Matrix integrals and generating functions for enumerating rooted hypermaps by vertices, edges and faces for a given number of darts
MMatrix integrals and generating functions for enumeratingrooted hypermaps by vertices, edges and faces for a givennumber of darts
Jacob P Dyer1st April 2019 [email protected] of Mathematics, University of York, York YO10 5DD, UK
Abstract
A recursive method is given for finding generating functions which enumerate rooted hy-permaps by number of vertices, edges and faces for any given number of darts. It makes useof matrix-integral expressions arising from the study of bipartite quantum systems. Directevaluation of these generating functions is then demonstrated through the enumeration ofall rooted hypermaps with up to 13 darts.
Keywords: enumeration, rooted hypermap, bipartite quantum system, matrix integral, gener-ating function, divergent power series
This paper is an extension of work we carried out in a previous paper [1]. In that paper weshowed how the mean value of traces of integer powers of the reduced density operator of a finite-dimensional bipartite quantum system are proportional to generating functions for enumeratingone-face rooted hypermaps. We then used this relation to derive a matrix integral expressionfor these generating functions, and found closed form expressions for them.Matrix integral expressions derived from finding the average of a function of the reduceddensity operator have been studied for some time [2, 3, 4, 5, 6], so numerous methods forevaluating them have been described. In particular, Lloyd and Pagels were able to reduce thematrix integral to an integral over the space of eigenvalues with the density function [2] P ( p , . . . , p m ) dp . . . dp m ∝ δ (cid:32) − m (cid:88) i =1 p (cid:33) ∆ ( p , . . . , p m ) m (cid:89) k =1 p n − mk dp k where ∆( p , . . . , p m ) is the Vandermonde determinant of the eigenvalues of the reduced densityoperator. Using this, in conjunction with the work of Sen [6], we were able to evaluate aclosed-form expression for the one-face generating functions.In this paper we extend these methods to derive expressions for generating functions whichenumerate all rooted hypermaps by number of vertices, edges and faces for a given number of1 a r X i v : . [ m a t h . C O ] N ov arts (we give a definition of rooted hypermaps in Section 2). These generating functions aredefined recursively in terms of another expression called F ( m, n, λ ; x ) , which we define in Section3 and is itself evaluated using a matrix integral as above.Previous work already exists on enumeration of hypermaps [7, 8, 9], and in particular Walshmanaged to enumerate all rooted hypermaps with up to 12 darts by number of edges, verticesand darts, and genus [9]. But as far as we are aware this is the first time that generatingfunctions for enumerating all rooted hypermaps by these properties have been found withoutdirect computation of the hypermaps themselves. By avoiding having to generate the hypermapsindividually, we are able to vastly speed up the process of enumeration (there are more than r ! hypermaps with r darts [1], so generating them all is a very slow process).We will give an overview of our previous work in Section 2.1, before showing how best togeneralise it to multiple faces in Section 2.2. We will then use this to study the global generatingfunction for rooted hypermaps H ( m, n, λ ; x ) in Sections 3 and 4, before looking at the processof evaluating these functions and extracting hypermap counts from them in Section 5. A thorough discussion of hypermaps can be found in [10].A hypermap is a generalisation of a map (a graph embedded on an orientable surface sothat its complement consists only of regions which are homeomorphic to the unit disc) in whichthe edges are capable of having any positive number of connections to vertices instead of theusual two. Hypermaps can be thought of as equivalent to bipartite bicoloured maps on the samesurface (with the two colours of vertices in the map representing the vertices and edges of thehypermap) [11]. Each edge-vertex connection (the edges in the eqivalent bipartite map) is calleda dart , and a rooted hypermap is a hypermap where one of the darts has been labelled as the root , making it distinct from the others.The embedding of a hypergraph (the analogue of a graph) on an orientable surface to producea hypermap can be represented in other ways which do not require explicit consideration of thesurface involved. These are called combinatorial embeddings , and one such method uses anobject called a 3-constellation:
Definition 1.
