1 Introduction and statement of results

We begin by defining the infinite finitary symmetric and alternating groups and their corresponding wreath products, and then we state our results regarding growth series identities.

For an infinite set X, the finitary symmetric group \(\text {Sym}(X)\) is the group of permutations of X with finite support. The permutational wreath product of a group H with \(\text {Sym}(X)\) is the group \(H\wr _X\text {Sym}(X):=H^{(X)}\rtimes \text {Sym}(X)\) defined as follows:

  1. (i)

    The group \(H^{(X)}\) is the group of functions from X to H with finite support.

  2. (ii)

    The action of permutations \(f\in \text {Sym}(X)\) on functions \(\psi \in H^{(X)}\) is defined by

    $$\begin{aligned} \psi \mapsto f(\psi ):=\psi \circ f^{-1}. \end{aligned}$$
  3. (iii)

    Multiplication in the semi-direct product is defined for \(\varphi ,\psi \in H^{(X)}\) and \(f,g\in \text {Sym}(X)\) by

    $$\begin{aligned} (\varphi ,f)(\psi ,g)=(\varphi f(\psi ),fg). \end{aligned}$$

The finitary alternating group \(\text {Alt}(X)\) is the subgroup of \(\text {Sym}(X)\) of permutations with even signature, and the permutational wreath product \(H\wr _X\text {Alt}(X)\) is defined as above. In this paper, we only consider permutational wreath products with finite group H. We now define some general terminology. For any group G generated by a set S, the word length \(\ell _{G,S}(g)\) of any element \(g\in G\) is the smallest nonnegative integer n such that there exist \(s_1,\dots ,s_n\in S\cup S^{-1}\) with \(g=s_1\cdots s_n\). The conjugacy length \(\kappa _{G,S}(g)\) is the smallest word length appearing in the conjugacy class of g. If n is any natural number, we denote by \(\gamma _{G,S}(n)\in \mathbb {N}\cup \{0\}\cup \{\infty \}\) the number of conjugacy classes in G with smallest word length n. If \(\gamma _{G,S}(n)\) is finite for all n, then we may define the conjugacy growth series of a group G with generating set S to be the following q-series:

$$\begin{aligned} C_{G,S}(q):=\sum \limits _{[g]\in \text {Conj}(G)}q^{\kappa _{G,S}(g)}=\sum \limits _{n=0}^\infty \gamma _{G,S}(n)q^n, \end{aligned}$$

where the first sum is over representatives of conjugacy classes of G. Bacher and de la Harpe [1] prove conjugacy growth series identities for sufficiently largeFootnote 1 generating sets S of \(\text {Sym}(X)\), \(S'\) of \(\text {Alt}(X)\), and \(S^{\left( W_S\right) }\) of \(W_S=H_S\wr _X\text {Sym}(X)\) relating the finitary permutation groups and their wreath products to the partition function. Explicitly, we have the fascinating identities

$$\begin{aligned} C_{\text {Sym}(X),S}(q)=\sum \limits _{n=0}^\infty p(n)q^n=\prod _{n=1}^\infty \frac{1}{1-q^n} \end{aligned}$$
(1.1)

for the finitary symmetric group,

$$\begin{aligned} C_{\text {Alt}(X),S'}(q)=\left( \sum \limits _{n=0}^\infty p(n)q^n\right) \left( \sum \limits _{m=0}^\infty p_e(m)q^m\right) =\frac{1}{2}\prod \limits _{n=1}^\infty \frac{1}{(1-q^n)^2}+\frac{1}{2}\prod \limits _{n=1}^\infty \frac{1}{1-q^{2n}} \end{aligned}$$
(1.2)

for the finitary alternating group,Footnote 2 and

$$\begin{aligned} C_{W_S,S^{\left( W_S\right) }}(q)=\sum \limits _{n=0}^\infty \gamma _{W_S,S^{\left( W_S\right) }}(n)q^n=\prod \limits _{n=1}^\infty \frac{1}{\left( 1-q^n\right) ^{M_S}} \end{aligned}$$
(1.3)

for wreath products \(W_S=H_S{\wr }_X\text {Sym}(X)\), where \(M_S\) is the number of conjugacy classes of \(H_S\). From now on, we denote \(\gamma _{W_S}(n):=\gamma _{W_S,S^{\left( W_S\right) }}(n)\) for convenience.

Remark

Recall Dedekind’s eta function \(\eta (\tau )=q^{1/24}\prod _{n\ge 1}(1-q^n)\) for \(\tau \in \mathcal {H}\), where \(\mathcal {H}\) denotes the upper half complex plane and \(q:=e^{2\pi i\tau }\). Equation (1.2) can be written as the linear combination of eta-quotients

$$\begin{aligned} C_{\text {Alt}(X),S'}(q)=\frac{1}{2}\cdot \frac{q^{1/12}}{\eta (\tau )^2}+\frac{1}{2}\cdot \frac{q^{1/12}}{\eta (2\tau )}, \end{aligned}$$

which is essentially the sum of a modular form of weight \(-1\) and a modular form of weight \(-\frac{1}{2}\), up to multiplication by \(q^{1/12}\). Studying such linear combinations may shed light on properties of sums of mixed weight modular forms.

