Redfield-Pólya theorem in $\mathrm{WSym}$

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Redfield-Pólya theorem in WSym

Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, Olivier Mallet

To cite this version:

Jean-Paul Bultel, Ali Chouria, Jean-Gabriel Luque, Olivier Mallet. Redfield-Pólya theorem in WSym.

25th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2013),

2013, Paris, France. pp.563-574. �hal-01229736�

Word symmetric functions and the Redfield-P ´olya theorem ^†

Jean-Paul Bultel and Ali Chouria and Jean-Gabriel Luque and Olivier Mallet

Laboratoire LITIS, EA 4108, Universit´e de Rouen, Saint- ´Etienne-du-Rouvray, France

Abstract.We give noncommutative versions of the Redfield-P´olya theorem inWSym, the algebra of word symmetric functions, and in other related combinatorial Hopf algebras.

R´esum´e.Nous donnons des versions non-commutatives du th´eor`eme d’´enum´eration de Redfield-P´olya dansWSym, l’alg`ebre des fonctions sym´etriques sur les mots, ainsi que dans d’autres alg`ebres de Hopf combinatoires.

Keywords:Symmetric functions, Redfield-P´olya theorem, Combinatorial Hopf algebras, Operads

1 Introduction

The Redfield-P´olya enumeration theorem (R-P theorem) is one of the most exciting results in combina- torics of the twentieth century. It was first published by John Howard Redfield (18) in 1927 and indepen- dently rediscovered by George P´olya ten years later (17). Their motivation was to generalize Burnside’s lemma on the number of orbits of a group action on a set (seee.g(3)). Note that Burnside attributed this result to Frobenius (8) and it seems that the formula was prior known to Cauchy. Although Redfield found the theorem before P´olya, it is often attributed only to P´olya. This is certainly due to the fact that P´olya popularized the result by providing numerous applications to counting problems and in particular to the enumeration of chemical compounds. The theorem is a result of group theory but there are important im- plications in many disciplines (chemistry, theoretical physics, mathematics — in particular combinatorics and enumerationetc.) and its extensions lead to Andr´es Joyal’s combinatorial species theory (1).

Consider two setsXandY (X finite) and letGbe a finite group acting onX. For a mapf : X →Y, define the vectorvf = (#{x:f(x) =y)})_y∈Y ∈N^Y. The R-P theorem deals with the enumeration of the mapsf having a givenvf =v(vfixed) up to the action of the groupG. The reader can refer to (11) for proof, details, examples and generalizations of the R-P theorem.

Algebraically, the theorem can be pleasantly stated in terms of symmetric functions (seee.g. (15, 12));

†This paper is partially supported by the ANR project PhysComb, ANR-08-BLAN-0243-04, the CMEP-TASSILI project 09MDU765, the ANR project CARMA ANR-12-BS01-0017 and the PEPS project COGIT.

‡email: jean-paul.bultel@univ-rouen.fr, ali.chouria1@univ-rouen.fr,

jean-gabriel.luque@univ-rouen.fr,olivier.mallet@univ-rouen.fr

1365–8050 c2013 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

the cycle index polynomial is defined in terms of power sum symmetric functions and, whence writing in the monomial basis, the coefficients count the number of orbits of a given type. From the seminal paper (9), many combinatorial Hopf algebras have been discovered and investigated. The goal is to mimic the combinatorics and representation theory related to symmetric functions in other contexts. This paper asks the question of the existence of combinatorial Hopf algebras in which the R-P theorem can naturally arise.

In this sense, the article is the continuation of (6, 7) in which the authors investigated some Hopf algebras in the aim to study the enumeration of bipartite graphs up to the permutations of the vertices.

Each functionf :X →Y can be encoded by a word of size#Xon an alphabetAY ={ay :y ∈Y}.

Hence, intuitively, we guess that the R-P theorem arises in a natural way in the algebra of wordsChAi.

The Hopf algebra of word symmetric functions WSymhas been studied in (19, 2, 10). In S Section 2, we recall the basic definitions and properties related to this algebra and propose a definition for the specialization of an alphabet using the concept of operad (13, 14). In Section 3, we construct and study other related combinatorial Hopf algebras. In Section 4, we investigate the analogues of the cycle index polynomials in these algebras and give two noncommutative versions of the R-P theorem. In particular, we give a word version and a noncommutative version. Finally, in Subsection 4.5, we propose a way to raise Harary-Palmer type enumerations (the functions are now enumerated up to an action ofGonXand an action of another groupHonY) inWSym. For this last equality, we need the notion of specialization defined in Section 2.

