Bijective evaluation of the connection coefficients of the double coset algebra

HAL Id: hal-00755301

https://hal.inria.fr/hal-00755301v2

Submitted on 13 Oct 2015

HAL is a multi-disciplinary open access archive for the deposit and dissemination of sci- entific research documents, whether they are pub- lished or not. The documents may come from teaching and research institutions in France or abroad, or from public or private research centers.

L’archive ouverte pluridisciplinaire HAL, est destinée au dépôt et à la diffusion de documents scientifiques de niveau recherche, publiés ou non, émanant des établissements d’enseignement et de recherche français ou étrangers, des laboratoires publics ou privés.

double coset algebra

Alejandro H. Morales, Ekaterina A. Vassilieva

To cite this version:

Alejandro H. Morales, Ekaterina A. Vassilieva. Bijective evaluation of the connection coefficients of the double coset algebra. 23rd International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2011), 2011, Reykjavik, Iceland. pp.681-692. �hal-00755301v2�

Bijective evaluation of the connection coefficients of the double coset algebra

Alejandro H. Morales

and Ekaterina A. Vassilieva

1Department of Mathematics, Massachusetts Institute of Technology, Cambridge, MA, USA

2Laboratoire d’Informatique de l’Ecole Polytechnique, Palaiseau, France

Abstract.This paper is devoted to the evaluation of the generating series of the connection coefficients of the double cosets of the hyperoctahedral group. Hanlon, Stanley, Stembridge (1992) showed that this series, indexed by a partitionν, gives the spectral distribution of some random matrices that are of interest in random matrix theory.

We provide an explicit evaluation of this series whenν = (n)in terms of monomial symmetric functions. Our development relies on an interpretation of the connection coefficients in terms of locally orientable hypermaps and a new bijective construction between partitioned locally orientable hypermaps and some permuted forests.

R´esum´e.Cet article est d´edi´e `a l’´evaluation des s´eries g´en´eratrices des coefficients de connexion des classes doubles (cosets) du groupe hyperocta´edral. Hanlon, Stanley, Stembridge (1992) ont montr´e que ces s´eries index´ees par une partitionνdonnent la distribution spectrale de certaines matrices al´eatoires jouant un rˆole important dans la th´eorie des matrices al´eatoires. Nous fournissons une ´evaluation explicite de ces s´eries dans le casν = (n)en termes de monˆomes sym´etriques. Notre d´eveloppement est fond´e sur une interpr´etation des coefficients de connexion en termes d’hypercartes localement orientables et sur une nouvelle bijection entre les hypercartes localement orientables partitionn´ees et certaines forˆets permut´ees.

Keywords:double coset algebra, connection coefficients, locally orientable hypermaps, forests

1 Introduction

In what follows, we denote byλ = (λ1, λ2, . . . , λk)_≥ ` nan integer partition ofnand`(λ) = kthe number of parts ofλ. Ifni(λ)is the number of parts ofλthat are equal toi(by conventionn0(λ) = 0), then we writeλas1^n1(λ)2^n2(λ). . .and letAutλ = Q

ini(λ)!. Also, ifλ ` n, letλλand2λbe the partitions of2n(λ1, λ1, λ2, λ2, . . .)and(2λ1,2λ2, . . .)respectively. Let[m] = {1, . . . , m}andSmbe the symmetric group onmelements,e.g.on[m], and letCλbe the conjugacy class inSmof permutations wwith cycle typetype(w) =λ`m.

We look atperfect pairingsof the set[n]∪[n] =b {1, . . . n,b1, . . . ,bn}which we view as fixed point free involutions inS2n([n]∪[n]). Note that forb f, g∈S2n, the disjoint cycles of the productf◦ghave repeated lengthsi.e.f◦g∈ Cλλ. Also, forw∈S_2n, letf_wbe the pairing(w⁻¹(1), w⁻¹(ˆ1))· · ·(w⁻¹(n), w⁻¹(ˆn)).

LetBnbe the hyperoctahedral group which we view as the centralizer inS2n of the involutionf? = (1b1)(2b2)· · ·(nn). Thenb |Bn|= 2ⁿn!, and it is well known that the double cosets ofBninS2nare indexed by partitionsνofn, and consist of permutationsw∈S_2nsuch that the cycle type off_?◦f_wisνν[8, Ch.