It is natural to consider the number \(\gamma _{W_S}(n)\) as a function of the number of conjugacy classes \(M_S\) in order to study properties of the coefficients of the above q-series. Here we obtain a universal recurrence for these numbers. This result requires the ordinary divisor function \(\sigma _k(n)=\sum _{d\mid n}d^k\). We also must define, for \(n\ge 2\), the polynomial

$$\begin{aligned} \begin{aligned}&\widehat{F}_n(x_1,\dots ,x_{n-1})\\&\quad :=\sum _{\begin{array}{c} m_1,\dots ,m_{n-1}\ge 0\\ m_1+\cdots +(n-1)m_{n-1}=n \end{array} }(-1)^{m_1+\cdots +m_{n-1}}\cdot \frac{(m_1+\cdots +m_{n-1}-1)!}{m_1!\cdots m_{n-1}!}\cdot x_1^{m_1}\cdots x_{n-1}^{m_{n-1}}.\end{aligned} \end{aligned}$$

Remark

The polynomials \(\widehat{F}_n\) are fairly straightforward to compute using only the partitions of n; the first few are listed below.

$$\begin{aligned} \widehat{F}_2(x_1)&=\frac{1}{2}x_1^2,\\ \widehat{F}_3(x_1,x_2)&=-\frac{1}{3}x_1^3+x_1x_2,\\ \widehat{F}_4(x_1,x_2,x_3)&=\frac{1}{4}x_1^4-x_1^2x_2+\frac{1}{2}x_2^2+x_1x_3,\\ \widehat{F}_5(x_1,x_2,x_3,x_4)&=-\frac{1}{5}x_1^5+x_1^3x_2-x_1^2x_3-x_1x_2^2+x_1x_4+x_2x_3,\\ \widehat{F}_6(x_1,x_2,x_3,x_4,x_5)&=\frac{1}{6}x_1^6-x_1^4x_2+x_1^3x_3+\frac{3}{2}x_1^2x_2^2-x_1^2x_4-2x_1x_2x_3+x_1x_5-\frac{1}{3}x_2^3\\&\quad +x_2x_4+\frac{1}{2}x_3^2. \end{aligned}$$

Remark

These polynomials have been used in earlier work [2, 4] on divisors of modular forms and the Rogers–Ramanujan identities.

Theorem 1

Let \(\widehat{F}_n(x_1,\dots ,x_{n-1})\) be defined as above. Let \(H_S\) be a finite group with \(M_S\) conjugacy classes, X an infinite set, and \(W_S=H_S\wr _XSym (X)\) a wreath product generated by a sufficiently large set \(S^{\left( W_S\right) }\). Then we have

$$\begin{aligned} C_{W_S,S^{\left( W_S\right) }}(q)=\sum \limits _{n=0}^\infty \gamma _{W_S}(n)q^n=\prod \limits _{n=1}^\infty \left( 1-q^n\right) ^{-M_S}, \end{aligned}$$

where \(\gamma _{W_S}(n)\) satisfies the recurrence relation

$$\begin{aligned} \gamma _{W_S}(n)=\widehat{F}_n\big (\gamma _{W_S}(1),\dots ,\gamma _{W_S}(n-1)\big )+\frac{M_S}{n}\cdot \sigma _1(n). \end{aligned}$$

In recent work, Nekrasov and Okounkov obtained a different formula for the infinite products in Theorem 1 in terms of hook lengths of partitions. Let \(\lambda \vdash L\) denote that \(\lambda \) is a partition of the number L. The hook length of a partition \(\lambda =(\lambda _1,\dots ,\lambda _n)\vdash L\) is defined using the Ferrers diagram of \(\lambda \). For example, Fig. 1 below is a Ferrers diagram of the partition \(\lambda =(6,4,3,1,1)\vdash 15\), Fig. 2 represents a hook length of 4, and Fig. 3 shows all hook lengths associated to \(\lambda \).

Fig. 1
figure 1

Partition

Full size image
Fig. 2
figure 2

Hook length

Full size image
Fig. 3
figure 3

Hook lengths

Full size image

More generally, for each box v in the Ferrers diagram of a partition \(\lambda \), its hook length \(h_v(\lambda )\) is defined as the number of boxes u such that

  1. (i)

    \(u=v\),

  2. (ii)

    u is in the same column as v and below v, or

  3. (iii)

    u is in the same row as v and to the right of v.

The hook length multi-set \(\mathcal {H}(\lambda )\) is the set of all hook lengths of \(\lambda \). Theorem 1 implies the following formula for \(\gamma _{W_S}(n)\) in terms of hook lengths.

Corollary 2

We have that

$$\begin{aligned} \gamma _{W_S}(n)= & {} \widehat{F}_n\big (\gamma _{W_S}(1),\dots ,\gamma _{W_S}(n-1)\big )+\frac{M_S}{n}\cdot \sigma _1(n)\\= & {} \sum \limits _{\lambda \vdash n}\prod \limits _{h\in \mathcal {H}(\lambda )}\left( 1+\frac{M_S-1}{h^2}\right) . \end{aligned}$$

Remark

Kostant observed [6] that the coefficients of the Nekrasov-Okounkov hook length identity are polynomials in the variable \(z=1-M_S\), but he did not give an explicit formula for computing them.

Following Bacher’s and de la Harpe’s proofs of Eqs. (1.1), (1.2), and (1.3), we proveFootnote 3 the corresponding growth series identity for the wreath product \(W_A=H_A\wr _X\text {Alt}(X)\) with sufficiently large generating set \(S^{\left( W_A\right) }\), namely

$$\begin{aligned} C_{W_A,S^{\left( W_A\right) }}(q)=\sum \limits _{n=0}^\infty \gamma _{W_A,S^{\left( W_A\right) }}(n)q^n=\left( \frac{1}{2}\prod _{n=1}^\infty \frac{1}{(1-q^n)^2}+\frac{1}{2}\prod _{n=1}^\infty \frac{1}{1-q^{2n}}\right) ^{M_A}, \end{aligned}$$
(1.4)

where \(M_A\) is the number of conjugacy classes of \(H_A\). We provide the proof of Eq. (1.4) in Sect. 2. From now on, we denote \(\gamma _{W_A}(n):=\gamma _{W_A,S^{\left( W_A\right) }}(n)\) for convenience.

In analogy with Theorem 1, one may ask if the coefficients \(\gamma _{W_A}(n)\) in the alternating case can be seen as a function of the number of conjugacy classes \(M_A\). We obtain a similar recurrence relation in this case.