2 Word symmetric functions

2.1 Basic definitions and properties

Consider the familyΦ := {Φ^π}π whose elements are indexed by set partitions of{1, . . . , n}(we will denote π n). The algebra WSym (19) is generated by Φ for the shifted concatenation product:

Φ^πΦ^π0 = Φ^ππ0[n] whereπ and π⁰ are set partitions of {1, . . . , n} and {1, . . . , m}, respectively, and π⁰[n]means that we addnto each integer occurring inπ⁰. Other bases are known, for example, the word monomial functions defined by

Φ^π= X

π≤π⁰

M_π⁰

whereπ≤π⁰indicates thatπis finer thanπ⁰,i.e., that each block ofπ⁰is a union of blocks ofπ.

WSym is a Hopf algebra when endowed with the shifted concatenation product and the following coproduct, wherestd(π)means that for alli, we replace theith smallest integer inπbyi:

∆M_π= X

π⁰∪π⁰⁰=π π⁰∩π⁰⁰=∅

M_std(π0)⊗M_std(π00).

Note that the notion of standardization makes sense for more general objects. IfSis a total ordered set, the standardizedstd(`), for any list`ofnelements ofS, is classicaly the permutationσ= [σ1, . . . , σn] verfyingσ[i]> σ[j]if`[i]> `[j]or if`[i] =`[j]andi > j. Now if the description of an objectocontains a list`, the standardizedstd(o)is obtained by replacing`bystd(`)ino.

LetAbe an infinite alphabet. The algebraWSymis isomorphic toWSym(A), the subalgebra ofChAi defined by Rosas and Sagan (19) and constituted by the polynomials which are invariant by permutation of the letters of the alphabet. The explicit isomorphism sends eachΦ^πto the polynomialΦ^π(A) := P

ww

where the sum is over the wordsw =w₁· · ·w_n (w₁, . . . , w_n ∈ A) such that ifiandjare in the same block ofπthenw_i =w_j. Under this isomorphism, eachM_π is sent toM_π(A) =P

wwwhere the sum is over the wordsw = w₁· · ·w_n (w₁, . . . , w_n ∈ A) such thatw_i =w_j if and only ifiandjare in the same block ofπ. In the sequel, when there is no ambiguity, we will identify the algebrasWSymand WSym(A). With the realization explained above, the coproduct ofWSymconsists of identifying the algebraWSym⊗WSymwithWSym(A+B), whereAandBare two noncommutative alphabets such thatAcommutes withB, by settingf(A)g(B)∼f ⊗g(see. It is a cocommutative coproduct for which the polynomialsΦ^{1,...,n}are primitive. Endowed with this coproduct,WSymhas a Hopf structure which has been studied by Hivertet al.(10) and Bergeronet al.(2).

2.2 What are virtual alphabets in WSym?

We consider the setCof set compositions together with additional elements{o_m:m >1}(we will also seto0 = []) and a unity1. This set is a naturally bigraded set: ifC^m_n denotes the set of compositions of {1, . . . , n} into msubsets, we have C = {1} ∪S

n,m∈NC^m_n ∪ {om : m > 1}.We will also use the notationsCn(resp. C^m) to denote the set of compositions of{1, . . . , n}(resp. the set compositions intomsubsets together withom) with the special case: 1 ∈ C¹. The formal spaceC[C^m]is naturally endowed with a structure of rightC[Sm]-module; the permutations acting by permuting the blocks of each composition and lettingominvariant. For simplicity, we will denote also byCthe collection (S- module, seee.g.(13))[C[C⁰],C[C¹], . . . ,C[C^m]].

For each1≤i≤k, we define partial compositions◦i:C^k×C^k0 →C^k+k0−1by:

1. IfΠ = [π₁, . . . , π_k]andΠ⁰= [π⁰₁, . . . , π_k⁰0]then Π◦iΠ⁰=

[π₁, . . . , π_i−1, π⁰₁[π_i], . . . , π_k⁰0[π_i], π_i+1, . . . , π_k] ifΠ⁰∈C_#πi

o_k+k⁰₋₁ otherwise,

whereπ_j⁰[πi] ={ij₁, . . . , ij_p}ifπ_j⁰ ={j1, . . . , jp}andπi={i1, . . . , ik}withi1<· · ·< ik; 2. Π◦_io_k⁰=o_k◦_iΠ⁰ =o_k+k⁰₋₁for eachΠ∈C^kandΠ⁰ ∈C^k0;

3. 1◦1Π⁰= Π⁰andΠ◦i1= Πfor eachΠ∈C^kandΠ⁰∈C^k0.

