Refined Counting of Necklaces in One-loop $\mathcal{N}=4$ SYM

We compute the grand partition function of $\mathcal{N}=4$ SYM at one-loop in the $SU(2)$ sector with general chemical potentials, extending the results of P\'olya's theorem. We make use of finite group theory, applicable to all orders of $1/N_c$ expansion. We show that only the planar terms contribute to the grand partition function, which is therefore equal to the grand partition function of an ensemble of XXX$_\frac12$ spin chains. We discuss how Hagedorn temperature changes on the complex plane of chemical potentials.


Introduction
The N = 4 super Yang-Mills theory (SYM) has attracted a lot of attention owing to its simple and profound structure. Besides being the primary example of the AdS/CFT correspondence [1], this theory is believed to be integrable in the planar limit [2]. The integrability enables us to predict various observables at any values of the 't Hooft coupling; see [3] for a review.
As a parallel development, alternative methods have been developed to uncover the nonplanar structure of N = 4 SYM with the gauge group U (N c ) or SU (N c ), based on finite group theory [4]. New bases of gauge-invariant operators have been discovered, which diagonalize the tree-level two-point functions at finite N c [5,6,7,8]; see [9] for a review.
With numerous approaches at hand to study individual operators, let us ask questions complementary to the above line of development. We reconsider the statistical property of N = 4 SYM, namely the grand partition function including perturbative 1/N c corrections.
In [10], the tree-level partition function of N = 4 SYM on R × S 3 was computed to investigate its phase space structure. The free energy has the expansion F = N 2 c F 0 +F 1 +. . . , where F 0 = 0 in the confined or low-temperature phase, and F 0 > 0 in the deconfined or hightemperature phase. In the confined phase, the density of states increases exponentially as the energy increases, leading to the singularity of the partition function at a finite temperature. There the vacuum undergoes the so-called Hagedorn transition to the deconfined phase [11,12]. In the dual supergravity, it is argued that a thermal scalar in AdS 5 × S 5 becomes tachyonic at a finite temperature, and condensates into the AdS blackhole [13].
Below we consider the grand partition function of N = 4 SYM in the low-temperature phase at one-loop. It amounts to summing up the one-loop anomalous dimensions of all gauge-invariant operators. The problem simplifies a lot by noticing that we do not need to take an eigenbasis of the dilatation operator to compute the trace. The main problem is how to take the trace efficiently in the general setup. The one-loop partition function without chemical potential has been obtained by using Pólya enumeration theorem in [14]. The Hagedorn transition in the pp-wave/BMN limit was studied in [15,16]. The grand partition function with a one-parameter family of chemical potential was given in [17], and the phase space near the critical chemical potentials was studied in [18].
However, at one-loop the Pólya-type formulae are known only for single-variable cases, which makes it difficult to obtain the grand partition function with general chemical potentials. In this paper, we incorporate the fully general chemical potentials in the SU (2) sector, by using finite group theory which is valid to all orders of perturbative 1/N c expansion. 