Theorem 3

Let \(\widehat{F}_n(x_1,\dots ,x_{n-1})\) be defined as above. Let \(H_A\) be a finite group with \(M_A\) conjugacy classes, X an infinite set, and \(W_A=H_A\wr _XAlt (X)\) a wreath product generated by a sufficiently large set \(S^{\left( W_A\right) }\). Then we have

$$\begin{aligned} C_{W_A,S^{\left( W_A\right) }}(q)=\sum \limits _{n=0}^\infty \gamma _{W_A}(n)q^n=\left( \frac{1}{2}\prod _{n=1}^\infty \frac{1}{\left( 1-q^n\right) ^2}+\frac{1}{2}\prod _{n=1}^\infty \frac{1}{1-q^{2n}}\right) ^{M_A}, \end{aligned}$$

where \(\gamma _{W_A}(n)\) satisfies the recurrence relation

$$\begin{aligned} \begin{aligned}\gamma _{W_A}(n)&=\frac{1}{2^{M_A}}\sum _{k=0}^{M_A}\left( {\begin{array}{c}M_A\\ k\end{array}}\right) \Bigg (\widehat{F}_n\big (a_k(1),\dots ,a_k(n-1)\big )\\&\quad \quad \quad -\sum _{\delta \mid n}\delta \cdot \left[ (-1)^\delta (k-M_A)-(k+M_A)\right] \Bigg ), \end{aligned} \end{aligned}$$

and the \(a_k\) are defined by their generating function

$$\begin{aligned} \sum _{n=0}^\infty a_k(n)q^n:=\prod _{n=1}^\infty \frac{1}{\left( 1-q^n\right) ^{2k}\left( 1-q^{2n}\right) ^{M_A-k}}. \end{aligned}$$

Remark

It may be possible to interpret the coefficients \(\gamma _{W_A}(n)\) in terms of hook lengths from formulas of Han [5] or others, as in the symmetric case. The author does not make this connection here.

It is also natural to study the modified exponential rate of conjugacy growth Footnote 4 of a group G generated by a set S, namely

$$\begin{aligned} \widetilde{H}_{G,S}^{\text {conj}}=\limsup _{n\rightarrow \infty }\frac{\log \gamma _{G,S}(n)}{\sqrt{n}}. \end{aligned}$$

It is useful to notice that \(\text {exp}\left( \widetilde{H}_{G,S}^{\text {conj}}\right) \) is the radius of convergence of the conjugacy growth series \(C_{G,S}(q)\). For permutational wreath products, we apply a theorem of Cotron, Dicks and Fleming [3] on the asymptotic behavior of the generalized partition function defined in Sect. 2 [see Eqs. (2.1) and (2.2)]. Let \(W_S=H_S\wr _X\text {Sym}(X)\) be a wreath product where \(H_S\) is a finite group, \(M_S\) is the number of conjugacy classes of \(H_S\), and X is an infinite set. It is easy to see from Eq. (1.3) that the conjugacy growth series of such a wreath product is the generating function of the generalized partition function \(p(n)_\mathbf e \) for the vector \(\mathbf e =(M_S)\). This implies the following corollary.

Corollary 4

Let \(W_S=H_S\wr _X\text {Sym}(X)\) be a wreath product where \(H_S\) is a finite group, \(M_S\) is the number of conjugacy classes of \(H_S\), and X is an infinite set. If \(S^{\left( W_S\right) }\) is a sufficiently large generating set of \(W_S\), then we have

$$\begin{aligned} \gamma _{W_S}(n)\sim \left( \frac{M_S^{\frac{1+M_S}{4}}}{2^{\frac{5+3M_S}{4}}3^{\frac{1+M_S}{4}}n^{\frac{3+M_S}{4}}}\right) e^{\pi \sqrt{\frac{2nM_S}{3}}}. \end{aligned}$$

We now give the modified exponential rate of conjugacy growth for wreath products in the symmetric case using this asymptotic formula.

Corollary 5

The modified exponential rate of conjugacy growth for the group \(W_S=H_S\wr _X\text {Sym}(X)\) defined above is

$$\begin{aligned} \widetilde{H}^{\text {conj}}_{W_S}=\pi \sqrt{\frac{2nM_S}{3}}+\frac{1}{\sqrt{n}}\cdot \log \left( \frac{M_S^{\frac{1+M_S}{4}}}{2^{\frac{5+3M_S}{4}}3^{\frac{1+M_S}{4}}n^{\frac{3+M_S}{4}}}\right) . \end{aligned}$$

We can also apply the theorem to wreath products in the alternating case using Eq. (1.4).

Corollary 6

Let \(W_A=H_A\wr _X\text {Alt}(X)\) be a wreath product where \(H_A\) is a finite group, \(M_A\) is the number of conjugacy classes of \(H_A\), and X is an infinite set. If \(S^{\left( W_A\right) }\) is a sufficiently large generating set of \(W_A\), then we have

$$\begin{aligned} \gamma _{W_A}(n)\sim \left( \frac{M_A^{\frac{1+2M_A}{4}}}{2^{1+2M_A}3^{\frac{1+2M_A}{4}}n^{\frac{3+2M_A}{4}}}\right) e^{2\pi \sqrt{\frac{nM_A}{3}}}. \end{aligned}$$

We also give the modified exponential rate of conjugacy growth in the alternating case using the above asymptotic formula.