Proposition 2.1 The S-moduleC(i.e. each graded componentCn is aSn-module (13)) endowed with the partial compositions◦iis an operad in the sense of Martin Markl (14), which means the compositions satisfy:

1. (Associativity) For each1≤j≤k,Π∈C^k,Π⁰ ∈C^k0andΠ⁰⁰∈C^k00:

(Π◦jΠ⁰)◦iΠ⁰⁰=







(Π◦iΠ⁰⁰)◦j+k⁰⁰−1Π⁰, for1≤i < j, Π◦J(Π⁰◦_i−j+1Π⁰⁰), forj≤i < k⁰+j,

(Π◦_i−k⁰+1Π⁰⁰)◦jΠ⁰, forj+k⁰≤i < k+k⁰−1,

2. (Equivariance) For each1 ≤i ≤m,τ ∈ Smand σ ∈ Sn, letτ◦iσ ∈ Sm+n−1be given by inserting the permutationσat theith place inτ. IfΠ∈ C^k andΠ⁰ ∈C^k0 then(Πτ)◦i(Π⁰σ) = (Π◦τ(i)Π⁰)(τ◦iσ).

3. (Unitality)1◦1Π⁰= Π⁰andΠ◦i1= Πfor eachΠ∈C^k andΠ⁰∈C^k0. LetV =L

nVnbe a graded space overC. We will say thatV is a (symmetric)C-module, if there is an action ofConV which satisfies:

1. EachΠ∈C^macts as a linear applicationV^m→V.

2. (Compatibility with the graduation)[π₁, . . . , π_m](V_j1, . . . , V_jm) =0if#π_i6=j_ifor some1≤i≤ mandom(Vj₁, . . . , Vj_m) = 0. Otherwise, [π1, . . . , πm] ∈ Cn sendsV#π₁ × · · · ×V#π_m toVn. Note also the special case1(v) =vfor eachv∈V.

3. (Compatibility with the compositions)

(Π◦iΠ⁰)(v1, . . . , vk+k⁰−1) = Π(v1, . . . , v_i−1,Π⁰(vi, . . . , vi+k⁰−1), vi+k⁰, . . . , vk+k⁰−1).

4. (Symmetry)Π(v1, . . . , vk) = (Πσ)(v_σ(1), . . . , v_σ(k)).

If V is generated (as a C-module) by {v_n : n ≥ 1} with v_n ∈ V_n, then setting v^{{π1,...,πm}} = [π₁, . . . , π_m](v_#π1, . . . , v_#πm), we haveV_n = span{v^π : π n}. Note that the existence ofv^π fol- lows from the point 4 of the definition of aC-module.

Example 2.2 IfAis a noncommutative alphabet, the algebraChAican be endowed with a structure of C-module by setting

[π₁, . . . , π_m](w₁, . . . , w_m) =

[π₁,...,π_m](w₁, . . . , w_m) ifw_i∈A^#πifor each1≤i≤m

0 otherwise

wherew₁, . . . , w_m ∈A^∗ and [π₁,...,π_m](w₁, . . . , w_m) =a₁. . . a_nis the only word ofA^{#π1+···+#πm} such that for each1≤i≤m, ifπ_i={j₁, . . . , j_{`</sub>},a<sub>j1</sub>. . . a<sub>j`} =w_j.

Note thatC[A]has also a structure ofC-module defined by[π₁, . . . , π_m](x₁, . . . , x_m) =x₁. . . x_mifx_i is a monomial of degree#πi.

Proposition 2.3 WSymis aC-module.

Proof:We define the action ofCon the power sums by [π1, . . . , πm](Φⁿ¹, . . . ,Φ^nm) =

Φ^{{π1,...,πm}} ifn_i= #π_ifor each1≤i≤m

0 otherwise.

and extend it linearly to the spacesWSym_n. Since this action is compatible with the realization:

[π₁, . . . , π_m](Φⁿ¹, . . . ,Φ^nm)(A) = _{[π1,...,πm]}(Φⁿ¹(A), . . . ,Φ^nm(A))

(the definition of π is given in Example 2.2) and WSym is obviously stable by the action of C, WSym(A)is a sub-C-module ofChAi. Hence,WSymis aC-module.

A morphism of C-module is a linear map ϕfrom a C-module V1 to another C-module V2 satisfying ϕΠ(v1, . . . , vm) = Π(ϕ(v1), . . . , ϕ(vm))for eachΠ∈C^m. In this context, a virtual alphabet (or a spe- cialization) is defined by a morphism ofC-moduleϕfromWSymto aC-moduleV. The image of a word symmetric functionf will be denoted byf[ϕ]. TheC-moduleWSymis free in the following sense:

Proposition 2.4 LetV be generated byv₁ ∈ V₁, v₂ ∈ V₂, . . . , v_n ∈V_n, . . . as aC-module (or equiva- lently,V_n = span{v^π : πn}). There exists a morphism ofC-moduleϕ: WSym →V, which sends Φⁿtov_n.

Proof:This follows from the fact that{Φ^π:πn}is a basis ofWSym_n. Letϕ: WSym_n→Vnbe the linear map such thatϕ(Φ^π) =V^π; this is obviously a morphism ofC-module.

For convenience, we will writeWSym[ϕ] =ϕWSym.