1 In the planar limit, the dilatation operator of N = 4 SYM at one-loop in the SU (2) sector is the Hamiltonian of XXX 1 2 spin chain. The (canonical) partition function of the spin chain can be computed by using string hypothesis in the thermodynamic limit. Our work provides alternative derivation based on the microscopic counting of states, if we take care of subtle differences to be discussed in Section 3.4. 1 A similar quantity was computed in [19,20], that is the grand partition function with general chemical potential at one-loop in the SU (2) sector including O(N 2 c ) term. This result involves a multi-dimensional integral, and less explicit than our counting formula. Another formula was obtained in [21], namely the trace of the one-loop dimension over the product of two fundamental representations of psu (2, 2|4). This quantity is a building block in the Pólya-type formula, but not identical to the grand partition function.
Main results. Let us briefly summarize the main results of this paper. We consider the grand partition function of N = 4 SYM in the confined phase, with the gauge group U (N c ) up to one-loop in the SU (2) sector. The theory is put on R × S 3 , where R is the radial direction of R 4 and the radius of S 3 is set to unity. We write the grand partition function as Z(β, ω) = tr e −βD+ i ω i J i , D = D 0 + λD 2 + . . . , (1.1) where D is the dilatation operator and J i are the R-charges. We take the trace over all gauge-invariant local operators in the SU (2) sector, made out of complex scalars {W, Z}. The grand partition function (1.1) has the weak coupling expansion where the operators W m Z n are weighted by x m y n . We apply finite group theory to compute (the perturbative part of) Z MT 0 and Z MT 2 . It will turn out that only the planar term contributes at one-loop in this setup, even though our methods are valid to all orders of perturbative 1/N c corrections.
On top of 1/N c corrections, there are non-perturbative corrections coming from finite N c constraints. At finite N c , certain combinations of operators vanishes when their canonical dimension of an operator exceeds N c . Let us write the exact grand partition function as where the second sum represents the subtraction at finite N c . If we set x = y = e −β , then the second sum is of order e −βNc at large β , which are non-perturbative in view of 1/N c expansion. Our methods capture the first sum in (1.3), which may contain the corrections of order 1/N c at -loop. To simplify the notation, we write e.g. Z MT 0 (x, y) instead of Z MT(p) 0 (x, y) throughout the paper.
We obtain two expressions of Z MT 2 in Section 3, which we call Partition form and Totient form. The Z MT 2 in Partition form is written as a sum over partitions of operator length L , r = [1 r 1 , 2 r 2 , . . . , L r L ] : where θ > (x) = 1 for x > 1, and vanishes otherwise. The Z MT 2 in Totient form involves Euler's totient function Tot(d), which counts the number of relatively prime positive integers less than d : x km y k(L−m) δ(gcd(m, L), 1) .
The equivalence of two results can be checked by expanding both series around the zero temperature, corresponding to the limit of small x, y. When the gauge group is SU (N c ), an overall factor of (1 − x)(1 − y) should be multiplied. It is straightforward to compute the Hagedorn temperature by using Totient form (1.5). The Hagedorn temperature T H (λ) in the SU (2) sector is a function of the chemical potentials (ω 1 , ω 2 ) = (log x + 1 T , log y + 1 T ), which is given by (1.6) which is valid at large N c , due to the second sum in (1.3). Roughly said, the first line represents the deconfinement of W and Z, whereas the second line that of W 2 and Z 2 . This result shows that the complex chemical potentials change the location of the Hagedorn transition, as will be discussed further in Section 4.