Corollary 7

The modified exponential rate of conjugacy growth for the group \(W_A=H_A\wr _X\text {Alt}(X)\) defined above is

$$\begin{aligned} \widetilde{H}^{\text {conj}}_{W_A}=2\pi \sqrt{\frac{nM_A}{3}}+\frac{1}{\sqrt{n}}\cdot \log \left( \frac{M_A^{\frac{1+2M_A}{4}}}{2^{1+2M_A}3^{\frac{1+2M_A}{4}}n^{\frac{3+2M_A}{4}}}\right) . \end{aligned}$$

We are interested in finding relationships between wreath products of \(\text {Sym}(X)\) and wreath products of \(\text {Alt}(X)\). Let \(W_S=H_S\wr _X\text {Sym}(X)\) and \(W_S'=H_S'\wr _X\text {Sym}(X)\) be two wreath products of \(\text {Sym}(X)\), where \(H_S,H_S'\) are finite groups with \(M_S,M_S'\) conjugacy classes respectively. Let \(W_A=H_A\wr _X\text {Alt}(X)\) and \(W_A'=H_A'\wr _X\text {Alt}(X)\) be two wreath products of \(\text {Alt}(X)\), where \(H_A, H_A'\) are finite groups with \(M_A, M_A'\) conjugacy classes respectively.

Question 1

What is the asymptotic behavior of the following ratios?

$$\begin{aligned} (1)\quad \quad \frac{\gamma _{W_S}(n)}{\gamma _{W_S'}(n)}, \quad \quad (2)\quad \quad \frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}, \quad \quad (3) \quad \quad \frac{\gamma _{W_A}(n)}{\gamma _{W_S}(n)},\quad \quad (4)\frac{\gamma _{W_A}(n)}{\gamma _{W_A'}(n)}. \end{aligned}$$

In particular, when do the ratios approach some nonzero finite number?

The asymptotic behavior of the ratios follows from Corollaries 4 and 6.

Corollary 8

Let \(W_S,W_S',W_A,\) and \(W_A'\) be groups as above. Then as \(n\rightarrow \infty \), we have

$$\begin{aligned} (1) \quad \quad \frac{\gamma _{W_S}(n)}{\gamma _{W_S'}(n)}\sim & {} \left( \frac{M_S^{\frac{1+M_S}{4}}}{M_S'^{\frac{1+M_S'}{4}}}\right) \left[ 2^{\frac{3}{4}\left( M_S'-M_S\right) }(3n)^{\frac{M_S'-M_S}{4}}\right] e^{\pi \sqrt{\frac{2n}{3}}\left( \sqrt{M_S}-\sqrt{M_S'}\right) }.\\ (2) \quad \quad \frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\sim & {} \left( \frac{M_S^{\frac{1+M_S}{4}}}{M_A^{\frac{1+2M_A}{4}}}\right) \left[ 2^{\frac{8M_A-3M_S-1}{4}}(3n)^{\frac{2M_A-M_S}{4}}\right] e^{\pi \sqrt{\frac{2n}{3}}\left( \sqrt{M_S}-\sqrt{2M_A}\right) }.\\ (3) \quad \quad \frac{\gamma _{W_A}(n)}{\gamma _{W_S}(n)}\sim & {} \left( \frac{M_A^{\frac{1+2M_A}{4}}}{M_S^{\frac{1+M_S}{4}}}\right) \left[ 2^{\frac{1+3M_S-8M_A}{4}}(3n)^{\frac{M_S-2M_A}{4}=}\right] e^{\pi \sqrt{\frac{2n}{3}}\left( \sqrt{2M_A}-\sqrt{M_S}\right) }.\\ (4) \quad \quad \frac{\gamma _{W_A}(n)}{\gamma _{W_A'}(n)}\sim & {} \left( \frac{M_A^{\frac{1+2M_A}{4}}}{M_A'^{\frac{1+2M_A'}{4}}}\right) \left[ 4^{\left( M_A'-M_A\right) }(3n)^{\frac{M_A'-M_A}{2}}\right] e^{2\pi \sqrt{\frac{n}{3}}\left( \sqrt{M_A}-\sqrt{M_A'}\right) }. \end{aligned}$$

We now observe for which pairs \((M_S,M_S'),(M_S,M_A),(M_A,M_S),\) and \((M_A,M_A')\) these ratios asymptotically approach zero, infinity, or some nonzero finite number. Corollary 9 follows from the asymptotic behavior of the exponential functions in Corollary 8.

Corollary 9

Let \(W_S,W_S',W_A,\) and \(W_A'\) be groups as above. Then as \(n\rightarrow \infty \), we have the following asymptotic behavior.

  1. (1)

    If \(M_S<M_S'\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_S'}(n)}\sim 0\). If \(M_S>M_S'\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_S'}(n)}\sim \infty \). If \(M_S=M_S'\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_S'}(n)}\sim 1\).

  2. (2)

    If \(M_S<2M_A\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\sim 0\). If \(M_S>2M_A\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\sim \infty \). If \(M_S=2M_A\), then \(\frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\sim 2^{M_A}\).

  3. (3)

    If \(2M_A<M_S\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_S}(n)}\sim 0\). If \(2M_A>M_S\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_S}(n)}\sim \infty \). If \(2M_A=M_S\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_S}(n)}\sim \frac{1}{2^{M_A}}\).

  4. (4)

    If \(M_A<M_A'\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_A'}(n)}\sim 0\). If \(M_A>M_A'\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_A'}(n)}\sim \infty \). If \(M_A=M_A'\), then \(\frac{\gamma _{W_A}(n)}{\gamma _{W_A'}(n)}\sim 1\).

Moreover, the converses of all of the above statements hold as well.

Given any two wreath products of \(\text {Sym}(X)\) or \(\text {Alt}(X)\), Corollary 9 guarantees the asymptotic behavior of the ratios between the coefficients of their conjugacy growth series. In other words, for any two wreath products W and \(W'\), we know the expected relationship between the number of conjugacy classes of H in W and the number of conjugacy classes of \(H'\) in \(W'\) with minimal word length n for any n.

Remark

Although we know the asymptotic behavior of the above ratios, this does not mean that the ratios of the coefficients are always exactly equal to the above values.