Example 2.5 1. LetA be an infinite alphabet. The restriction toWSym(A) of the morphism of algebraϕ: ChAi → C[A]defined byϕ(a) = ais a morphism ofC-module sendingWSym(A) toSym(A)(the algebra of symmetric functions on the alphabetA). This morphism can be defined without the help of alphabets, considering thatSymis generated by the power sumsp1, . . . , pn, . . . with the action[π₁, . . . , π_m](p_#π1, . . . , p_#πm) =p_#π1. . . p_#πm.

2. LetAbe any alphabet (finite or not). IfV is a sub-C-module ofChAigenerated by the homogeneous polynomialsP_n ∈C[Aⁿ]as aC-module, the linear map sending, for eachπ={π₁, . . . , π_m},Φ^π to _Π[P_#π1, . . . , P_#πm], whereΠ = [π₁, . . . , π_m], is a morphism ofC-module.

3 The Hopf algebra of set partitions into lists

3.1 Set partitions into lists

A set partition into listsis an object which can be constructed from a set partition by ordering each block. For example,{[1,2,3],[4,5]}and{[3,1,2],[5,4]}are two distinct set partitions into lists of the set {1,2,3,4,5}. The number of set partitions into lists of ann-element set (or set partitions into lists of size n) is given by Sloane’s sequence A000262 (20). IfΠis a set partition into lists of{1, . . . , n}, we will write Πn. We will denote bycycle support(σ)thecycle supportof a permutationσ,i.e., the set partition associated to its cycle decomposition. For instance,cycle support(325614) ={{135},{2},{4,6}}. A set partition into lists can be encoded by a set partition and a permutation in view of the following easy result:

Proposition 3.1 For alln, the set partitions into lists of sizenare in bijection with the pairs(σ, π)where σis a permutation of sizenandπis a set partition which is less fine than or equal to the cycle support of σ.

Indeed, from a set partitionπand a permutationσ, we obtain a set partition into listsΠby ordering the elements of each block ofπso that they appear in the same order as inσ.

Example 3.2 Starting the set partitionπ={{1,4,5},{6},{3,7},{2}}and the permutationσ= 4271563, we obtain the set partition into listsΠ ={[4,1,5],[7,3],[6],[2]}.

3.2 Construction

LetΠnandΠ⁰n⁰be two set partitions into lists. Then, we setΠ]Π⁰ = Π∪ {[l1+n, . . . , lk+n] : [l1, . . . , lk] ∈ Π⁰} n+n⁰. LetΠ⁰ ⊂Π n, since the integers appearing inΠ⁰ are all distinct, the standardizedstd(Π⁰)ofΠ⁰is well defined as the unique set partition into lists obtained by replacing the ith smallest integer inΠbyi. For example,std({[5,2],[3,10],[6,8]}) ={[3,1],[2,6],[4,5]}.

Definition 3.3 The Hopf algebraBWSym is formally defined by its basis (Φ^Π)where the Π are set partitions into lists, its productΦ^ΠΦ^Π0 = Φ^Π]Π0 and its coproduct∆(Φ^Π) = PΦ^std(Π0)⊗Φ^std(Π00), where the sum is over the(Π⁰,Π⁰⁰)such thatΠ⁰∪Π⁰⁰= ΠandΠ⁰∩Π⁰⁰=∅.

Following Section 3.1 and for convenience, we will use alternativelyΦ^Π andΦ(^σπ)to denote the same object.

We defineMΠ =M(^σ_π)by settingΦ(^σπ) = X

π≤π⁰

M(^σ_π)⁰.The formula being diagonal, it definesMΠfor anyΠ and proves that the family(MΠ)Πis a basis of BWSym. Consider for instance{[3,1],[2]} ∼ 321

{{1,3},{2}}

, we haveΦ{[3,1],[2]}=M{[3,1],[2]}+M_{[3,2,1]}.

For any set partition into listsΠ, lets(Π)be the corresponding classical set partition. Then, the linear applicationφdefined byφ(Φ^Π) = Φ^s(Π)is obviously a morphism of Hopf algebras. As an associative al- gebra,BWSymhas also algebraical links with the algebraFQSym(4). Recall that this algebra is defined by its basis(E^σ)whose product isE^σE^τ =E^σ/τ,whereσ/τis the word obtained by concateningσand the word obtained fromτby adding the size ofσto all the letters (for example321/132 = 321465).

The subspaceV ofWSym⊗FQSymlinearly spanned by theΦ^π⊗E^σsuch that the cycle supports ofσ is finer thanπis a subalgebra, and the linear application sendingΦ^π,σtoΦ_π⊗E^σis an isomorphism of algebras. Moreover, when the set of cycle supports ofσis finer thanπ,M_π⊗E^σalso belongs toV and is the image ofM(^σ_π).