1365–8050 c2011 Discrete Mathematics and Theoretical Computer Science (DMTCS), Nancy, France

VII.2]. If we denote such double coset byK_ν and pick from it a fixed elementw_ν, then letb^ν_λ,µbe the number of ordered factorizationsu₁·u₂ofw_νwhereu₁∈K_λandu₂∈K_µ. We provide a combinatorial formula forb^ν_λ,µwhen ν = (n) = n(sayw_(n) = (123. . . n)(bn n−1b n−2b . . .b1)) by interpreting these factorizations aslocally orientable (partitioned) unicellular hypermaps.

Theorem 1.1 Letbⁿ_λ,µ be the number of ordered factorizationsu₁·u₂ of w_(n) whereu₁ ∈ K_λ and u₂∈K_µ. Ifp_λ(x)andm_λ(x)are the power and monomial symmetric functions then

1 2ⁿn!

X

λ,µ`n

bⁿ_λ,µp_λ(x)p_µ(y) =

X

λ,µ`n

AutλAutµmλ(x)mµ(y) X

A∈M_λ,µ

N(A) A!

(n−q−2r)!(n−p−2r)!

(n+ 1−p−q−2r)!

p⁰!q⁰! (r−p⁰)! (r−q⁰)!

2^2r−p0−q0

Y

i,j,k

i−1 j, k, j+k

^(P+Q)i,j,k i−1 j, k, j+k−1

^(P0+Q0)i,j,k

(1)

Where,M_λ,µis the set of4-tuplesA= (P, P⁰, Q, Q⁰)of tridimensional arrays of non negative integers withp=|P|=P

i,j,k≥0Pijk6= 0, p⁰ =`(λ)−p=|P<sup>0</sup>|, q=|Q|,q<sup>0</sup> =`(µ)−q=|Q⁰|, and ni(λ) = P

j,kPijk+P_ijk⁰ , ni(µ) = P

j,k≥0Qijk+Q⁰_ijk,

r = P

i,j,k(j+k)(Pijk+P_ijk⁰ ), r = P

i,j,k(j+k)(Qijk+Q⁰_ijk), q⁰ = P

i,j,kj(P_ijk+P_ijk⁰ ), p⁰ = P

i,j,kj(Q_ijk+Q⁰_ijk).

whereA! =Q

i,j,kPijk!P_ijk⁰ !Qijk!Q⁰_ijk!. AndN(A) =P

t,u,vtPtuvifq⁰ = 0, otherwise ifq⁰6= 0then

N(A) = 1 q⁰

X

t−2u−2v>0

tPtuv

t−2u−2v



(t−2u−2v)



 δp⁰6=0

p⁰ X

i,j,k

jQX

i,j,k

jP⁰+ P

i,j,k(i−1−2j−2k)QP

i,j,kjP n−q−2r





+u



 δp⁰6=0

p⁰ X

i,j,k

(i−2j−2k)P⁰X

i,j,k

jQ⁰+X

i,j,k

(i−2j−2k)Q⁰1 +P

i,j,k(i−1−2j−2k)P n−q−2r







.

Remark 1.2 For some limit values of the4-tupleA, the summand definition inP

A∈Mλ,µ· · · on the RHS of Equation(1)is slightly different as detailed in Appendix 6 of the paper.

1.1 Background on connection coefficientsb^ν_λ,µ

By abuse of notation, let the double cosetKνalso represent the sum of its elements in the group algebra CS2n. ThenKνform a basis of a commutative subalgebra ofCS2n(theHecke algebraof theGelfand pair (S2n, Bn)) and one can check thatKλ·Kµ =P

νb^ν_λ,µKν. Thus,{b^ν_λ,µ}are theconnection coefficients of this double coset algebra. We useZλ(x)to denote thezonal polynomialindexed byλwhich can be viewed as an analogue of the Schur functionsλ(for more information on these polynomials see [8, Ch.

VII]). In terms ofp_µ:s_λ(x) =P

µz⁻¹_µ χ^λ_µp_µ(x)wherez_λ=Aut_λQ

ii^ni(λ),χ^λ_µare the irreducible char- acters of the symmetric group; andZ_λ(x) = _|B¹

n|

P

µ`nϕ^λ(µ)p_µ(x)whereϕ^λ(µ) =P

w∈Kµχ^2λ_type(w).