For example, consider the wreath products \(W_S=H_S\wr _X\text {Sym}(X)\) and \(W_A=H_A\wr _X\text {Alt(X)}\), where \(H_S,H_A\) are finite groups with \(M_S=10,M_A=5\) conjugacy classes respectively. We expect the ratio of the coefficients of \(W_S\) to the coefficients of \(W_A\) to be asymptotic to \(2^5=32\). We compute the following coefficients with Maple.

n

\(\gamma _{W_S}(n)\)

\(\gamma _{W_A}(n)\)

\(\frac{\gamma _{W_S}(n)}{\gamma _{W_A}(n)}\)

1

10

5

2

10

1605340

176963

9.071613840

100

\(0.2333013623\times 10^{28}\)

\(0.7541087996\times 10^{26}\)

30.93736108

200

\(0.1067904403\times 10^{42}\)

\(0.3346942881\times 10^{40}\)

31.90686071

300

\(0.4721905614\times 10^{52}\)

\(0.1476229954\times 10^{51}\)

31.98624714

400

\(0.5248644122\times 10^{61}\)

\(0.1640339890\times 10^{60}\)

31.99729613

500

\(0.5369981415\times 10^{69}\)

\(0.1678152777\times 10^{68}\)

31.99935959

2 Proofs

We give the proofs of Eq. (1.4) and Theorems 1 and 3 here. We also explain what it means for a generating set to be sufficiently large and give remarks on Corollaries 2 and 6.

A set S of transpositions of a set X is called partition-complete (PC) [1] if

  1. (i)

    The transposition graph \(\Gamma (S)\) is connected, and

  2. (ii)

    For every partition \(\lambda =(\lambda _1,\dots ,\lambda _k)\vdash L\), \(\Gamma (S)\) contains a forest of k trees with \(\lambda _1+1,\dots ,\lambda _k+1\) vertices respectively.

For the corresponding property of partition-complete for wreath products (PCwr) [1], we must first establish more notation. Let X be an infinite set, H a group, and \(W=H\wr _X\text {Sym}(X)\). The group W acts naturally on the set \(H\times X\); namely, for \((\varphi ,f)\in W\), the action is defined by

$$\begin{aligned} (h,x)\mapsto \big (\varphi (f(x))h,f(x)\big ). \end{aligned}$$

For \(a\in H\setminus \{1\}\) and \(u\in X\), we let \(\varphi _u^a\in W\) denote the permutation that maps \((h,x)\in H\times X\) to (ah, u) if \(x=u\), and to (h, x) otherwise. Then \((\varphi _u^a)_{a\in H\setminus \{1\},\,u\in X}\) generates the group \(H^{(X)}\). Now, let \(H_u:=\{\varphi _u^a\mid a\in H\setminus \{1\}\}\), and define the subsets

$$\begin{aligned} T_H:= & {} \bigcup _{u\in X}H_u\subseteq H^{(X)},\\ T_X:= & {} \{(x\,\,\,y)\in \text {Sym}(X):x,y\in X\text { are distinct}\}\subseteq \text {Sym}(X). \end{aligned}$$

Let \(S_H\subset T_H\) and \(S_X\subset T_X\) be subsets, and let \(S=S_H\sqcup S_X\subseteq W\). Such a set S is said to be PCwr if

  1. (i)

    The transposition graph \(\Gamma (S_X)\) is connected, and

  2. (ii)

    For all \(L\ge 0\) and partitions \(\lambda =(\lambda _1,\dots ,\lambda _k)\vdash L\), \(\Gamma (S_X)\) contains a forest of k trees \(T_1,\dots ,T_k\), with \(T_i\) having \(\lambda _i\) vertices, including one vertex \(x^{(i)}\) such that \(\varphi _{x^{(i)}}^a\in S_H\) for all \(a\in H{\setminus }\{1\}\).

Remark

The conditions PC and PCwr essentially require the generating set S to contain “enough” transpositions to represent all possible partitions in its transposition graph.

Proof of equation (1.4)

This proof followsFootnote 5 from the proofs of eqs. (1.2) and (1.3) in [1]. For each \(w=(\phi , \sigma )\in W_A = H_A \wr _{X} \text {Alt}(X)\), we can split \(\sigma \) into a product of an even number of cycles of even length, denoted \(\sigma _{e}\), and a product of cycles of odd length, denoted \(\sigma _{o}\), so that \(w = (\phi , \sigma _{e} \sigma _{o})\). Let \({(H_A)}_{*}\) denote the set of conjugacy classes of \(H_A\); we write \(1\in {(H_A)}_{*}\) for the class \(\{1\}\in H_A\). To each conjugacy class in \(W_A\) we associate an \({(H_A)}_{*}\)-indexed family of partitions. Using the same notation as in [1], we associate the conjugacy classes in \(H_A\) to the family of partitions

$$\begin{aligned} \left( \lambda ^{(1)}, \nu ^{(1)}; \left( \mu ^{(\eta )}, \gamma ^{(\eta )}\right) _{\eta \in {(H_A)}_{*} \setminus 1} \right) , \end{aligned}$$

where \(\nu ^{(1)}\) and \(\gamma ^{(\eta )}\) each have an even number of positive parts, in the following way.

Let \(X^{(w)}\) be the finite subset of X that is the union of the supports of \(\phi \) and \(\sigma \). Let \(\sigma \) be the product of the disjoint cycles \(c_{1}, ..., c_{k}\), where \(c_{i} = \left( x_{1}^{(i)}, x_{2}^{(i)}, ..., x_{v_{i}}^{(i)}\right) \) with \(x_{j}^{(i)} \in X^{(w)}\) and \(v_{i} = \text { length}(c_{i})\). We include cycles of length 1 for each \(x \in X\) such that \(x \in \text {sup}(\phi )\) and \(x \notin \text {sup}(\sigma )\), so that

$$\begin{aligned} X^{(w)} = \bigsqcup _{1\le i \le k}\text {sup}(c_{i}). \end{aligned}$$