The linear application fromBWSymtoFQSymwhich sendsM_{[σ]}toE^σandMΠto0ifcard(Π)>1, is also a morphism of algebras.

3.3 Realization

LetA^(j) = {a^(j)_i | i > 0} be an infinite set of bi-indexed noncommutative variables, with the order relation defined bya^(j)_i < a^(j)_i0 ifi < i⁰. LetA = S

jA^(j). Consider the set partition into listsΠ = {L¹, L², . . .} ={[l₁¹, l¹₂, . . . , l¹_n

1],[l²₁, l²₂, . . . , l_n²

2], . . .} n. Then, one obtains apolynomial realization BWSym(A)by identifyingΦ^ΠwithΦ^Π(A), the sum of all the monomialsa₁. . . a_n(where thea_iare in A) such thatk=k⁰impliesa_lk

t, a_lk0

s ∈A^(j)for somej, and for eachk,std(a_i1. . . a_ink) = std(l^k₁. . . l^k_n

k) with{l₁^k, . . . , l^k_nk}={i1<· · ·< in_k}(The “B” ofBWSymis for “bi-indexed letters”). The coproduct

∆can be interpreted by identifying∆(Φ^Π)withΦ^Π(A+B)as in the case ofWSym. Here, ifa∈Aand b∈Bthenaandbare not comparable.

Proposition 3.4 The Hopf algebrasBWSymandBWSym(A)are isomorphic.

Now, letM_Π⁰(A)be the sum of all the monomialsa1. . . an,ai ∈A, such thata_lk

t anda_lk0

s belong in the sameA^(j)if and only ifk=k⁰, and for eachk,std(ai₁. . . ai_nk) = std(l₁^k. . . l^k_nk)with{l^k₁, . . . , l^k_nk}= {i₁ <· · · < i_nk}. For example, the monomiala⁽¹⁾₁ a⁽¹⁾₁ a⁽¹⁾₂ appears in the expansion ofΦ{[1,3],[2]}, but not in the one ofM{[1,3],[2]}⁰ . TheM_Π⁰ form a new basis(M_Π⁰)ofBWSym. Note that this basis is not the same as(M_Π). For example, one has

Φ^{[1],[2]}=M_{[1],[2]}+M_{[1,2]}=M_{[1],[2]}⁰ +M_{[1,2]}⁰ +M_{[2,1]}⁰ .

Consider the basisFσofFQSymdefined in (4). The linear application, fromBWSymtoFQSym, which sendsM_{[σ]}⁰ toF_(σ−1), andM_Π⁰ to0ifcard(Π)>1, is a morphism of algebras.

3.4 Related Hopf algebras

By analogy with the construction ofBWSym, we define a “B” version for each of the algebrasSym, QSym andWQSym. In this section, we sketch briefly how to construct them; the complete study is deferred to a forthcoming paper.

As usual whenL= [`<sub>1</sub>, . . . , `_k]andM = [m₁, . . . , m₂]are two lists, the shuffle product is defined recur- sively by[ ] L=L [ ] ={L}andL M = [`<sub>1</sub>].([`₂, . . . , `_k] M)∪[m₁].(L [m₂, . . . , m_k]). The algebra of biword quasi-symmetric functions (BWQSym) has its bases indexed by set compositions into lists. The algebra is defined as the vector space spanned by the formal symbolsΦ_Π, whereΠis a composi- tion into lists of the set{1, . . . , n}for a givenn, together with the productΦ_ΠΦ_Π⁰=P

Π⁰⁰∈Π Π⁰[n]Φ_Π⁰⁰, whereΠ⁰[n]means that we addnto each of the integers in the lists ofΠ⁰ andΠis a composition into lists of{1, . . . , n}. Endowed with the coproduct defined by∆(ΦΠ) = P

Π⁰.Π⁰⁰=ΠΦ_std(Π⁰₎⊗Φ_std(Π⁰⁰₎, BWQSymhas a structure of Hopf algebra. Note thatBWQSym =L

nBWQSym_nis naturally graded;

the dimension of the graded componentBWQSym_nis2ⁿ⁻¹n!(see sequence A002866 in (20)).

The algebraBSym =L

nBSym_nis a graded cocommutative Hopf algebra whose bases are indexed by multisets of permutations. Formally, we setBSym_n= span{φ^{{σ1,...,σk}}:σi∈Sn_i, n1+· · ·+nk =n}, φ^S1.φ^S2 =φ^S1∪S2 and for any permutationσ,φ^{σ}is primitive. The dimensions of the graded compo- nents are given by the sequence A077365 of (20).

Finally,BQSym = L

nBQSym_n is generated byφ[σ₁,...,σ_k], its product isφLφL⁰ = P

L⁰⁰∈L L⁰φL⁰⁰

and its coproduct is∆(φ_L) =P

L=L⁰.L⁰⁰φ_L⁰⊗φ_L⁰⁰. The dimension of the graded componentBQSym_n is given by Sloane’s sequence A051296 (20).