A 3-constellation is an ordered triple { ξ, η, χ } of permutations acting on someset R , satisfying the following two properties:1. The group generated by { ξ, η, χ } acts transitively on R .2. The product ξηχ equals the identity.A hypermap H with r darts can be expressed using a 3-constellation on the set R = [1 . . . r ] .If the elements of R are associated with the darts in H , then the actions of ξ , η and χ are to cyclethe darts around their adjacent faces, edges and vertices respectively. For our purposes here,the important result is that the number of faces, edges and vertices in a hypermap H ≡ { ξ, η, χ } are given by the number of cycles in ξ , η and χ respectively [10, p 43].Two 3-constellations { ξ, η, χ } and { ξ (cid:48) , η (cid:48) , χ (cid:48) } are isomorphic to each other if they are relatedby the bijection τ : { ξ, χ, η } → { ξ (cid:48) , χ (cid:48) , η (cid:48) } = { τ ξτ − , τ ητ − , τ χτ − } (2.1)for some permutation τ [10, p 8]. are isomorphic to each other (the action of τ on the hypermapas given above simply involves a reordering of the darts in the set R without changing the2igure 2.1: A diagrammatic representation of a one-face rooted hypermap H ≡ { ξ, η, χ } , with ξ = (12 . . . r ) and η = (1453)(2)(67) , referred to as a ladder diagram . Mapping from black towhite, the single dashed lines represent ξ − and the double (solid and dashed) lines represent η . The number of edges in the hypermap equals the number of closed solid loops, while thenumber of vertices equals the number of closed dashed loops (the double lines count as eithersolid or dashed). Also shown are the correspondence between the nodes and the terms in (2.2),which is used in the evaluation of P r ( m, n ) .connectivity). With this representation of hypermap isomorphism established, we define rootedhypermaps as hypermaps with the aditional property that they are only equivalent under theaction of τ only when τ (1) = 1 (i.e. choosing for the root dart to have the label ). In our previous paper paper, we used the 3-constellation representation of hypermap embeddingto define a diagrammatic representation of rooted hypermaps [1] (see Figure 2.1). If we define ξ = (12 . . . r ) , then the set of rooted hypermaps with one face is equivalent to the set of permutations η on [1 . . . r ] (i.e. the symmetric group Sym r ) through the then any rooted hypermap with one faceis equivalent to , then the set of nonisomorphic rooted hypermaps with r darts is equivalent tothe set of permutations all η on [1 . . . r ] (i.e. the symmetric group Sym r ) through the bijection η → H η ≡ { ξ, η, η − ξ − } (as ξ is fixed, no two choices of η will result in equivalent rooted hypermaps). The diagrammaticrepresentation in Figure 2.1 (here referred to as a ladder diagram) allows us to quickly count thenumber of vertices and edges in H η by counting closed loops (the numbers of which are equalto the number permutations in η and ξη ).We showed that these diagrams also arise in the evaluation of the function P r ( m, n ) = ∂∂α a b . . . ∂∂α a r b r ( α a b . . . α a r b ) (cid:12)(cid:12)(cid:12)(cid:12) α =0 , (2.2)where α is an m × n real matrix: when the multiderivative in (2.2) is fully expanded out, it hasthe form P r ( m, n ) = (cid:88) η ∈ Sym r r (cid:89) i =1 δ [ a i , a η ( i ) ] δ [ b i , b ξη ( i ) ] = (cid:88) η ∈ Sym r m cyc ( η ) n cyc ( ξη ) cyc ( σ ) is the number of cycles in the permutation σ . As cyc ( η ) and cyc ( ξη ) are respectivelythe number of edges and vertices in the rooted hypermap H η ≡ { ξ, η, η − ξ − } , and the numberof faces in H η is cyc ( ξ ) = 1 , P r is therefore the generating function for enumerating one-facerooted hypermaps with r darts by number of edges and vertices. P r can also be computed usingladder diagrams as above, then, each diagram contributing a single m cyc ( η ) n cyc ( ξη ) term.We also showed, through Gaussian integration, that P r ( m, n ) = ˆ C mn d mn xe − x · x x a b x ∗ a b . . . x a r b r x ∗ a r b = Γ( mn + r )Γ( mn ) (cid:104) Tr [(ˆ ρ A ) r ] (cid:105) , (2.3)where ˆ ρ A is the reduced density operator of an m -dimensional subsystem of an mn -dimensionalbipartite quantum system, and the mean is being taken over all possible pure states of theoverall bipartite system. What this means is explained in more detail in [1], but the facts ofmost relevance here are that (2.3) is symmetric in m and n , and, when n ≥ m , the mean canbe represented as an integral over the eigenvalues ( p , . . . , p m ) of ˆ ρ A with the density function[2, 12] P ( p , . . . , p m ) dp . . . dp m ∝ δ (cid:32) − m (cid:88) i =1 p (cid:33) ∆ ( p , . . . , p m ) m (cid:89) k =1 p n − mk dp k , where ∆ ( p , . . . , p m ) is the Vandermonde discriminant of the eigenvalues, giving (cid:104) Tr [(ˆ ρ A ) r ] (cid:105) ∝ ˆ δ (cid:32) − m (cid:88) i =1 p (cid:33) ∆ ( p , . . . , p m ) m (cid:89) k =1 ( p n − mk dp k ) m (cid:88) j =1 p rj . Using a co-ordinate substitution given in [3], we multiply this by the factor mn + r ) ˆ ∞ λ mn + r − e − λ dλ and define q i = λp i , integrating over λ in order to remove the δ function, giving (cid:104) Tr [(ˆ ρ A ) r ] (cid:105) ∝ mn + r ) ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 ( e − q k q n − mk dq k ) m (cid:88) j =1 q rj . Finally, we normalise this by using the fact that, as n ≥ m , (cid:104) Tr [(ˆ ρ A ) ] (cid:105) = (cid:104) Tr [ I m ] (cid:105) = m , giving (cid:104) Tr [(ˆ ρ A ) r ] (cid:105) = Γ( mn )Λ mn Γ( mn + r ) ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 ( e − q k q n − mk dq k ) m (cid:88) j =1 q rj where Λ mn = ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 ( e − q k q n − mk dq k ) . We now need to generalise both the concept of ladder diagrams and the closely connected P r ( m, n ) functions in order to enumerate hypermaps with more than one face, and we will definethese generalisations in the next section. 4igure 2.2: A ladder diagram of a three-face rooted hypermap H ≡ { ξ, η, χ } . As in Figure 2.1, η = (1453)(2)(67) , but in this case ξ = (12)(3456)(7) . The double edges have been greyed outfor clarity. Summation over possible permutations η in this case generates P , , ( m, n ) In Figure 2.1, there was just one loop consisting only of single lines (i.e. single solid lines andsingle dotted lines, but not the solid/dotted paired lines), which corresponded to the single cyclein ξ , and therefore to the face in the associated rooted hypermap. It follows that a hypermapwith multiple faces would have a diagram with multiple such loops (i.e. ξ has multiple cycles,one for each face). An example of such a diagram is shown in Figure 2.2.In these diagrams, the solid lines in combination with the dotted lines defined by ξ can bethought of as a fixed backbone, on which the double lines given by η are superimposed. Wedescribed in Section 2.1 how, when ξ = (12 . . . r ) , we can sum over all possible η and in each casecount the solid and dotted loops in order to get a generating function for enumerating rootedone-face hypermaps with r darts. We also showed that this function was equivalent to (2.2) and(2.3).We can apply the same procedure to diagrams with other backbones. Looking at (2.2) and(2.3), we can see that the single cycle of length r in ξ corresponds to a term Tr [(ˆ ρ A ) r ] in thequantum expression for P r . By extension it follows that if consists of N cycles with lengths r , r , . . . , r N (e.g. the ξ used in Figure (2.2) corresponds to N = 3 , { r , r , r } = { , , } ),summing over all ladder diagrams with such a backbone and following the same procedure asin section 2.1, we get the function P r r ...r N ( m, n ) = Γ( mn + Σ Ni =1 r i )Γ( mn ) (cid:42) N (cid:89) j =1 Tr [(ˆ ρ A ) r j ] (cid:43) = 1Λ mn ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 ( e − q k q n − mk dq k ) × N (cid:89) i =1 m (cid:88) j =1 q r i j , (2.4)again valid when m ≤ n .These functions are not yet useful generating functions, however, for two reasons: the sumover diagrams used to calculate them can include disconnected diagrams (hypermaps are ne-cessarily connected, so cannot correspond to disconnected ladder diagrams), and any two hy-permaps which are related through cyclic permutation of one of the cycles in ξ are equivalent,producing a degeneracy. We will overcome these issues in the following sections; first we willdefine some additional functions in terms of the various P r ... in Section 3 which will account forthe presence of disconnected diagrams, and then we will use these to construct global generatingfunctions for counting rooted hypermaps in Section 4.5 Connected diagrams