Define \(\eta _{*}^{w}(c_{i}) \in {(H_A)}_{*}\) to be the conjugacy class of the product \(\phi \left( x_{v_{i}}^{(i)}\right) \phi \left( x_{v{i}-1}^{(i)}\right) \cdots \phi \left( x_{1}^{(i)}\right) \in H_A\). For \(\eta \in {(H_A)}_{*}\) and \(\ell \ge 1\), let \(m_{\ell }^{w,\eta }\) denote the number of cycles c in \(\{c_{1}, ..., c_{k} \}\) such that \(\text {length}(c)=\ell \) and \(\eta _{*}^{w}(c) = \eta \). Let \(\mu ^{w, \eta } \vdash n^{w, \eta }\) be the partition with \(m_{\ell }^{w, \eta }\) parts equal to \(\ell \), for all \(\ell \ge 1\). Note that

$$\begin{aligned} \sum _{\eta \in {(H_A)}_{*}} n^{w, \eta } = \sum _{\eta \in {(H_A)}_{*},\,\,\ell \ge 1} \ell m_{\ell }^{w, \eta } = |X^{(w)}|. \end{aligned}$$

Also observe that the partition \(\mu ^{w, 1}\) does not contain parts of size 1, because if \(v_{i} = 1\), then \(\eta _{*}^{w}(c_{i}) \ne 1\). Using the same notation as above, let \(\lambda ^{w, 1}\) be the partition with \(m_{\ell }^{w, 1}\) parts equal to \(\ell - 1\). We can write \(\sigma = \sigma _{e} \sigma _{o}\) as above, so \(\mu ^{w,1}\) and \(\lambda ^{w,1}\) each split into two partitions, one of which has an even number of parts. Define the type of w to be the family \(\left( \lambda ^{(1)}, \nu ^{(1)}; \left( \mu ^{(\eta )}, \gamma ^{(\eta )}\right) _{\eta \in {(H_A)}_{*} \setminus 1} \right) \). Then two elements in \(W_A\) are conjugate if and only if they have the same type. Thus, each \({(H_A)}_{*}\)-indexed family of partitions \(\left( \lambda ^{(1)}, \nu ^{(1)}; \left( \mu ^{(\eta )}, \gamma ^{(\eta )}\right) _{\eta \in {(H_A)}_{*} \setminus 1} \right) \) is the type of one conjugacy class in \(W_A\).

Consider an \({(H_A)}_{*}\)-indexed family of partitions \(\left( \lambda ^{(1)}, \nu ^{(1)}; \left( \mu ^{(\eta )}, \gamma ^{(\eta )}\right) _{\eta \in {(H_A)}_{*}\setminus 1} \right) \) and the corresponding conjugacy class in \(W_A\). Let \(u^{(1)}, v^{(1)}, u^{(\eta )}, v^{(\eta )}\) be the sums of the parts of \(\lambda ^{(1)}, \nu ^{(1)}, \mu ^{(\eta )}, \gamma ^{(\eta )}\) respectively, and let \(k^{(1)}, t^{(1)}, k^{(\eta )}, t^{(\eta )}\) be the number of parts of \(\lambda ^{(1)}, \nu ^{(1)}, \mu ^{(\eta )}, \gamma ^{(\eta )}\) respectively.

Choose a representative \(w = (\phi , \sigma )\) of this conjugacy class such that

$$\begin{aligned} \sigma = \prod _{i=1}^k c_{i} = \prod _{i=1}^k \left( x_{1}^{(i)}, x_{2}^{(i)}, ..., x_{\mu _{i}}^{(i)}\right) \end{aligned}$$

and

$$\begin{aligned} \phi \left( x_{j}^{(i)}\right)= & {} 1 \in H_A \quad \text { for all } j \in \{1, ..., \mu _{i} \} \quad \text { when } \eta _{*}^{w}(c_{i}) = 1,\\ \phi \left( x_{j}^{(i)}\right)= & {} {\left\{ \begin{array}{ll} 1 &{}\text { for all } j \in \{1, ..., \mu _{i} - 1\} \\ h \ne 1 &{}\text { for } j = \mu _{i} \end{array}\right. } \quad \text { when } \eta _{*}^{w}(c_{i}) \ne 1. \end{aligned}$$

Observe that

$$\begin{aligned} k= & {} k^{(1)} + t^{(1)} + \sum _{\eta \in {(H_A)}_{*}\setminus 1} \left( k^{(\eta )} + t^{(\eta )} \right) ,\\ \Big |X^{(w)}\Big |= & {} u^{(1)} + k^{(1)} + v^{(1)} + t^{(1)} + \sum _{\eta \in {(H_A)}_{*}\setminus 1} \left( u^{(\eta )} + v^{(\eta )} \right) . \end{aligned}$$

Hence, the contribution to \(C_{W_A,S^{(W_A)}}(q)\) from \(\left( \lambda ^{(1)}, \nu ^{(1)}; \left( \mu ^{(\eta )}, \gamma ^{(\eta )}\right) _{\eta \in {(H_A)}_{*}\setminus 1} \right) \) is

$$\begin{aligned} \left( q^{u^{(1)}}q^{v^{(1)}} \prod _{\eta \in {(H_A)}_{*}\setminus 1} q^{u^{(\eta )}}q^{v^{(\eta )}} \right) . \end{aligned}$$