Tree-level counting
We introduce two methods of computing the generating function of the number of gaugeinvariant operators in N = 4 SYM with U (N c ) gauge group. This generating function is equal to the grand partition function at tree-level.

Permutation basis of gauge-invariant operators
General gauge-invariant local operators of N = 4 SYM can be specified by an element of permutation group. We introduce the elementary fields of N = 4 SYM and their polynomials There is an equivalence relation coming from the relabeling (a i , Each of the equivalence class uniquely specifies a gauge-invariant operator. In the SU (2) sector, we restrict W A to the elementary fields to a pair of complex scalars {W, Z}. We use the gauge degrees of freedom (2.3) to set The residual gauge degrees of freedom is This equivalence class uniquely specifies a gauge-invariant operator in the SU (2) sector. The number of such multi-trace operators at a fixed (m, n) is given by summing over all solutions of (2.5), 2 The generating function of the number of multi-trace operators is defined by which will be computed below.

Sum over partitions
First we evaluate N MT m,n in (2.6). Let us write γ = γ W · γ Z with γ W ∈ S m and γ Z ∈ S n . Suppose that γ W , γ Z have the cycle structure p m, q n, respectively. We also define r k = p k + q k , r m + n. (2.10) We look for the general solutions of the condition α −1 γαγ −1 = 1 for a fixed γ, which we call stabilizer Stab(γ). Let us parametrize γ by h,k ) is the cyclic permutation defined in Appendix A. The identity (A.4) gives h,k )), (2.12) and the stabilizer condition is solved by σ(h),τ h (k) , σ ∈ S r k , τ h ∈ Z k , (h = 1, 2, . . . , r k ). (2.13) Thus, for each γ, α should belong to the direct product of the wreath product groups, (2.14) The symbol |G| means the order of the group G.
The number of permutations in S m with the cycle structure p m is given by the orbit-stabilizer theorem, We can rewrite N MT m,n in (2.6) as Consider the generating function (2.8). The double sum m p m can be transformed to an infinite product ∞ k=1 ∞ p k =0 , and thus The first few terms read Z MT 0 (x, y) = 1 + (x + y) + 2 x 2 + xy + y 2 + 3x 3 + 4x 2 y + 4xy 2 + 3y 3 + 5x 4 + 7x 3 y + 10x 2 y 2 + 7xy 3 + 5y 4 + . . . . (2.18) The series gives the number of multi-trace operators in the SU (2) sector of N = 4 SYM with U (N c ) gauge group. For SU (N c ) theories, we subtract the terms with p 1 > 0 or q 1 > 0 in (2.17), which givesZ At finite N c , fewer terms contribute to the generating function (2.8), which modifies the expansion (2.18). The precise expression will be reviewed in Section 2.5. Our formula (2.17) is valid up to the order x m y n with m + n ≤ N c .

Power enumeration theorem
We review another derivation of the tree-level generating function based on Pólya Enumeration Theorem [10].
Consider a single-trace operator with length p. We define the domain D = {1, 2, . . . , p} and the range R = {Z, W }. A single-trace operator is graphically equivalent to a necklace, that is the map D → R modulo the action of the cyclic group Z p acting on D, Single-trace operator ↔ Necklace = Map (Z p \D → R) . (2.20) We associate the weights in R by c(x, y) = x + y, where x m y n corresponds to the operator W m Z n . Then, Pólya Enumeration Theorem says that the generating function of the number of graphs (2.20) is given by The first few terms read Z ST 0 (x, y) = (x + y) + x 2 + xy + y 2 + x 3 + x 2 y + xy 2 + y 3 + x 4 + x 3 y + 2x 2 y 2 + xy 3 + y 4 + x 5 + x 4 y + 2x 3 y 2 + 2x 2 y 3 + xy 4 + y 5 + . . . . (2.25) The generating function of multi-trace operator is given by the plethystic exponential of the single-trace generating function, where we used j|d Tot(j) = d. This result agrees with (2.17). For SU (N c ) theories, we subtract the p = 1 term in (2.21), in agreement with (2.19).

Counting single-traces
For later purposes, we rederive the generating function of the number of single-trace operators by counting the solutions (2.6) under the constraint α ∈ Z L , with L = m + n.
We will obtain which is derived as follows. Suppose m and n are divisible by a positive integer d, This set of (μ, κ, α) is the general solution to the conditions Let us parametrize µ =μ as The number of possibleμ chosen from S dm × S dn is 3 For eachμ , we sum over α as parametrized in (2.30). For this purpose we identify {a 1 , . . . , a } with some of {m hk } in (2.32). In order to avoid double counting, we fix a 1 =m 11 and choose The number of choices of a 2 . . . a is 4 , and the order of the latter group is given by the orbit-stabilizer theorem (2.15). 4 In other words, we remove the redundancy coming from the overall translation of α ∈ Z L .
The number of possible κ is Tot(d). Thus, the number of possible α, divided by |S m × S n | is which is (2.28). This result is formally correct when m n = 0 thanks to d|L Tot(d) = L. Therefore, the tree-level generating function is given by To simplify it, we apply the formulae which is (2.24).