As stated in the previous section, the functions P r ... defined in (2.4) are generating functionseach of which count over a set of ladder diagrams. As defined, however, they include disconnecteddiagrams in this count, whereas we require generating functions which count only over connecteddiagrams. In this section we will define such functions.For any given P r ...r N , define ¯ P r ...r N to be a generating function defined as a summationover the same set of diagrams as P r ... except with any disconnected diagrams excluded. In theone-loop case, P r = ¯ P r as all one-loop ladder diagrams are connected. When there is more thanone loop present, P r ... may be factorised in terms of ¯ P r ... using the fact that any disconnectedladder diagram can be split into a number of disjoint connected subdiagrams. We write thisfactorisation P rr ...r N = ¯ P rr ...r N + ¯ P r P r ...r N + u,v (cid:54) = ∅ (cid:88) u ∪ v = { r ...r N } ¯ P ru ... P v ... , (3.1)where the summation is over all partitions of the ordered multiset { r . . . r N } into two disjointnon-empty subfamilies. This factorisation works by breaking up each diagram in the summationinto its disjoint connected subdiagrams and considering which subdiagram the loop of length r is in. This loop is factored out in a ¯ P term. As an example, when N = 3 , P rabc = ¯ P rabc + ¯ P r P abc + ¯ P ra P bc + ¯ P rb P ac + ¯ P rc P ab + ¯ P rab P c + ¯ P rac P b + ¯ P rbc P a . If the definition of P r... is extended to include P ( m, n ) = 1 , which is consistent with (2.4), then (3.1) can be written more simply as P rr ...r N = (cid:88) u ∪ v = { r ...r N } ¯ P ru ... P v ... , (3.2)where u and v are now allowed to be empty.(3.1) and (2.4) can be used recusively to construct integral expressions for any given ¯ P r... .However, when constructing the global generating function in Section 4, it will be more usefulto work with the functions Π ( N ) r ( m, n ; x ) = ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N =1 x r N r N P rr ...r N ( m, n ) (3.3) ¯Π ( N ) r ( m, n ; x ) = ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N =1 x r N r N ¯ P rr ...r N ( m, n ) (3.4) Σ ( N ) ( m, n ; x ) = ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N =1 x r N r N P r ...r N ( m, n ) . (3.5)These are specifically defined as formal power series in x ; in general these series will be diver-gent if treated as functions of a finite parameter x . Noting the special cases Π (0) r ( m, n ; x ) = Π (0) r ( m, n ; x ) = P r ( m, n ) and Σ (0) ( m, n ; x ) = 1 , we use (3.2) to derive the recursion relation Π ( N ) r ( m, n ; x ) = ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N =1 x r N r N P rr ...r N ( m, n )= ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N =1 x r N r N (cid:88) u ∪ v = { r ...r N } ¯ P ru ... ( m, n ) P v ... ( m, n )= N (cid:88) k =0 (cid:18) Nk (cid:19) ¯Π ( k ) r ( m, n ; x )Σ ( N − k ) ( m, n ; x ) , (3.6)with the sum over partitions in (3.2) becoming a sum over the different possible sizes of thepartitions instead.In addition to these three sets of series, we will need one more series to be defined: F ( m, n, λ ; x ) = ∞ (cid:88) N =0 λ N N ! Σ ( N ) ( m, n ; x ) . (3.7)This series’ derivative satisfies x ∂∂x F ( m, n, λ ; x ) = x ∂∂x ∞ (cid:88) N =0 λ N N ! ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N =1 x r N r N P r ...r N ( m, n )= ∞ (cid:88) N =0 λ N N ! · N ∞ (cid:88) r =1 x r ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N x r N r N P r ...r N ( m, n )= ∞ (cid:88) N =1 λ N ( N − ∞ (cid:88) r =1 x r Π ( N − r ( m, n ; x )= ∞ (cid:88) N =0 λ N +1 N ! ∞ (cid:88) r =1 x r Π ( N ) r ( m, n ; x ) , (3.8)and when x is set to zero, F ( m, n, λ ; 0) = ∞ (cid:88) N =0 λ N N ! Σ ( N ) ( m, n ; 0)= Σ (0) ( m, n ; 0)= 1 . (3.9)With these various functions and series defined, we can proceed to define the global generat-ing function for enumerating rooted hypermaps in terms F . After doing this in the next section,we will return to F in Section 5 and discuss methods for evaluating it. Let us define H ( m, n, λ ; x ) as the generating function for enumerating all rooted hypermaps inthe form H ( m, n, λ ; x ) = (cid:88) e,v,f,r H vefr m v n e λ f x r , (4.1)7igure 4.1: A cyclic permutation one place to the left applied to the second loop in a ladderdiagram. As the two diagrams are isomorphic they are equivalent to the same hypermap, butthey contribute separately to the function ¯ P as the diagrams themselves are distinct. Thetotal degeneracy in this case is r · r = 4 (there is no degeneracy associated with the first loopas it contains the root, and is therefore fixed against permutation).where H vefr is the number of rooted hypermaps with v edges, e vertices, f faces and r darts.As with the expressions used in Section 3, this generating function is strictly speaking a formalpower series in x which will be dievergent in general. However, if we write H ( m, n, λ ; x ) = ∞ (cid:88) r =0 H r ( m, n, λ ) x r , then the individual H r will be well-behaved polynomial functions enumerating all rooted hyper-maps with r darts. Ultimately our aim will be to compute these.It is worth noting the symmetry properties of these functions: Theorem 1.
Each H r is completely symmetric in its three parameters, or, equivalently, H r ( m, n, λ ) = H r ( n, m, λ ) = H r ( m, λ, n ) . Proof.