In [5, Lemma 3.3] a formula for the connection coefficients in terms ofχ^λ_µwas given:

b^ν_λ,µ= 1

|K_ν| X

β`n

1

H_2νϕ^β(ν)ϕ^β(λ)ϕ^β(µ), (2)

where|Kν| = |Bn||Cν|2^n−`(ν)[2, Lemma 2.1], andH2λ is the product of all thehook-lengthsof the partition2λ.

LetΨ^ν(x,y) = _|B¹

n|

P

λ,µb^ν_λ,µpλ(x)pµ(y). SoΨ⁽ⁿ⁾is the LHS of (1). Equation (2) immediately implies thatΨ^ν(x,y) = _|K¹

ν|

P

λ`n

|B_n|

H_2λϕ^λ(ν)Zλ(x)Zµ(y). Moreover, if for ann×nmatrixX we say thatpk(X) = trace(X^k), then in [5, Thm. 3.5] it was shown that Ψ^ν is also the expectation of pν(XU Y U^T)overU, whereUaren×nmatrices whose entries are independent standard normal random realvariables andX, Y are arbitrary but fixed real symmetric matrices.

2 Combinatorial formulation

2.1 Unicellular locally orientable hypermaps

From a topological point of view, alocally orientable hypermapof n edges can be defined as a connected bipartite graph with black and white vertices. Each edge is composed of two half edges both connecting the two incident vertices. This graph is embedded in a locally orientable surface such that if we cut the graph from the surface, the remaining part consists of connected components called faces or cells, each homeomorphic to an open disk. The map can be represented as a ribbon graph on the plane keeping the incidence order of the edges around each vertex. In such a representation, two half edges can be parallel or cross in the middle. A crossing (or a twist) of two half edges indicates a change of orientation in the map and that the map is embedded in a non orientable surface (projective plane, Klein bottle,...). We say a hypermap isrootedif it has a distinguished half edge. In [2], it was shown that rooted hypermaps admit a natural formal description involving triples of perfect pairings(f1, f2, f3)on the set of half edges where:

• f3associates half edges of the same edge,

• f1associates immediately successive (i.e. with no other half edges in between) half edges moving around the white vertices, and

• f₂associates immediately successive half edges moving around the black vertices.

Formally we label each half edge with an element in[n]∪[n] =b {1, . . . , n,b1, . . . ,bn}, labelling the rooted half edge by1. We then define(f₁, f₂, f₃)as perfect pairings on this set. Combining the three pairings gives the fundamental characteristics of the hypermap since:

• The cycles of f3◦f1 give the succession of edges around the white vertices. Iff3◦f1 ∈ Cλλ

then the degree distribution of the white vertices isλ(counting only once each pair of half edges belonging to the same edge),

• The cycles of f3◦f2 give the succession of edges around the black vertices. If f3◦f2 ∈ Cµµ

then the degree distribution of the black vertices isµ(counting only once each pair of half edges belonging to the same edge),

• The cycles off1◦f2encode the faces of the map. Iff1◦f2∈ Cννthen the degree distribution of the faces isν

In what follows, we consider the numberLⁿ_λ,µof rootedunicellular, or one-face, locally orientable hy- permaps with face distributionν = (n) =n¹, white vertex distributionλ, and black vertex distribution µ.

Letf₁be the pairing(1bn)(2b1)(3b2). . .(n n−1)b andf₂=f_?= (1b1)(2b2). . .(nn). We haveb f₁◦f₂= (123. . . n)(nnb −1b n−2b . . .b1)∈ C_(n)(n). Then by the above description, one can show that

Lⁿ_λ,µ=| {f₃pairings inS_2n([n]∪[bn])|f₃◦f₁∈ C_λλ, f₃◦f₂∈ C_µµ} |. (3) Moreover, the following relation betweenLⁿ_λ,µandbⁿ_λ,µholds [2, Cor 2.3]

Lⁿ_λ,µ= 1

2ⁿn!bⁿ_λ,µ. (4)

Thus we can encode the connection coefficients as numbers of locally orientable hypermaps.