Partition function at finite N c
The exact tree-level partition function of N = 4 SYM at finite N c can be computed precisely in various methods. We briefly review these arguments, to understand the finite N c corrections in (1.3).
A straightforward method is to compute the grand partition function is to evaluate the path integral of N = 4 SYM action [12]. Let a 0 be the zero-mode of the gauge field A 0 and U = exp(iβa 0 ). The grand partition function in the complete P SU (2, 2|4) sector is where ζ B (w), ζ F (w) are functions of chemical potentials, and dU SU (Nc) is the SU (N c ) Haar measure. 5 The complete partition function (2.43) can be reduced to the one in the SU (2) sector by setting . It turns out that the resulting expression is identical to the Molien-Weyl formula which gives the Hilbert-Poincaré series of GL(N c ) invariants [23]. For example, the Molien-Weyl formula for the gauge group U (N c ) with q variables, corresponding to the SU (q) sector, can be written as 6 where U is the counterclockwise contour of unit radius. 7 It can also be written as [26,5] Z SU (q) Nc where C(R, R, Λ) is the Clebsch-Gordan multiplicity defined by R ⊗ R = ⊕Λ C(R,R,Λ) as S Lmodules, and s Λ (x 1 , . . . x q ) is the Schur polynomial. We sum over the partitions Λ having at most q rows, since Λ is related to the SU (q) global symmetry. At finite N c we should sum over the partitions R having at most N c rows in (2.46). One can evaluate the formula (2.45) or (2.46) explicitly when N c and q are small. At (N c , q) = (2, 2) we obtain in agreement with [27] for q = 2. The formula also reproduces the q > 2 cases in [18].

One-loop counting
We compute the sum of anomalous dimensions at one-loop in the SU (2) sector in two ways, which we call Partition form and Totient form. The corresponding generating function gives the partition function at one-loop.

Mixing matrix
The dilatation operator of N = 4 SYM is given by [28,29], Let H m,n be the Hilbert space of all gauge-invariant operators in the SU (2) sector with the R-charges (m, n). We define the mixing matrix as On the permutation basis introduced in Section 2.1, the mixing matrix inside H m,n takes the form [30] ( where we introduced the notation L = m + n, and denoted the equivalence class by (3.5) We will evaluate the sum of one-loop dimensions at a fixed (m, n), Since the gauge-invariant operator is in one-to-one correspondence with the equivalence class (3.5), we can rewrite the sum (3.6) as According to (3.3), the mixing matrix is invariant inside the same conjugacy class. By writing γ −1 2 γ 1 ≡ γ, we find  34)(23)(12). The number of transpositions defines the parity of a permutation, which is conserved at any orders of perturbative 1/N c expansion. 8 In particular, odd powers of transpositions cannot become the identity, and only the planar term i = α(j) contributes in (3.8). Thus, where L = m + n. The generating function of the sum of one-loop dimensions is defined by (3.10)

Partition form
We evaluate the sum of dimensions (3.9) by generalizing the methods used in Section 2.2.

First term
Consider the first term of (3.9), We denote the cycle type of µ by p m, q n and define r k = p k + q k . We parametrize µ by µ = m+n as in (2.11). The condition µα −1 µ −1 α = 1 imposes that α should belong to the stabilizer of µ.
Suppose that i and j are part of the cycle of µ of length-a and length-b, respectively. There are L(L − 1) choices of {i, j}, which can be written as From (2.13) we see that α ∈ Stab(µ) permutes {1, 2, . . . , L} only among those having the same cycle length in µ. Thus, the condition i = α(j) results in a = b, so we neglect the terms a = b in (3.12). Define the number of solutions of the two δ-functions in (3.11) for a given (i, j, µ) by This can be rewritten as (3.14) If we introduce α = α 0 (ij), then Since the group Stab(µ) acts transitively on S ra [Z a ] ⊂ S ara , the isotropy group satisfies the property Thus, the number of solutions in (3.11) for a given µ ∈ T p × T q is where The symbol a b means that the number a appears b times.