It follows that

$$\begin{aligned} C_{W_A,S^{\left( W_A\right) }}(q)= & {} \left[ \left( \prod _{u_{1}=1}^\infty \frac{1}{1-q^{u_{1}}} \right) \left( \frac{1}{2} \prod _{v_{1}=1}^\infty \frac{1}{1-q^{v_{1}}} + \frac{1}{2} \prod _{v_{1}=1}^\infty \frac{1}{1+q^{v_{1}}} \right) \right] \\&\times \prod _{\eta \in {(H_A)}_{*}\setminus 1} \left[ \left( \prod _{u_{\eta }=1}^\infty \frac{1}{1-q^{u_{\eta }}} \right) \left( \frac{1}{2} \prod _{v_{\eta }=1}^\infty \frac{1}{1-q^{v_{\eta }}} + \frac{1}{2} \prod _{v_{\eta }=1}^\infty \frac{1}{1+q^{v_{\eta }}} \right) \right] \\= & {} \left[ \left( \frac{1}{2} \prod _{n_{1}=1}^\infty \frac{1}{1-q^{2 n_{1}}} + \frac{1}{2} \prod _{n_{1}=1}^\infty \frac{1}{(1-q^{n_{1}})^{2}} \right) \right] \\&\times \prod _{\eta \in {(H_A)}_{*}\setminus 1} \left[ \left( \frac{1}{2} \prod _{n_{\eta }=1}^\infty \frac{1}{1-q^{2 n_{\eta }}} + \frac{1}{2} \prod _{n_{\eta }=1}^\infty \frac{1}{(1-q^{n_{\eta }})^{2}} \right) \right] \\= & {} \left( \frac{1}{2} \prod _{k=1}^\infty \frac{1}{1-q^{2 k}} + \frac{1}{2} \prod _{k=1}^\infty \frac{1}{(1-q^{k})^{2}} \right) ^{\left| {(H_A)}_{*}\right| }. \end{aligned}$$

The equality between the first and second line is given in the appendix of [1]. \(\square \)

The generalized partition function \(p(n)_\mathbf e \) is defined for the vector \(\mathbf e =(e_1,\dots ,e_k)\in \mathbb {Z}^k\) by its generating function

$$\begin{aligned} \sum \limits _{n=0}^\infty p(n)_\mathbf e q^n=\prod _{n=1}^\infty \frac{1}{(1-q^n)^{e_1}\cdots (1-q^{kn})^{e_k}}. \end{aligned}$$
(2.1)

The following theorem gives an asymptotic formula for the generalized partition function, which was obtained by using properties of modular forms.Footnote 6

Theorem  (Cotron-Dicks-Fleming [3]) Let \(\mathbf e =(e_1,\dots ,e_k)\) be any nonzero vector with nonnegative integer entries, and let \(d:=\gcd \{m:e_m\ne 0\}\). Define the quantities

$$\begin{aligned} \gamma :=\gamma (\mathbf e )=\sum _{m=1}^ke_{dm}\text {and}\delta :=\delta (\mathbf e )=\sum _{m=1}^k\frac{e_{dm}}{m}. \end{aligned}$$

Then as \(n\rightarrow \infty \), we have that

$$\begin{aligned} p(dn)_\mathbf{e }\sim \frac{\lambda A^{\frac{1+\gamma }{4}}}{2\sqrt{\pi }n^{\frac{3+\gamma }{4}}}e^{2\sqrt{An}}, \end{aligned}$$
(2.2)

where

$$\begin{aligned} \lambda :=\prod _{m=1}^k\left( \frac{m}{2\pi }\right) ^{\frac{e_{dm}}{2}}\text {and}A:=\frac{\pi ^2\delta }{6}. \end{aligned}$$

Corollaries 4, 6, 8, and 9 all follow from the above theorem, since the conjugacy growth series for permutational wreath products correspond to the generalized partition function with vectors \((M_S)\) in the symmetric case and \((2k,M_A-k),0\le k\le M_A,\) in the alternating case.

A Remark on Corollary 6

By the binomial theorem applied to the conjugacy growth series in Eq. (1.4), we find that

$$\begin{aligned} \gamma _{W_A}(n)\sim \frac{1}{2^{M_A}}\sum _{k=0}^{M_A}\left[ \frac{(4M_A-3k)^{\frac{1+2M_A-k}{4}}}{2^{\frac{4M_A-3k+3}{2}}3^{\frac{1+2M_A-k}{4}}n^{\frac{3+2M_A-k}{4}}}\cdot e^{2\pi \sqrt{\left( \frac{4M_A-3k}{12}\right) n}}\right] . \end{aligned}$$

But, intuitively, the summands corresponding to \(k>0\) grow much more slowly than the summand corresponding to \(k=0\), since the instance of k in the exponential function is negative. Therefore, the above sum is asymptotic to the \(k=0\) term, so we have

$$\begin{aligned} \gamma _{W_A}(n)\sim \frac{M_A^{\frac{1+2M_A}{4}}}{2^{1+2M_A}3^{\frac{1+2M_A}{4}}n^{\frac{3+2M_A}{4}}}\cdot e^{2\pi \sqrt{\frac{nM_A}{3}}}. \end{aligned}$$

\(\square \)

We now introduce the proofs of Theorems 1 and 3. In a paper by Bruinier, Kohnen, and Ono [2], the universal polynomial \(F_n\) is defined as

$$\begin{aligned}&F_n\left( x_1,\dots ,x_{n-1}\right) :=-\frac{2x_1\sigma _1(n-1)}{n-1}\\&\qquad +\sum _{\begin{array}{c} m_1,\dots ,m_{n-2}\ge 0 \\ m_1+\cdots +(n-2)m_{n-2}=n-1 \end{array} }(-1)^{m_1+\cdots +m_{n-2}}\cdot \frac{(m_1+\cdots +m_{n-2}-1)!}{m_1!\cdots m_{n-2}!}\cdot x_2^{m_1}\cdots x_{n-1}^{m_{n-2}}, \end{aligned}$$

and it is used to define a recursion relation for coefficients of meromorphic modular forms on \(SL_2(\mathbb {Z})\). Frechette and the author [4] modify this polynomial to the above \(\widehat{F}_n\) and use it to define a recursion relation for coefficients of quotients of Rogers-Ramanujan-type q-series. Their proof surprisingly only requires properties of logarithmic derivatives applied to a q-series infinite product identity. The proof below is adapted from the proof in [4] and can be applied to any q-series infinite product identity, including the famous identity of Nekrasov and Okounkov [7].