Example 2.1 Figure 1 depicts a locally orientable unicellular hypermap inLⁿ_λ,µwithλ= 1¹2²3¹4¹and µ= 3¹4¹5¹(at this stage we disregard the geometric shapes around the vertices).

!"

!"

#"

#"

$"

$"

%"

%" &"

&"

'"

'"

("

("

)"

)"

*"

*"

!+"

!+"

!!"

!!"

!#"

!#"

,"

,"

,"

,"

,"

,"

,"

,"

,"

,"

,"

,"

Fig. 1:A unicellular locally orientable hypermap

2.2 Partitioned locally orientable hypermaps

Next, we consider locally orientable hypermaps where we partition the white vertices (black resp.). In terms of the pairings, this means we “color” the cycles off₃◦f₁(f₃◦f₂resp.) allowing repeated colors but imposing that the two cycles corresponding to each white (black resp.) vertex have the same color.

The following definition in terms of set partitions of[n]∪[bn]makes this more precise.

Definition 2.2 (Locally orientable partitioned hypermaps) We consider the setLPⁿ_λ,µof triples of the form(f3, π1, π2)wheref3is a pairing on[n]∪[n],b π1andπ2are sets partitions on[n]∪[n]b with blocks of even size and of respective types2λand2µ(orhalf typesλandµ) with the constraint thatπi(i= 1,2)is stable byfiandf3. Any such triple is called alocally orientable partitioned hypermapof type(λ, µ).

In addition, letLP_λ,µⁿ =| LPⁿ_λ,µ|.

Remark 2.3 The analogous notion of partitioned or colored map is common in the study of orientable maps. e.g. see [7],[4] for maps. Recently Bernardi in [1] extended the approach in [7] to find a bijection between locally orientable partitioned maps and orientable partitioned maps with a distinguishedplanar submap. As far as we know [1, Sect. 7] this technique does not extend to locally orientable hypermaps.

Lemma 2.4 The number of hat numbers in a block is equal to the number of non hat numbers

Proof: If a non hat numberi belongs to block π^k₁ then f₁(i) = i−1b also belongs to π₁^k. The same

argument applies to blocks ofπ2withf2(i) =bi. 2

Example 2.5 As an example, the locally orientable hypermap on Figure 1 is partitioned by the blocks:

π1 = {{c12,1,b3,4,b7,8,c11,12};{b1,2,b6,7,b8,9};{b2,3,c10,11};{b4,5,b5,6,b9,10}}

π2 = {{1,b1,3,b3,6,b6,10,c10}; {2,b2,7,b7,11,c11}; {4,b4,5,b5,8,b8,9,b9,12,c12}}

(blocks are depicted by the geometric shapes around the vertices, all the vertices belonging to a block have the same shape).

LetRλ,µ be the number of unordered partitionsπ = {π¹, . . . , π^p} of the set[`(λ)]such thatµj = P

i∈π^jλi for1 ≤ j ≤ `(µ). Then for the monomial and power symmetric functions,mλandpλ, we have:pλ=P

µλAutµRλ,µmµ. We use this to obtain a relation betweenLⁿ_λ,µandLP_λ,µⁿ Proposition 2.1 For partitionsρ, `nwe haveLP_ν,ρⁿ =P

λ,µRλνRµρLⁿ_λ,µ, whereλandµare refine- ments ofνandρrespectively.

Proof: Let (f3, π1, π2) ∈ LPⁿ_ν,ρ. Iff3◦f1 ∈ Cλλ andf3 ◦f2 ∈ Cµµthen by definition of the set partitions we have thatλandµare refinements oftype(π1) =νandtype(π2) =ρrespectively. Thus, we can classify the elements ofLPⁿ_ν,ρby the cycle types off3◦f1andf3◦f2.i.e.LPⁿ_ν,ρ=S

λ,µLPⁿ_ν,ρ(λ, µ) where

LPν,ρ(λ, µ) ={(f3, π1, π2)∈ LPⁿ_ν,ρ|(f3◦f1, f3◦f2)∈ Cλλ× Cµµ}.

IfLP_µρⁿ (λ, µ) =|LPⁿ_µρ(λ, µ)|then it is easy to see thatLP_µ,ρⁿ (λ, µ) =R_λνR_µρLⁿ_λµ.