This result follows easily from considering a rooted hypermap as a 3-constellation { ξ, χ, η } .The mapping T ef : { ξ, χ, η } → { χ − , ξ − , η − } maps rooted hypermaps with r darts onto each other, and specifically maps a rooted hypermapwith v vertices, e edges and f faces onto one with v vertices, f faces and e edges. As T ef isbijective (it is its own inverse), this means that H vefr = H vfer , and so H r ( m, n, λ ) = (cid:88) v,e,f H vefr m v n e λ f = H r ( m, λ, n ) . Similarly, the mapping T ve : { ξ, χ, η } → { ξ − , η − , χ − } is a bijection which swaps the number of edges and vertices in each rooted hypermaps, meaning H vefr = H evfr and H r ( m, n, λ ) = H r ( n, m, λ ) . While we cannot evaluate H directly, we are able to define it in relation to the series F defined previously: 8 heorem 2. The generating function H satisfies the relation H ( m, n, λ ; x ) F ( m, n, λ ; x ) = x ∂∂x F ( m, n, λ ; x ) . Proof.
Each ¯ P r ...r N is a generating function for a set of rooted hypermaps with N faces (onefor each of the loops in the associated ladder diagrams), and if all possible ¯ P r ...r N for fixed r + . . . + r N = r are summed over, then the resulting function will include terms for everyrooted hypermap with N faces and r darts, as any such hypermap has at least one associatedladder diagram which contributes a term to one of the ¯ P r ...r N . However, each such hypermapwith have a total of ( N − r r . . . r N such ladder diagrams (the N − loops of length r through r N can be put in any order to get distinct diagrams to get a degeneracy of ( N − , andeach of these loops can have its nodes permuted cyclically – see Figure 4.1 – giving a degeneracyof r r . . . r N ; in both cases the r loop is fixed because it is associated with the root), so inorder to get a generating function which only counts each rooted hypermap once, each ¯ P r ...r N must be divided by this degeneracy.We therefore write out H explicitly by summing over all P r ...r N , dividing each by ( N − r r . . . r N , and multiplying each by λ N x r + ... + r N in order to index the enumeration by num-ber of faces and darts as well. The resulting expression is H ( m, n, λ ; x ) = ∞ (cid:88) N =1 λ N ( N − ∞ (cid:88) r =1 x r ∞ (cid:88) r =1 x r r ! . . . ∞ (cid:88) r N =1 x r N r N ¯ P r r ...r N ( m, n )= ∞ (cid:88) N =0 λ N +1 N ! ∞ (cid:88) r =1 x r ∞ (cid:88) r =1 x r r ! . . . ∞ (cid:88) r N =1 x r N r N ¯ P rr ...r N ( m, n ) . We simplify this by substituting in (3.4): H ( m, n, λ ; x ) = ∞ (cid:88) N =0 λ N +1 N ! ∞ (cid:88) r =1 x r ¯Π ( N ) r ( m, n ; x ) . (4.2)Now, mutliplying this by F as defined in (3.7), we get H ( m, n, λ ; x ) F ( m, n, λ ; x ) = ∞ (cid:88) N =0 λ N +1 N ! ∞ (cid:88) r =1 x r ¯Π ( N ) r ( m, n ; x ) ∞ (cid:88) k =0 λ k k ! Σ ( k ) ( m, n ; x )= ∞ (cid:88) r =1 x r ∞ (cid:88) k =0 ∞ (cid:88) N =0 λ N + k +1 N ! k ! ¯Π ( N ) r ( m, n ; x )Σ ( k ) ( m, n ; x )= ∞ (cid:88) r =1 x r ∞ (cid:88) k =0 ∞ (cid:88) N = k λ N +1 ( N − k )! k ! ¯Π ( N − k ) r ( m, n ; x )Σ ( k ) ( m, n ; x )= ∞ (cid:88) r =1 x r ∞ (cid:88) N =0 N (cid:88) k =0 λ N +1 ( N − k )! k ! ¯Π ( N − k ) r ( m, n ; x )Σ ( k ) ( m, n ; x )= ∞ (cid:88) N =0 λ N +1 N ! ∞ (cid:88) r =1 x r N (cid:88) k =0 (cid:18) Nk (cid:19) ¯Π ( N − k ) r ( m, n ; x )Σ ( k ) ( m, n ; x ) . H ( m, n, λ ; x ) F ( m, n, λ ; x ) = ∞ (cid:88) N =0 λ N +1 N ! ∞ (cid:88) r =1 x r Π ( N ) r ( m, n ; x )= x ∂∂x F ( m, n, λ ; x ) . (4.3)If F and H were both well-behaved functions, this expression would be sufficient to evaluate H given F . As both are formal power series, however, it is only meaningful to consider thisexpression in terms of the terms in these series. Defining the functions F r ( m, n, λ ) such that F ( m, n, λ ; x ) = ∞ (cid:88) r =0 F r ( m, n, λ ) x r , (4.3) becomes r (cid:88) k =0 H r − k ( m, n, λ ) F k ( m, n, λ ) = rF r ( m, n, λ ) . (4.4)When r = 0 this simply gives H ( m, n, λ ) = 0 , and then we can recursively construct other H r for r > . For example, the first few are H ( m, n, λ ) = F ( m, n, λ ) H ( m, n, λ ) = 2 F ( m, n, λ ) − [ F ( m, n, λ )] H ( m, n, λ ) = 3 F ( m, n, λ ) − F ( m, n, λ ) F ( m, n, λ ) + [ F ( m, n, λ )] , where we have made use of the fact that F ( m, n, λ ) = F ( m, n, λ ; 0) = 0 as shown in (3.9).All that remains, then, is to evaluate the various F r . We will do this in the next section. F r We now have the generating function H defined in terms of the series F . The problem of eval-uating terms in the x -series expansion of H is therefore equivalent to the problem of evaluatingthe the terms in F . In this section we will establish an integral representation of F and thendiscuss the use of this to explicitly evaluate the terms F r in F . Theorem 3.
