. We define a teeooted map to be a ooted map with a maked spanning tee; see Figue 3b. We say that a bipatite teeooted map is l,l labelled if it has l black vetices labelled with distinct labels in [l], and l white vetices labelled with distinct labels in [l ]. It was shown in [1] that fo any compositions γ = γ 1,...,γ l, δ = δ 1,...,δ l of n, the set B γ,δ is in bijection with the set of l,l labelled bipatite teeooted maps such that the black esp. white vetex labelled i has degee γ i esp. δ i. Fom this bijection, it is not too had to deive the enumeative fomula 7 see Remak 21. We now adapt the bijection established in [1] to the sets Tγ,δ α of sepaated coloed factoizations. Fo a composition α = α 1,...,α k, a l,l labelled bipatite maps is said to be αmaked if α i edges incident to the white vetex labelled i ae maked fo all i in [k]. Theoem 20. The bijection in [1] extends into a bijection between the set Tγ,δ α and the set of αmaked l,l labelled bipatite teeooted maps with n edges such that the black esp. white vetex labelled i has degee γ i esp. δ i. We will now show that the bijection given by Theoem 20 easily implies #T α γ = #T β γ, 31 wheneve the compositions α and β have the same length and size. Obseve that, in tun, 31 eadily implies Coollay 7. By Theoem 20, the set Tγ α specified by Definition 2 is in bijection with the set T γ α of αmaked l, k + labelled bipatite teeooted maps with n edges such that the black vetex labelled i has degee γ i. We will now descibe a bijection between the sets T γ α β and T γ when α and β have the same length and size. Fo this pupose it is convenient to intepet maps as gaphs endowed with a otation system. A otation system of a gaph G is an assignment fo each vetex v of G of a cyclic odeing of the halfedges incident to v. Any map M defines a otation system ρm of the undelying gaph: the cyclic odeings ae given by the clockwise ode of the halfedges aound the vetices. This coespondence is in fact bijective see e.g. [7]: fo any connected gaph G the mapping ρ gives a bijection between maps having undelying gaph G and the otation systems of G. Using the otation system intepetation, any map can be epesented in the plane with edges allowed to coss each othe by choosing the clockwise ode of the halfedges aound each vetex to epesent the otation system; this is the convention used in Figues 4 and 5. * * e * * 4 * 2 4 e 1 * * * * e 1 * ϕ 1,3 1 4 Figue 4: Left: a 3, 1, 1maked 4, 5labelled bipatite teeooted map. Right: the 2, 1, 2 maked 4,5labelled bipatite teeooted map obtained by applying the mapping ϕ 1,3. In this figue, maps ae epesented using the otation system intepetation, so that the edgecossings ae ielevant. The spanning tees ae dawn in thick lines, the maked edges ae indicated by stas, and the oot halfedge is indicated by an aow. 16
17 We now pove 31 it is sufficient to establish a bijection between the sets T α γ and T β γ in the case α = α 1,...,α k, β = β 1,...,β k with β i = α i 1, β j = α j + 1 and α s = β s fo s i,j. Let M be an αmaked l,l labelled bipatite teeooted map. We conside the path joining the white vetices i and j in the spanning tee of M. Let e i and e j be the edges of this path incident to the white vetices i and j espectively; see Figue 4. We conside the fist maked edge e i following e i in clockwise ode aound the vetex i note that e i e i since α i = β i + 1 > 1. We then define ϕ i,j M as the map obtained by ungluing fom the vetex i the halfedge of e i as well as all the halfedges appeaing stictly between e i and e i, and gluing them in the same clockwise ode in the cone following e j clockwise aound the vetex j. Figue 4 illustates the mapping ϕ 1,3. It is easy to see that ϕ i,j M is a teeooted map, and that ϕ i,j and ϕ j,i ae evese mappings. Theefoe ϕ i,j M is a bijection between T α γ and T β γ. This poves 31. Remak 21. By an agument simila to the one used above to pove 31, one can pove that if γ,γ,δ,δ ae compositions of n such that lγ = lγ and lδ = lδ then B γ,δ = B γ,δ this is actually done in a moe geneal setting in [2]. Fom this popety one can compute the cadinality of B γ,δ by choosing the most convenient compositions γ, δ of length l and l. We take γ = n l + 1,1,1,...,1 and δ = n l + 1,1,1,...,1, so that #B γ,δ is the numbe of l,l  labelled bipatite teeooted maps with the black and white vetices labelled 1 of degees n l + 1 and n l +1 espectively, and all the othe vetices of degee 1. In ode to constuct such an object see Figue 5, one must choose the unooted plane tee 1 choice, the labelling of the vetices l 1!l 1! choices, the n l l +1 edges not in the tee n l n l n l l +1 n l l +1 n l l +1! choices, and lastly the oot n choices. This gives Figue 5: A teeooted map in B γ,δ, whee γ = 8,1,1,1,1, δ = 9,1,1,1. Hee the map is epesented using the otation system intepetation, so that the edgecossings ae ielevant. 5.3 A diect poof of Theoem 13 In Section 3 we obtained Theoem 13 as a consequence of Theoem 6. Hee we explain how to obtain it diectly. Fist of all, by a easoning identical to the one used to deive 5 one gets 2N m HN α t + k = =0 t #U α, 32 whee U α is the set of tiples π,a,c 2 whee π is a fixedpoint fee involution of [2N], A is in A α n and c 2 is a cycle coloing of the poduct π 1,2,...,2N in [k + ] such that evey colo in [k + ] is used and fo all i in [k] the elements in the subset A i ae coloed i. In ode to enumeate U α one consides fo each composition γ = γ 1,...,γ l the set M γ of pais π,c 2, whee π is a fixedpointfee involution of [2N] and c 2 is a cycle coloing of the 17