2 By the change of basis equation betweenp_λandm_λ, this immediately relates the generating seriesΨⁿ and the generating series forLP_λ,µⁿ in monomial symmetric functions.i.e.Ψⁿ(x,y)is

X

λ,µ`n

Lⁿ_λ,µpλ(x)pµ(y) = X

λ,µ`n

AutλAutµLP_λ,µⁿ mλ(x)mµ(y). (5)

Definition 2.6 LetLP(A)be the set, of cardinalityLP(A), of partitioned locally orientable hypermaps withnedges whereA= (P, P⁰, Q, Q⁰)are tridimensional arrays such that fori, j, k≥0:

• Pijk(resp.P_ijk⁰ ) is the number of blocks ofπ1of half sizeisuch that:

(i) either1belongs to the block or its maximumnon-hatnumber is paired to ahatnumber byf3

(resp. blocks ofπ1not containing1such that the maximumnon-hatnumber of the block is paired to anon-hatnumber byf3),

(ii) the block containsjpairs{t, f3(t)}wheretis the maximumhatnumber of a block ofπ2such thatf3(t)is also ahatnumber, and,

(iii) the block contains as a wholej+kpairs{u, f3(u)} where bothuandf3(u)arenon-hat numbers.

• Q_ijk(resp.Q⁰_ijk) is the number of blocks ofπ₂of half sizeisuch that:

(i) the maximumhatnumber of the block is paired to anon-hat(resp.hat) number byf₃, (ii) the block containsjpairs{t, f₃(t)}wheretis the maximumnon-hatnumber of a block ofπ₁

non containing1and such thatf₃(t)is also anon-hatnumber, and

(iii) the block contains as a wholej+kpairs{u, f3(u)}where bothuandf3(u)arehatnumbers.

As a direct consequence, for LP(A)to be non zero A has to check the conditions of Theorem 1.1.

Furthermore:

LPⁿ_λ,µ= X

A∈Mλ,µ

LP(A) (6)

Example 2.7 The partitioned hypermap on Figure 1 belongs toLP(A)forP =E_4,1,0+E_3,0,1+E_2,0,0, P⁰=E_3,0,1,Q=E_5,0,1+E_4,1,0,Q⁰ =E_3,0,1whereE_t,u,v, the elementary array withE_t,u,v= 1and 0elsewhere.

One can notice that a hypermap is orientable if and only iff₃(t)is a hat number whent is a non hat number (we go through each edge of the map in both directions and there are no changes of direction during the map traversal). As a result, a hypermap inLP(A)is orientable if and only if:

• p⁰=q⁰=r= 0and

• Pijk=Qijk= 0ifj >0and/ork >0.

In this particular case, we have the following values forN(A)andA!:

• N(A) =P

i,i,kiPijk =P

tini(λ) =n

• A! =Q

iPi,0,0!Qi,0,0! =AutλAutµ

If we denotecⁿ_λ,µthe number of such orientable maps, by Theorem 1.1, Equation 6, Lemma 5 and Relation 4 we recover the following combinatorial result from [9, Thm. 1]:

Corollary 2.8 [9, Thm. 1]

X

λ,µ`n

cⁿ_λ,µp_λ(x)p_µ(y) =n X

λ,µ`n

(n−`(λ))!(n−`(µ))!

(n+ 1−`(λ)−`(µ))!m_λ(x)m_µ(y). (7) Remark 2.9 Note that{cⁿ_λ,µ}λ,muare better known as the connection coefficients of the symmetric group which count the number of ordered factorizations w1 ·w2 of the long cycle(1,2, . . . , n)inSn where w1∈ Cλandw2∈ Cµ.

2.3 Permuted forests and reformulation of the main theorem

We show that partitioned locally orientable hypermaps admit a nice bijective interpretation in terms of some recursive forests defined as follows:

Definition 2.10 (Rooted bicolored forests of degree A) In what follows we consider the setF(A) of permuted rooted forests composed of:

• a bicolored identified orderedseed treewith a white root vertex,

• other bicolored ordered trees, callednon-seed treeswith either a white or a black root vertex,

• each vertex of the forest has three kind of ordered descendants : tree-edges(connecting a white and a black vertex),thorns(half edges connected to only one vertex) andloopsconnecting a vertex to itself. The twoextremitiesof the loop are part of the ordered set of descendants of the incident vertex and therefore the loop can be intersected by thorns, edges and other loops as well.