The series F has the integral representation F ( m, n, λ ; x ) = 1Λ mn ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 (cid:32) e − q k q n − mk dq k ∞ (cid:88) a =0 Γ( λ + a ) a !Γ( λ ) q ak x a (cid:33) (5.1) for positive integers m , n , λ and x satisfying m ≤ n , where ∆( q , . . . , q m ) is the Vandermondedeterminant, the integral is over the range ≤ q k < ∞ for all ≤ k ≤ m , Λ mn = ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 e − q k q n − mk dq k . roof. From (3.7) and (3.5) we have that F ( m, n, λ ; x ) = ∞ (cid:88) N =0 λ N N ! ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N =1 x r N r N P r ...r N ( m, n ) . (5.2)If we then substitute (2.4) into this, we get that, when m ≤ n , F ( m, n, λ ; x ) = 1Λ mn ∞ (cid:88) N =0 λ N N ! ∞ (cid:88) r =1 x r r . . . ∞ (cid:88) r N =1 x r N r N ˆ ∆ ( q , . . . , q m ) × m (cid:89) k =1 ( e − q k q n − mk dq k ) N (cid:89) i =1 m (cid:88) j =1 q r i j = 1Λ mn ∞ (cid:88) N =0 λ N N ! ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 ( e − q k q n − mk dq k ) × N (cid:89) i =1 m (cid:88) j =1 ∞ (cid:88) r i =1 q r i j x r i r i = 1Λ mn ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 ( e − q k q n − mk dq k ) × ∞ (cid:88) N =0 λ N N ! m (cid:88) j =1 ∞ (cid:88) r =1 q rj x r r N . (5.3)This expression is divergent for any given non-zero x , as almost all of the domain of integ-ration has at least one q j such that | q j x | > , making ∞ (cid:88) r =1 q rj x r r diverge. However, we are still able to make more progress by considering (5.3) as a formal powerseries in x again. we have the identity ∞ (cid:88) N =0 λ N N ! m (cid:88) j =1 ∞ (cid:88) r =1 q rj x r r N = m (cid:89) j =1 ∞ (cid:88) a j =0 Γ( λ + a j ) a j !Γ( λ ) q a j j x a j for λ > (see Theorem 6 in Appendix 1), so we rewrite (5.3) as F ( m, n, λ ; x ) = 1Λ mn ˆ ∆ ( q , . . . , q m ) m (cid:89) k =1 e − q k q n − mk dq k ∞ (cid:88) a k =0 Γ( λ + a k ) a k !Γ( λ ) q a k k x a k . This expression still bears similarities to expressions used in past work [3, 4, 5, 6, 1]. Toevaluate the integral, we will use a method similar to that used by Foong [4].11 heorem 4. F ( m, n, λ ; x ) = ∞ (cid:88) a =0 · · · ∞ (cid:88) a m − =0 (cid:89) ≤ i 1) Γ( n − m + 2 + a · · · Γ( n + am )( a n − m + 1 + a 1) ( a n − m + 2 + a · · · ( am + m )Γ( n + am )( a n − m + 2 + a 1) ( a n − m + 3 + a · · · ( am + m )Γ( n + 1 + am ) ... ... ... ... ( a n − a 1) ( a n + a · · · ( am + m )Γ( n + m − am ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . 13. Subtract ( n − m ) times the second row from the third, ( n − m + 1) times the third rowfrom the fourth etc. to give Q = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( n − m + 1 + a 1) Γ( n − m + 2 + a · · · Γ( n + am )( a n − m + 1 + a 1) ( a n − m + 2 + a · · · ( am + m )Γ( n + am )( a n − m + 1 + a 1) ( a n − m + 2 + a · · · ( am + m )2Γ( n + am ) ... ... ... ... ( a n − a 1) ( a n − a · · · ( am + m )2Γ( n + m − am ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) . 3. Continue to repeat this process, starting a row further down each time, until Q = (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) Γ( n − m + 1 + a 1) Γ( n − m + 2 + a · · · Γ( n + am )( a n − m + 1 + a 1) ( a n − m + 2 + a · · · ( am + m )Γ( n + am )( a n − m + 1 + a 1) ( a n − m + 2 + a · · · ( am + m )2Γ( n + am ) ... ... ... ... ( a m − n − m + 1 + a 1) ( a m − n − m + 2 + a · · · ( am + m ) m − n + am ) (cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12)(cid:12) = ∆( a + 1 , . . . , a m + m ) m (cid:89) k =1 Γ( n − m + a k + k )= (cid:89) is also symmetric, order r in each parameter, and zero when m , n or λ are zero.Therefore, we can compute