4 On the R-P theorem

4.1 R-P theorem and symmetric functions

Consider two setsXandY such thatXis finite (#X =n), together with a groupG⊂S_nacting onXby permuting its elements. We consider the setY^Xof the mapsX →Y. The type of a mapfis the vector of the multiplicities of its images; more precisely,type(f)∈N^Y withtype(f)y= #{x∈X :f(x) =y}.

For instance, considerX ={a, b, c, d, e},Y ={0,1,2},f(a) =f(c) =f(d) = 1,f(b) = 2,f(c) = 0:

we havetype(f) = [1⁰,3¹,1²]. The action ofGonXinduces an action ofGonY^X. Obviously, the type of a function is invariant for the action ofG. Then all the elements of an orbit ofGinY^Xhave the same type, so that the type of an orbit will be the type of its elements. The question is: how to count the number nI of orbits for the given typeI? Note that, ifλI denotes the (integer) partition obtained by removing all the zeros inIand reordering its elements in the decreasing order,λI =λI⁰impliesnI =nI⁰; it suffices to understand how to computenλwhenλis a partition. The Redfield-P´olya theorem deals with this problem and its main tool is the cycle index:

ZG := 1

#G X

σ∈G

pcycle type(σ),

wherecycle type(σ)is the (integer) partition associated to the cycle ofσ(for instanceσ = 325614 = (135)(46),cycle type(σ) = [3,2,1]). Whenλ= [λ1, . . . , λk]is a partition,p^λdenotes the (commuta- tive) symmetric functionp^λ=pλ1. . . pλ_kandpnis the classical power sum symmetric function.

The Redfield-P´olya theorem states:

Theorem 4.1 The expansion ofZGon the basis(mλ)of monomial symmetric functions is given by Z_G =X

λ

n_λm_λ.

Example 4.2 Suppose that we want to enumerate the non-isomorphic non-oriented graphs on three ver- tices. The symmetric groupS3acting on the vertices induces an action of the group

G:={123456,165432,345612,321654,561234,543216} ⊂S₆

on the edges. The construction is not unique. We obtain the groupGby labelling the6edges from1to 6. Hence, to each permutation of the vertices corresponds a permutation of the edges. Here, the1labels the loop from the vertex1to itself,2labels the edge which links the vertices1and2,3is the loop from the vertex2to itself,4labels the edge from the vertex2to the vertex3,5is the loop from the vertex3to itself, finally,6links the vertices1and3. The cycle index ofGis

ZG = ¹₆(p⁶₁+ 3p²₂p²₁+ 2p²₃) =m6+ 2m51+ 4m42+ 6m411+ 6m33+. . .

The coefficient4ofm₄₂means that there exists4non-isomorphic graphs with4edges coloured in blue and2edges coloured in red.

4.2 Word R-P theorem

Ifσis a permutation, we defineΦ^σ := Φcycle support(σ).Now for our purpose, a mapf ∈ Y^X will be encoded by a wordw: we consider an alphabetA={ay :y∈Y}, the elements ofXare relabelled by 1,2, . . . ,#X =nandwis defined as the wordb1. . . bn ∈Aⁿsuch thatbi=a_f(i). With these notations, the action ofGonY^X is encoded by the action of the permutations ofGon the positions of the letters in the words ofAⁿ.

It follows that for any permutationσ∈G, one has Φ^σ= X

wσ=w

w. (1)

The cycle support polynomial is defined byZG:=P

σ∈GΦ^σ.From (1) we deduce Z_G=X

w

#Stab_G(w)w

whereStab_G(w) ={σ∈G:wσ=w}is the subgroup ofGwhich stabilizesw. In terms of monomial functions, this yields :

Theorem 4.3

ZG=X

π

#StabG(wπ)Mπ

wherew_πis any worda₁. . . a_nsuch thata_i=a_jif and only ifi, j∈π_kfor some1≤k≤n.

Example 4.4 Consider the same example as in Example 4.2. Each graph is now encoded by a word a1a2a3a4a5a6: the lettera1 corresponds to the colour of the vertex1, the letter2 to the colour of the vertex2and so on.

The cycle support polynomial is

ZG := Φ{{1},{2},{3},{4},{5},{6}}+ Φ{{2,6},{3,5},{1},{4}}

+Φ{{1,3},{4,6},{2},{5}}+ Φ{{1,5},{2,4},{3},{6}}+ 2Φ{{1,3,5},{2,4,6}}.