The forests inF(A)also have the following properties:

(i) the root vertices of the non-seed trees have at least one descending loop with one extremity being the rightmost descendant of the considered vertex,

(ii) the total number of thorns (resp. loops) connected to the white vertices is equal to the number of thorns (resp. loops) connected to the black ones,

(iii) there is a bijection between thorns connected to white vertices and the thorns connected to black vertices. The bijection between thorns will be encoded by assigning the same symboliclatinlabels {a, b, c, . . .}to thorns associated by this bijection,

(iv) there is a mapping that associates to each loop incident to a white (resp. black) vertex, a black (resp.

white) vertexvsuch that the number of white (resp. black) loops associated to a fixed black (resp.

white) vertexvis equal to its number of incident loops. We will use symbolicgreeklabels{α, β, . . .}

to associate loops with vertices except for the maximal loop of a root vertexrof the non-seed trees.

In this case, we draw an arrow ( ) outgoing from the root vertexrand incoming to the vertex associated with the loop. Arrows are non ordered, and

(v) the ascendant/descendant structure defined by the edges of the forest and the arrows defined above is a tree structure rooted in the root of the seed tree.

Finally the degreeAof the forest is given in the following way:

(vii) Pijk(respP_ijk⁰ ) counts the number of non root white vertices (including the root of the seed tree) (resp. white root vertices excluding the root of the seed tree) of degreei, withj incoming arrows and a total ofj+kloops.

(viii) Qijk(respQ⁰_ijk) counts the number of non root black vertices (resp. black root vertices) of degree i, withjincoming arrows and totalj+kloops.

Example 2.11 As an example, Figure 2 depicts two permuted forests. The one on the left is of degree A= (P, P⁰, Q, Q⁰)forP =E4,1,0+E3,0,1+E2,0,0,P⁰ =E3,0,1,Q=E5,0,1+E4,1,0, andQ⁰=E3,0,1

while the one on the right is of degreeA⁽²⁾ = (P⁽²⁾, P⁰⁽²⁾, Q⁽²⁾, Q⁰⁽²⁾)forP⁽²⁾ = E7,0,3+E4,1,0, P⁰⁽²⁾={0}i,j,k,Q⁽²⁾ =E7,0,2, andQ⁰⁽²⁾=E4,0,2.

!" ^α"

α"

β"

β"

γ"

γ"

δ"

δ"

!"

#"

#"

α!

α!

β!

γ!

"!

"!

α!

β!

β!

γ! γ!

γ!

Fig. 2:Two Permuted Forests

Lemma 2.12 Using the Lagrange theorem for implicit functions, one can show thatF(A)equals

N(A) A!

(n−q−2r)!(n−p−2r)!

(n+ 1−p−q−2r)!

p⁰!q⁰! (r−p⁰)! (r−q⁰)!

2^2r−p0−q0

Y

i,j,k

i−1 j, k, j+k

^(P+Q)ijk i−1 j, k, j+k−1

^(P

0+Q⁰)_ijk

.

Reformulation of the main theorem

In order to show Theorem 1.1 the next sections are dedicated to the proof of the following stronger result:

Theorem 2.13 There is a bijectionΘA:LP(A)→ F(A) and soLP(A) =F(A).

3 Bijection between partitioned locally orientable unicellular hy- permaps and permuted forests

We proceed with the description of the bijective mappingΘAbetween partitioned locally orientable hy- permaps and permuted forests of degreeA. Let(f3, π1, π2)be a partitioned hypermap inLP(A). The first step is to define a set of white and black vertices with labeled ordered half edges such that:

• each white vertex is associated to a block ofπ1and each black vertex is associated to a block ofπ2,

• the number of half edges connected to a vertex is half the cardinality of the associated block, and

• the half edges connected to the white (resp. black) vertices are labeled with the non hat (resp. hat) integers in the associated blocks so that moving clockwise around the vertices the integers are sorted in increasing order.