the polynomial coefficients of H r (and therefore enumerate rootedhypermaps) by evaluating F r ( m, n, λ ) – and by extension H r ( m, n, λ ) – using (5.4) and (4.4) atall ≤ m ≤ n ≤ λ ≤ r and using polynomial interpolation.We used this method to compute the coefficients of H r for all ≤ r ≤ . Some of the outputis given in Appendix 2, and the results agree exactly with past computations, in particularWalsh’s enumeration of all rooted hypermaps up to r = 12 [9]. Running on a 2012 Dell XPS 12these calculations took 107 minutes, in comparison to the few days taken by Walsh’s algorithm. While no simple closed-form polynomial expressions are available for H r ( m, n, λ ) , there are afew special cases in which we can get more useful results.Consider the function H r (1 , m, n ) = (cid:88) v,e,f H vefr m e n f . (5.7)This is the generating function for enumerating rooted hypermaps with r darts by number ofedges and faces (with all possible numbers of vertices summed over). By the symmetry of H r ,(5.7) could also be used to enumerate by number of vertices and edges, summing over all numbersof faces etc. Theorem 5. For all r > , H r (1 , m, n ) = 1( r − m + r )Γ( m ) Γ( n + r )Γ( n ) − r − (cid:88) k =1 k ! Γ( m + k )Γ( m ) Γ( n + k )Γ( n ) H r − k (1 , m, n ) . (5.8) Proof. From (5.4) we have that F (1 , m, n ; x ) = ∞ (cid:88) a =0 Γ( m + a )Γ( m ) Γ( n + a )Γ( n ) x a a ! , so F r (1 , m, n ) = 1 r ! Γ( m + r )Γ( m ) Γ( n + r )Γ( n ) . 15e substitute this into (4.4) and rearrange to get H r (1 , m, n ) = rF r (1 , m, n ) − r − (cid:88) k =1 F k (1 , m, n ) H r − k (1 , m, n )= 1( r − m + r )Γ( m ) Γ( n + r )Γ( n ) − r − (cid:88) k =1 k ! Γ( m + k )Γ( m ) Γ( n + k )Γ( n ) H r − k (1 , m, n ) . In contrast to expressions such as (5.1) and (5.4), this expression obviously gives rise tosymmetric polynomial functions.Given (5.8), the following two results follow trivially: Corollary 1. For all r > , H r (1 , , m ) = r Γ( m + r )Γ( m ) − r − (cid:88) k =1 Γ( m + k )Γ( m ) H r − k (1 , , m ) . Corollary 2. For all r > , H r (1 , , 1) = r · r ! − r − (cid:88) k =1 k ! H r − k (1 , , . The second in particular allows us to count how many rooted hypermaps there are in totalwith r darts. The first few values are 1, 3, 13, 71, 461... We have demonstrated a method for computing generating functions to enumerate rooted hyper-maps by number of vertices, edges and faces for any given number of darts. This is an extensionof previous work where we derived closed form generating functions for counting enumeratingrooted hypermaps with one face [1], but in contrast to that case the method shown here definesthe generating function H r for r darts recursively in terms of H , . . . , H r − , and it only allows H r to be evaluated numerically, not expanded directly as a polynomial. We were still able toobtain a polynomial expansion, however, by using polynomial interpolation.This work is a further demonstration of the use of matrix integration as a tool for findinggenerating functions for enumerating sets of combinatoric objects. It specifically demonstratesthe link, first discussed in [1], between rooted hypermaps and the ensemble of reduced densityoperators on random states of a bipartite quantum system.We also discussed a number of related results. First we showed the symmetry of the gen-erating functions H r , arising from the symmetry of 3-constellations, and used this to speedup computation of H r by reducing the range over which H r needed to be evaluated to fix thepolynomial expansion. Then we looked at cases where one or more parameters in H r were setto unity, giving generating functions for enumerating larger sets of rooted hypermaps (such asall those with r darts and f faces, summing over all possible numbers of edges and vertices). Inparticular, this allowed us to easily count all rooted hypermaps with r darts and any number ofedges, vertices and faces. 