Totient form
We compute the generating function of the sum of one-loop dimensions in another way. First, we compute the one-loop generating function for single-trace operators by imposing α ∈ Z L , as done in Section 2.4. Then, we conjecture the generating function for multi-traces, by writing the plethystic exponential of the single-trace results.

First term
Let d ≥ 1 be a divisor of m and n as in (2.29). Specify the cycle type of µ to p = [d m ] and q = [d n ] and α to [L] simultaneously. Consider the first term of the one-loop mixing matrix: The number of solutions to the stabilizer condition α = µαµ −1 is given by (2.36). For each (j, µ =μ) and α given by (2.30), there is only one i ∈ {1, 2, . . . , L} satisfying i = α(j).
Hence the sum over i, j gives a factor of L, leading to Note that d = L is possible only when mn = 0. 9 Negative terms are needed to kill the coefficients of BPS terms.
We conjecture that the one-loop generating function for multi-traces is given by the plethystic exponential of the singe-trace results (3.30): x km y k(L−m) δ(gcd(m, L), 1) .
One can check that its expansion in small x, y agrees with (3.24). The first line is generalization of the single-variable case discussed in [14].

Comparison with Bethe Ansatz
and in the physical solutions the second Q-functionQ(v) must be a polynomial in v, as clarified in [31,32]. A solution of BAEs is called regular if no Bethe roots are located at infinity. Regular BAE solutions correspond to the SU (2) highest weight states. If we denote a state with W m Z n by |m, n , the highest weight states satisfy where {S ± , S 3 } are the SU (2) generators. We also need m ≥ n to count the BAE solutions correctly. We should include exceptional solutions whose energy superficially diverges due to the Bethe roots at v = ±i/2. In such cases, we should regularize the BAEs by introducing twists and by carefully taking the zero-twist limit. The results are shown in Table 2.
A bit of arithmetic is needed to compare the two sets of numbers in Table 2. First, consider the second row. At (m, n) = (3, 2) we have 10 = 4 + 6, where 6 comes from the    The (canonical) partition function of XXX 1 2 spin chain of length L has been computed in [33]. He also showed that the partition function gives the character of su(2) affine Kac-Moody algebra at level one in the large L limit, as conjectured in [34]. Their analysis slightly differs from ours in three points: we compute M 2 rather than e βM 2 , sum over the level-matched states, and consider the grand partition function by summing over L. 11

Hagedorn transition
We compute the grand partition function of N = 4 SYM in the SU (2) sector at one-loop at large N c based on the above results. Then we determine the Hagedorn temperature of the N = 4 SYM in the SU (2) sector, namely the smallest temperature T ≥ 0 at which the grand partition function diverges, as in (1.6). We will see that the Hagedorn temperature has numerous branches depending on the value of the chemical potential on the complex plane.