The coefficient ofM{{2,6},{3,5},{1},{4}}inZG is2because it appears only inΦ{{1},{2},{3},{4},{5},{6}}

andΦ{{2,6},{3,5},{1},{4}}. The monomials ofM{{2,6},{3,5},{1},{4}}are of the formabcdbcb, wherea, b, c anddare four distinct letters. The stabilizer ofabcdcbinGis the two-element subgroup{123456,165432}.

Note that the cycle support polynomial has already appeared in the literature on the work of Sagan and Gebhard (5) on a slightly different setting which is a special case of our purpose.

4.3 From word R-P theorem to R-P theorem

The aim of this section is to link the numbersnI of Section 4.1 and the numbers#StabG(w)appearing in Section 4.2.

Ifwis a word we will denote byorbG(w)its orbit under the action ofG. The Orbit-stabilizer theorem (seee.g.(3)) together with Lagrange’s theorem gives:

#G= #orbG(w)#StabG(w) (2)

Denote byΛ(π)the unique integer partition defined by (#π₁, . . . ,#π_k)if π = {#π1, . . . , π_k} with

#π₁≥#π₂≥ · · · ≥#π_k. Ifλ= (m^km, . . . ,2^k2,1^k1)we setλ^!=k_m!. . . k₂!k₁!. The shape of a word wis the unique set partitionπ(w)such thatwis a monomial ofM_π(w). Note that all the orbits of words with a fixed shapeπhave the same cardinality. Letπ = {π₁, . . . , π_k}andA_k = {a₁, . . . , a_k}be an alphabet of sizek. we will denote byS_π,Akthe set of wordswwith sizeksuch that the number of letter aiinwis#πi. All the words ofSπ,A_khave the same coefficients inZGand this set splits into _#orb(w^Λ(π)!

π)

orbits of size#orb(w_π)

nλ= X

Λ(π)=λ

λ^!

#orbG(wπ) = X

Λ(π)=λ

λ^!#Stab_G(w_π)

#G . (3)

If we consider the morphism of algebraθ: WSym →Symwhich sendsΦn topn, we haveθ(Mπ) = Λ(π)^!mλ. Hence, we have

1

#Gθ(ZG) =X

λ



 X

Λ(π)=λ

λ^!

#orbG(wπ)



mλ=X

λ

nλmλ

as expected by the Redfield-P´olya theorem (Theorem 4.1).

4.4 R-P theorem without multiplicities

Examining with more details Example 4.4, the coefficient2ofM{2,6},{3,5},{1},{4} inZG follows from the group{123456,165432}of order two which stabilizesabcdcb. In terms of set partitions into lists, this can be interpreted by M{[2,6],[3,5],[1],[4]}+M{[6,2],[5,3],[1],[4]}→2M{{2,6},{3,5},{1},{4}}.We deduce the following version (without multiplicities) of Theorem 4.3 inBWSym.

Theorem 4.5 LetGbe a permutation group. We have

ZG:= X

σ∈G

Φ(cycle support(σ)^σ ) =X

π

X

σ∈Stabπ(G)

M(^σ_π).

Consider again Example 4.4.

ZG= Φ({{1}{2}{3}{4}{5}{6}}¹²³⁴⁵⁶ ) + Φ({{1}{26}{35}{4}}¹⁶⁵⁴³² ) + Φ({{135}{246}}³⁴⁵⁶¹² ) + Φ({{13}{2}{46}{5}}³²¹⁶⁵⁴ ) + Φ({{135}{246}}⁵⁶¹²³⁴ ) + Φ({{15}{234}{6}}⁵⁴³²¹⁶ ).

When expanded in the monomialM basis, there are exactly2terms of the formM({{2,6},{3,5},{1},{4}}^σ ) (forσ= 123456andσ= 165432). Note that we can use another realization which is compatible with the space but not with the Hopf algebra structure. It consists to setΦ(e ^σπ) :=P

w σ w

, where the sum is over the wordsw=w1. . . wn(wi∈A) such that ifiandjare in the same block ofπthenwi=wj. If we consider the linear applicationψesendingΦ(^σπ)toΦ(e ^σπ),ψesendsM(^σ_π)toMf(^σ_π) :=P

w σ w

, where the sum is over the wordsw=w1. . . wn (wi ∈A) such thatiandj are in the same block ofπif and only ifwi=wj. Letwbe a word, the set of permutationsσsuch that _w^σ

appears in the expansion ofψ(e Z^G) is the stabilizer ofwinG. The linear application sending each biword _w^σ

towsendsΦ(e ^σπ)toΦ^π and P

σ∈StabG(w) σ w

to#StabG(w)w. Note that#StabG(w)is also the coefficient of the corresponding monomialM_π(w) in the cycle support polynomial ZG. For instance, we recover the coefficient 2 in Example 4.4 from the biwords ¹²³⁴⁵⁶_abcdcb

and ¹⁶⁵⁴³²_abcdcb

inψ(e ZG).