16 Acknowledgements The work in this paper was supported by an EPSRC research studentship at the University ofYork. Appendix 1 Theorem 6. For positive integer m and λ > , the formal power series ∞ (cid:88) N =0 λ N N ! m (cid:88) j =1 ∞ (cid:88) r =1 q rj x r r N = m (cid:89) j =1 ∞ (cid:88) a j =0 Γ( λ + a j ) a j !Γ( λ ) q a j j x a j , where q j are components of an m -dimensional real vector.Proof. Let L λ,q ( x ) = ∞ (cid:88) N =0 λ N N ! m (cid:88) j =1 ∞ (cid:88) r =1 q rj x r r N . (7.1)For any given positive integer a , we see by inspection that this series contains only a finitenumber of terms of order x a , as such terms can only come from cases where ≤ N ≤ a . Inaddition, there is only one constant term: the N = 0 case which equals unity. Therefore, L λ,q ( x ) can be written in the form L λ,q ( x ) = ∞ (cid:88) a =0 f a ( λ, q ) x a (7.2)where each f a ( λ, q ) is a polynomial in λ and q . L λ,q ( x ) converges when ( | q j x | ) < for all j to L λ ( x ) = ∞ (cid:88) N =0 λ N N ! − m (cid:88) j =1 ln(1 − q j x ) = exp[ − λ m (cid:88) j =1 ln(1 − q j x )]= m (cid:89) j =1 − q j x ) λ . This has a series expansion in x , also valid when | q j x | < for all j , of m (cid:89) j =1 ∞ (cid:88) a j =0 Γ( λ + a j ) a j !Γ( λ ) q a j j x a j . (7.3)This can also be rearranged into the form (7.2). (7.1) and (7.3) are therefore both Taylor serieswith the same radius of convergence, and they are equal to each other everywhere within it, soit follows from the uniqueness of Taylor series expansions of smuooth functions that they areequivalent, i.e. ∞ (cid:88) N =0 λ N N ! m (cid:88) j =1 ∞ (cid:88) r =1 q rj x r r N = m (cid:89) j =1 ∞ (cid:88) a j =0 Γ( λ + a j ) a j !Γ( λ ) q a j j x a j . ppendix 2 Numbers of rooted hypermaps with v vertices, e edges, f faces and r darts, calculated bycomputing the generating functions H r . Only the cases with v ≤ e ≤ f are given, as the restfollow from the symmetry of H r . The cases ≤ r ≤ are included for comparison with Walsh’sprevious computation [9], with all cases up to r = 12 agreeing with his computation. The newcase r = 13 is also shown. r = 1 : v e f N r = 2 : v e f N r = 3 : v e f N r = 4 : v e f N r = 5 : v e f N r = 6 : v e f N r = 7 : v e f N r = 13 : v e f N References [1] J. P. Dyer, Matrix integrals and generating functions for permutations and one-face rootedhypermaps, arXiv:1407.7774.[2] S. Lloyd, H. Pagels, Complexity as thermodynamic depth, Annals of Physics 188 (1) (1988)186–213.[3] D. Page, Average entropy of a subsystem, Physical Review Letters 71 (9) (1993) 1291–1294.[4] S. Foong, S. Kanno, Proof of Page’s conjecture on the average entropy of a subsystem,Physical Review Letters 72 (8) (1994) 1148–1151.195] J. Sánchez-Ruiz, Simple proof of Page’s conjecture on the average entropy of a subsystem,Phys Rev E 52 (5) (1995) 5653.[6] S. Sen, Average Entropy of a Quantum Subsystem, Physical Review Letters 77 (1) (1996)1–3.[7] D. Arquès, Hypercartes pointées sur le tore: Décompositions et dénombrements, Journalof Combinatorial Theory, Series B 43 (3) (1987) 275–286.[8] A. Mednykh, R. Nedela, Enumeration of unrooted hypermaps of a given genus, DiscreteMathematics 310 (3) (2010) 518–526.[9] T. R. S. Walsh, Generating nonisomorphic maps and hypermaps without storing them(2012).URL http://accueil.labunix.uqam.ca/~walsh_t/papers/GENERATINGNONISOMORPHIC.pdfhttp://accueil.labunix.uqam.ca/~walsh_t/papers/GENERATINGNONISOMORPHIC.pdf