Grand partition function
Consider the grand partition function of N = 4 SYM in (1.1), The trace is taken over the Hilbert space of all gauge-invariant operators in the SU (2) sector.
The partition function has the weak coupling expansion where we used (3.2). We assign the R-charge J i to complex scalars as The partition function (1.1) depends on (β, ω 1 , ω 2 ), whereas the generating functions in (4.2) depends on (x, y). The two sets of variables are related by 12 In particular, the low-temperature expansion corresponds to the expansion in small x, y. The computation below is valid at large N c , due to the non-perturbative corrections in (1.3).
We assume that the Hagedorn temperature and the partition function are expanded as (4.8) Let us expand the partition function (4.2) around the pole (4.6) with k = 1 and compare the result with the above expansion. We find that Consider the region outside R + . When we cross the linex +ỹ = 1, then T * becomes negative for all k. As we approachx → 0 keepingx +ỹ ≥ 1, all simple poles in (4.6) accumulate at T * = 1/ log(ỹ). Let us take eitherx andỹ negative, where the chemical potentials (4.4) are shifted by πi. When |x|, |ỹ| are large enough, the pole (4.6) with k = 2 becomes the closest to the origin among those giving T * > 0. Thus, in the region the one-loop Hagedorn temperature is given by More generally, if we put (x,ỹ) ∈ C 2 on Argx = 2π p 1 , Argỹ = 2π p 2 , p = lcm(p 1 , p 2 ), (p 1 , p 2 ∈ Z ≥1 , p N 2 c ), (4.12) the Hagedorn temperature is given by the pole (4.6) at k = p. When p = O(N 2 c ), the Hagedorn transition may take place around 1/ log |x +ỹ|, because the free energy becomes O(N 2 c ) without hitting the pole. In Figure 1, the plots of the grand partition function Ω ≡ −T log Z are shown as a function of (T,x,ỹ). The left figure at a fixed T shows that the singularity of Ω is associated with the boundary of R ± . By comparing the middle figure (T,x =ỹ) and the right figure (T,x = −ỹ), we find that the former is not invariant under the flipx ↔ −x, whereas the latter is invariant. This pattern is consistent with R ± .
For comparison with the literature, we vary T at a fixed (ω 1 = ω 1 /β , ξ = y/x). It follows that (4.13) The first line agrees with [14] when ξ = 1,ω 1 = 0, and with [17] when ξ = 1. 13 Let us make a few remarks on the Hagedorn transition. First, the physical partition function should not diverge. This means that the system turns into the deconfined phase around the Hagedorn temperature, when the free energy becomes O(N 2 c ). In order to inspect the details of the phase transition, we need to evaluate (2.43) in the large N c limit as in [10]. In the SU (2) sector, it is expected that the system is described by N 2 c + 1 harmonic oscillators around the Hagedorn temperature [18]. 14 Second, the parameter regionx < 0 orỹ < 0 can be interpreted as the insertion of the number operator (−1) N Z or (−1) N W to the grand partition function (1.1), which makes Z or W effectively a fermion. The pole at k = 1 disappears when the scalar becomes fermionic. The pole at k = 2 still contributes to the divergence because Z 2 or W 2 are bosonic. Similarly, when the transition takes place at k = p as in (4.12), Z p , W p are effectively bosonic. This pattern indicates that only effective bosons form a condensate inside which U (N c ) degrees of freedom are liberated from the confinement.
Third, the grand partition function at finite N c is a smooth function of the temperature, and no transition should happen [11]. This can be checked by evaluating Z SU (2) Nc (x, y) in (2.45). For example, Z

SU (2)
Nc=2 (x, y) with x = y = e −β in (2.47) is regular for any β > 0. More generally, it is conjectured that the denominator of Z SU (2) Nc (x, y) at any N c < ∞ is always a product of the factors (1 − x a y b ) for some integers a, b ≥ 0 [24]. Hence, the Hagedorn temperature is infinite at finite N c .

Conclusion and Outlook
In this paper, we computed the grand partition function of N = 4 SYM in the SU (2) sector at one-loop by making use of finite group theory. Only the planar terms contribute in this setup, though our result is valid to all orders of perturbative 1/N c expansion. We derived two expressions for the one-loop generating function, called Partition form and Totient form.
Based on Totient form we computed the Hagedorn temperature which depends on general values of the chemical potentials. We argued how the Hagedorn temperature changes when the chemical potentials are complex.
As future directions of research, one can consider the grand partition functions in more general situations, such as finite N c corrections at one-loop, larger sectors of N = 4 SYM, or higher order in λ in the SU (2) sector. It is also interesting to study superconformal field theories other than N = 4 SYM, such as β-deformed and γ-deformed theories [36], ABJM model [37], theories with 16 supercharges [38], and quiver gauge theories [39]. Our counting methods should be applicable to integrable models like q-deformed Hubbard model [40], which is a generalization of XXZ spin chain.
Another topic is to develop group-theoretical techniques to study multi-point functions. It is well known that the OPE limit of four-point functions in N = 4 SYM yields the sum of the anomalous dimensions of intermediate operators, weighted by the square of OPE coefficients. Such objects have been studied by conformal bootstrap [41] and integrability methods [42,43,44]. The effects of 1/N c corrections in such a limit is worth further investigation.