18 pemutation π 1,2,...,2N such that γ i elements ae coloed i fo all i [l]. One then uses the following analogue of 7: N2N l! #M γ = N l + 1! 2l N. 33 Using this esult in conjunction with Lemma 5, one then obtains the following analogue of 6: #U α N2N k! 2N + k 1 =. N k + 1! 2N m Plugging this esult in 32 completes the poof of Theoem 13. Similaly as 7, Equation 33 can be obtained bijectively. Indeed by a classical encoding, the set M γ is in bijection with the set of ooted oneface maps with vetices coloed in [l] in such a way that fo all i [l], thee ae exactly γ i halfedges incident to vetices of colo i. Using this intepetation, it was poved in [1] that the set M γ is in bijection with the set of teeooted maps with l vetices labelled with distinct labels in [l] such that the vetex labelled i has degee γ i. The latte set is easy to enumeate using symmety as in Remak 21 and one gets Concluding emaks: stong sepaation and connection coefficients Given a tuple A = A 1,...,A k of disjoint subsets of [n], a pemutation π is said to be stongly Asepaated if each of the subsets A i, fo i [k] is included in a distinct cycle of π. Given a patition λ of n and a composition α of m n, we denote by πλ α the pobability that the poduct ω ρ is stongly Asepaated, whee ω esp. ρ is a unifomly andom pemutation of cycle type λ esp. n and A is a fixed tuple in A α n. In paticula, fo a composition α of size m = n, one gets k πλ α = Kα λ,n i=1 α i 1!, n 1! #C λ whee Kλ,n α is the connection coefficient of the symmetic goup counting the numbe of solutions ω,ρ C λ C n, of the equation ω ρ = φ whee φ is a fixed pemutation of cycle type α. We now ague that the sepaation pobabilities {σλ α} α =m computed in this pape ae enough to detemine the pobabilities {πλ α} α =m. Indeed, it is easy to pove that σ α λ = β α R α,β π β λ, 34 whee the sum is ove the compositions β = β 1,...,β l of size m = α such that thee exists 0 = j 0 < j 1 < j 2 < < j k = l such that β ji 1 +1,β ji 1 +1,...,β ji is a composition of α i fo all i [k], and R α,β = k i=1 R i whee R i is the numbe of ways of patitioning a set of size α i into blocks of espective sizes β ji 1 +1,β ji 1 +1,...,β ji. Moeove, the matix R α,β α,β =m is invetible since the matix is uppe tiangula fo the lexicogaphic odeing of compositions. Thus, fom the sepaation pobabilities {σλ α} α =m one can deduce the stong sepaation pobabilities {πλ α} α =m and in paticula, fo m = n, the connection coefficients Kλ,n α of the symmetic goup. Acknowledgment: We thank Taedong Yun fo seveal stimulating discussions. 18
19 Refeences [1] O. Benadi. An analogue of the HaeZagie fomula fo unicellula maps on geneal sufaces. Adv. in Appl. Math., 481: , [2] O. Benadi and A.H. Moales. Bijections and symmeties fo the factoizations of the long cycle. Submitted, AXiv: , [3] M. Bóna and R. Flynn. The aveage numbe of block intechanges needed to sot a pemutation and a ecent esult of stanley. Infom. Pocess. Lett., 10916: , [4] J. Hae and D. Zagie. The Eule chaacteistic of the moduli space of cuves. Invent. Math., 853: , [5] D.M. Jackson. Some combinatoial poblems associated with poducts of conjugacy classes of the symmetic goup. J. Combin. Theoy Se. A, 492, [6] S.K. Lando and A.K. Zvonkin. Gaphs on sufaces and thei applications. SpingeVelag, [7] B. Moha and C. Thomassen. Gaphs on sufaces. J. Hopkins Univ. Pess, [8] A.H. Moales and E.A. Vassilieva. Bijective enumeation of bicoloed maps of given vetex degee distibution. In DMTCS Poceedings, 21st Intenational Confeence on Fomal Powe Seies and Algebaic Combinatoics FPSAC [9] G. Schaeffe and E.A. Vassilieva. A bijective poof of Jackson s fomula fo the numbe of factoizations of a cycle. J. Combin. Theoy, Se. A, 1156: , [10] R.P. Stanley. Enumeative combinatoics, volume 2. Cambidge Univesity Pess, [11] R.P. Stanley. Poducts of cycles, Slides fo the confeence Pemutation Pattens 2010, stan/tanspaencies/cyclepod.pdf. [12] E.A. Vassilieva. Explicit monomial expansions of the geneating seies fo connection coefficients. AXiv: , Olivie Benadi Depatment of Mathematics, Massachusetts Institute of Technology; Cambidge, MA USA Rosena R. X. Du Depatment of Mathematics, East China Nomal Univesity; Shanghai, China Alejando H. Moales Depatment of Mathematics, Massachusetts Institute of Technology; Cambidge, MA USA Richad P. Stanley Depatment of Mathematics, Massachusetts Institute of Technology; Cambidge, MA USA
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