4.5 WSym and Harary-Palmer type enumerations

Let A := {a1, . . . , am} be a set of formal letters and I = [i1, . . . , ik] a sequence of elements of {1, . . . , m}. We define the virtual alphabetA_I by

Φⁿ(A_I) := (a_i1. . . a_ik)^nk + (a_i2. . . a_ika_i1)^nk +· · ·+ (a_ika₁. . . a_ik−1)^nk,

ifkdividesnand0otherwise. Ifσ∈Smwe define the alphabetAσas the formal sum of the alphabets Acassociated to its cycles:

Φ^{1...n}[A_σ] := X

ccycle inσ

Φⁿ[A_c].

From Example 2.5.2, the set{Φ^{1,...,n}[Aσ] :n∈N}generates the sub-C-moduleWSym[Aσ]ofChAi (the compositionΠacting by Π).

LetH⊂SmandG⊂Snbe two permutation groups. We defineZ(H;G) :=P

τ∈HΦ^G[Aτ].

Proposition 4.6 We have:

Z(H;G) = X

w∈Aⁿ

#StabH,G(w)w

whereStabH,G(w)denotes the stabilizer ofwunder the action ofH×G(Hacting on the left on the names of the variablesai and Gacting on the right on the positions of the letters in the word); equivalently, Stab_H,G(a_i1. . . a_ik) ={(τ, σ)∈H×G:a_τ(iσ(j))for each1≤j≤n}.

Hence, from Burnside’s classes formula, sending each variable to1inZ(H;G), we obtain the number of orbits ofH×G.

Example 4.7 Consider the set of the non-oriented graphs without loop whose edges are labelled by three colours. Suppose that we consider the action of the groupH ={123,231,312} ⊂S_non the colours. We want to count the number of graphs up to permutation of the vertices (G=S₃) and the action ofHon the edges. There are three edges, and each graph will be encoded by a worda_i1a_i2a_i3 wherei_jdenotes the colour of the edgej. We first compute the specializationΦ^{1...n}[Aσ]for1≤n≤3andσ∈H. We find Φ^{1}[A123] =a1+a2+a3,Φ^{1}[A231] = Φ^{1}[A312] = 0,Φ^{1,2}[A123] =a²₁+a²₂+a²₃,Φ^{1,2}[A231] = Φ^{1,2}[A312] = 0,Φ^{1,2,3}[A123] = a³₁+a³₂+a³₃,Φ^{1,2,3}[A213] = a2a3a1+a3a1a2+a1a2a3, and Φ^{1,2,3}[A312] =a1a3a2+a3a2a1+a2a1a3. Now, we deduce the values of the otherΦ^π[Aσ]withπ3 andσ∈Hby the action of Π. For instance:

Φ{{1,2},{3}}[A123] = [{1,2},{3}] Φ^{1,2}[A123],Φ^{1}[A1]

= a³₁+a²₁a2+a²₁a3+a²₂a1+a³₂+a²₂a3+a²₃a1+a²₃a2+a³₃. We find also

Φ{{1,3},{2}}[A123] =a³₁+a1a2a1+a1a3a1+a2a1a2+a³₂+a2a3a2+a3a1a3+a3a2a3+a³₃, Φ{{1},{2,3}}[A123] =a³₁+a2a²₁+a3a²₁+a1a²₂+a³₂+a3a²₂+a1a²₃+a2a²₃+a³₃,

Φ{{1},{2},{3}}[A₁₂₃] = (a₁+a₂+a₃)³. The otherΦ^π[A_σ]are zero. Hence,

Z[H;S3] = Φ¹²³[A123] + Φ¹³²[A123] + Φ²¹³[A123] + Φ³²¹[A123]+

Φ²³¹[A231] + Φ²³¹[A312] + Φ³¹²[A231] + Φ³¹²[A312]

= 6(a³₁+a³₂+a³₃) + 2P

i6=ja²_iaj+ 2P

i6=jajaiaj+ 2P

i6=jaja²_i +3(a1a2a3+a2a3a1+a3a1a2) + 3(a1a3a2+a3a2a1+a2a1a3).

The coefficient3ofa1a2a3means that the word is invariant under the action of three pairs of permutations (here(123,123), (231,312), (312,231)). Settinga1=a2=a3= 1, we obtainZ[H,S3] = 18×4:18 is the order of the groupH×S3and4is the number of orbits:

{a³₁, a³₂, a³₃},{a²₁a2, a1a2a1, a2a²₁, a²₂a3, a2a3a2, a3a²₂, a²₃a1, a3a1a3, a1a²₃},{a²₂a1, a2a1a2, a1a²₂, a²₁a3, a1a3a1, a3a²₁, a²₃a2, a3a2a3, a1a²₃}, and{a1a2a3, a1a3a2, a2a1a3, a2a3a1, a3a1a2, a3a2a1}.

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