Then we define an ascendant/descendant structure on the vertices. A black vertexbis the descendant of a white onewif the maximum half edge label ofbbelongs to the block ofπ₁associated tow. Similar rules apply to define the ascendant of each white vertex except the one containing the half edge label1.

If black vertexb^d(resp. white vertexw^d) is a descendant of white vertexw^a(resp. black vertexb^a) and has maximum half edge labelmsuch thatf₃(m)is the label of a half edge ofw^a(resp.b^a), i.e.f₃(m^b)is a non hat (resp. hat) number, then we connect these two half edges to form an edge. Otherwisef3(m)is a hat (resp. non hat) number and we draw an arrow ( ) between the two vertices. Note that descending edges are ordered but arrows are not.

Lemma 3.1 The above construction defines a tree structure rooted in the white vertex with half edge1.

Proof:Let black verticesb1andb2associated to blocksπ^b₂¹andπ^b₂² be respectively a descendant and the ascendant of white vertexwassociated toπ₁^w. We denote bym^b1,m^b2andm^wtheir respective maximum half edge labels (hat, hat, and non hat) and assumem^b1 6= bn. Asπ^w₁ is stable byf₁, thenf₁(m^b1)is a non hat number inπ^w₁ not equal to1. It followsm^b1 < f₁(m^b1) ≤m^w < f₂(m^w). Then asπ^b₂² is stable byf₂, it containsf₂(m^w)andf₂(m^w)≤m^b2. Putting everything together yieldsm^b1 < m^b2. In a similar fashion, assume white verticesw₁andw₂are descendant and ascendant of black vertexb. If we notem^w1,m^w2andm^btheir maximum half edge labels (non hat, non hat, and hat) withm^b6=n, one canb show thatm^w1 < m^w2. Finally, asf1(bn) = 1, the black vertex with maximum half edgebnis descendant

of the white vertex containing the half edge label1. 2

Example 3.2 Using the hypermap of Figure 1 we get the set of vertices and ascendant/descendant struc- ture as described on Figure 3.

Next, we proceed by linking half edges connected to the same vertex if their labels are paired byf3to form loops. Ifiandf3(i)are the labels of a loop connected to a white (resp. black) vertex such that neitherinorf3(i)are maximum labels (except if the vertex is the root), we assign the same label to the loop and the black (resp. white) vertex associated to the block ofπ₂(resp.π₁) also containingiandf₃(i).

!

"#

$# %#

!&#

&# '#

(#

)# !!#

*# +#

!,#

$#

*# %#

(#

!&# &# '#

!!#

!#

)# +#

!,#

"# "#

"#

"# "# "#

"#

"#

"# "#

"#

!

"#

$# %#

!&#

&# '#

(#

)#

!!#

*# +#

!,#

$#

*# %#

(#

!&# &# '#

!!#

!#

)# +#

"# "# !,#

"#

"# "# "#

"#

"#

"# "#

"#

Fig. 3:Construction of the ascendant/descendant structure

As we assign at most one label from{α, β, . . .}to a given vertex, several loops may share the same label.

Lemma 3.3 The number of loops connected to the vertex labeledαis equal to its number of incoming arrows plus the number of loops labeledαincident to other vertices in the forest.

Proof:The result is a direct consequence of the fact that in each block the number of hat/hat pairs is equal

to the number of non hat/non hat pairs 2

As a final step, we define a permutation between the remaining half edges (thorns) connected to the white vertices and the one connected to the black vertices. If two remaining thorns are paired byf₃then these two thorns are given the same label from{a, b, . . .}. All the original integer labels are then removed.

We denote byFethe resulting forest.

Example 3.4 We continue with the hypermap from Figure 1 and perform the final steps of the construction as described on Figure 4.(Note that the geometric shapes are here for reference only, they do not play any role in the final objectFe).

!

"#

$# %#

!&#

&# '#

(#

)#

!!#

*# +#

!,#

$#

*# %#

(#

!&# &# '#

!!#

!#

)# +#

"# "# !,#

"#

"# "# "#

"#

"#

"# "#

"#

α# α#

β#

β#

γ#

γ#

δ#

δ#

-# ^α#

α#

β#

β#

γ#

γ#

δ#

δ#

-#

.#

.#

Fig. 4:Final steps of the permuted forest construction

As a direct consequence of definition 2.10,Febelongs toF(A).