B Details of derivation
The second term of (3.9) can be rewritten as This quantity will be computed below. The result is called Partition form when we sum over α ∈ S L , and Totient form when α ∈ Z L .

B.1 Second term in Partition form
We evaluate M 2 (2nd) m,n in the following steps: 1) Choose µ ∈ T p × T q ⊂ S m × S n 2) Generate µ 0 = (ij) µ by summing over (i, j) 3) Solve the two δ-function constraints simultaneously 1) We denote the cycle type of µ by r L. We have r k = p k + q k for 1 ≤ k ≤ L, and Thus, the transposition (ij) relates the cycle type of µ and µ 0 as Let us count the number of (i, j) corresponding to each line of (B.3). As for the first line, i.e. splitting, we choose a cycle of length a and split it into l + (a − l), for 1 ≤ l ≤ a − 1. There are ar a ways to choose i, and the choice of j is unique for a given (i, l). As for the second line, i.e. joining, we choose two different cycles of length a and b. If a = b, there are ab r a r b ways to identify i , j . And if a = b, there are a 2 r a (r a − 1) ways. In total, we have 15 Thus, we replace the sum over (i, j) in (B.1) by the sums over a, b, l shown above. Schematically, the sum of dimensions is given by to solve the δ-function constraint (B.1). The ζ belongs to either of the two groups in (B.3). Next, we solve the planarity condition i = α(j) for the three cases (B.4)-(B.6). Recall that i and j belong to the cycle of the same length in µ 0 to solve the conditions i = α(j) and α ∈ Stab(µ 0 ), as discussed in Section 3.2.1.
As for the splitting case (B.4), only the process r 2l → r l + r l can solve the condition i = α(j). The stabilizer of µ 0 is (B.10) There are 2l r 2l choices of i, j, including the interchange i ↔ j. 16 In order to solve i = α(j) when i, j appear in the cycles of length l, we write Here α 0 freezes the cycle of i but not of j, which restricts Stab(µ 0 ) split down to The number of solutions in the splitting process is given by As for the joining cases (B.5) and (B.6), any a, b can solve the condition i = α(j). The stabilizer of µ 0 is For a = b there are ab r a r b choices of i, j, and for a = b there are a 2 r a (r a − 1) choices, including the interchange i ↔ j. In order to solve i = α(j) when i, j appear in the cycle of length a + b, we restrict Stab(µ 0 ) join down to The number of solutions in the joining process is given by (B.16) 4) In total, we have Consider an example. Let µ be (1)(23)(456), having r = [1 1 , 2 1 , 3 1 ] 6. Then Stab(µ) = Z 1 · Z 2 · Z 3 , which has the order 1 × 2 × 3 = 6. The possible splitting process is We have (i, j) = (2, 3) or (3,2). The list of {α(j) | α ∈ Stab(µ 0 )} is {1 6 , 2 6 , 3 6 } for j = 2, 3, in the notation of Table 1. Thus, the number of solutions is and their stabilizers are The lists of α(j) is

B.2 Second term in Totient form
We evaluate M 2 (2nd) m,n in the following steps: Generate µ 0 = (ij) µ by reverting the last step 4) Solve the two δ-function constraints simultaneously 1) We denote the cycle type of µ ∈ S m × S n by p m and q n. Let m, n be divisible by d ≥ 1 as in (2.29).
2) Given µ 0 ∈ Z d parametrized as in (2.32), we generate µ = (ij) µ 0 . We classify two cases depending on whether i, j belong to the same or different cycles of µ 0 , Same : In terms of cycle types of µ 0 and µ, these processes can be written as We assume = 1 in the Same case and ≥ 2 in the Different case, because we will see later that other cases do not contribute. These conditions are equivalent to d = L and d < L, respectively.