Proof of Theorem 1

Define the q-series identity

$$\begin{aligned} F_r(q):=\sum \limits _{n=0}^\infty p_n(r)q^n:=\prod _{n=1}^\infty (1-q^n)^r \end{aligned}$$

so that \(p_n(r)=\gamma _{W_S}(n)\) and \(r=-M_S\). We take logarithms of both sides to obtain

$$\begin{aligned} \log \left( 1+\sum \limits _{n=1}^\infty p_n(r)q^n\right)= & {} \sum \limits _{n=1}^\infty r\log (1-q^n)\\= & {} -\sum \limits _{n=1}^\infty \sum \limits _{k=1}^\infty \frac{rq^{kn}}{k}, \end{aligned}$$

by the Taylor expansion for \(\log (1-x)\). Then we take the derivatives of both sides to obtain

$$\begin{aligned} \frac{\sum \nolimits _{n=1}^\infty np_n(r)q^{n-1}}{1+\sum \nolimits _{n=1}^\infty p_n(r)q^n}= & {} -\sum \nolimits _{n=1}^\infty \sum \limits _{d\mid n} rdq^{n-1}\\= & {} -\sum \limits _{n=1}^\infty r\sigma _1(n)q^{n-1}, \end{aligned}$$

so we have

$$\begin{aligned} \sum \limits _{n=1}^\infty np_n(r)q^n=\left( -\sum \limits _{n=1}^\infty r\sigma _1(n)q^n\right) \left( 1+\sum \limits _{n=1}^\infty p_n(r)q^n\right) . \end{aligned}$$

For convenience, define \(b(n):=r\sigma _1(n)\). Expanding the right hand side and equating coefficients, we now have

$$\begin{aligned} 0=b(n)+b(n-1)p_1(r)+b(n-2)p_2(r)+\cdots +b(1)p_{n-1}(r)+np_n(r). \end{aligned}$$

The symmetric power functions

$$\begin{aligned} s_i:=X_1^i+\cdots +X_n^i \end{aligned}$$

and the elementary symmetric functionsFootnote 7

$$\begin{aligned} \sigma _i=\sum \limits _{1\le j_1\le \cdots \le j_i\le n}X_{j_1}\cdots X_{j_i} \end{aligned}$$

exhibit a similar relationship; namely, we have the identity

$$\begin{aligned} 0=s_n-s_{n-1}\sigma _1+s_{n-2}\sigma _2-\cdots +(-1)^{n-1}s_1\sigma _{n-1}+(-1)^n\sigma _n. \end{aligned}$$
(2.3)

Evaluating Eq. (2.3) at \((X_1,\dots ,X_n)=(\lambda (1,n),\dots ,\lambda (n,n))\), where \(\lambda (j,n)\) are the roots of the polynomial

$$\begin{aligned} X^n+p_1(r)X^{n-1}+\cdots +p_{n-1}(r)X+p_n(r), \end{aligned}$$

we have that \(p_n(r)=(-1)^n\sigma _n\) for \(n\ge 1\). Then we have \(b(n)=s_n\). Using the fact that

$$\begin{aligned} s_n=n\sum _{\begin{array}{c} m_1,\dots ,m_n\ge 0 \\ m_1+\cdots +nm_n=n \end{array} }(-1)^{m_2+m_4+\cdots }\cdot \frac{(m_1+\cdots +m_n-1)!}{m_1!\cdots m_n!}\cdot \sigma _1^{m_1}\cdots \sigma _n^{m_n}, \end{aligned}$$

we see that \(\frac{b(n)}{n}\) is exactly \(\widehat{F}_n\big (p_1(r),\dots ,p_{n-1}(r)\big )\) plus the additional term with \(m_1=\dots =m_{n-1}=0\) and \(m_n=1\). We arrive at the recursion

$$\begin{aligned} p_n(r)=\widehat{F}_n\big (p_1(r),\dots ,p_{n-1}(r)\big )-\frac{r}{n}\sigma _1(n). \end{aligned}$$

Thus, we have

$$\begin{aligned} \gamma _{W_S}(n)=\widehat{F}_n\big (\gamma _{W_S}(1),\dots ,\gamma _{W_S}(n-1)\big )+\frac{M_S}{n}\sigma _1(n). \end{aligned}$$

\(\square \)

Theorem 1 gives a recurrence formula for the coefficients \(\gamma _{W_S}(n)\) of the conjugacy growth series of a permutational wreath product in which the group \(H_S\) has \(M_S\) conjugacy classes. Now, we consider the more general infinite product \(\prod _{n\ge 1}\left( 1-q^n\right) ^r\) for any complex number r, and we ignore its implications for finite groups. Then the above proof also applies to the coefficients of the Nekrasov-Okounkov hook length formula [7]

$$\begin{aligned} \sum \limits _{\lambda \in \mathcal {P}}x^{|\lambda |}\prod \limits _{h\in \mathcal {H}(\lambda )}\left( 1-\frac{z}{h^2}\right) =\prod \limits _{k\ge 1}\left( 1-x^k\right) ^{z-1} \end{aligned}$$

if we change variables \(z\mapsto 1+r\) and \(x\mapsto q:=e^{2\pi i\tau }\) for \(\tau \in \mathcal {H}\). The coefficients

$$\begin{aligned} \prod \limits _{h\in \mathcal {H}(\lambda )}\left( 1-\frac{z}{h^2}\right) =\prod \limits _{h\in \mathcal {H}(\lambda )}\left( 1-\frac{1+r}{h^2}\right) \end{aligned}$$

of the infinite product

$$\begin{aligned} \prod \limits _{k\ge 1}\left( 1-x^k\right) ^{z-1}=\prod \limits _{n\ge 1}\left( 1-q^n\right) ^r \end{aligned}$$

therefore satisfy the recurrence relation

$$\begin{aligned} \prod \limits _{h\in \mathcal {H}(\lambda )}\left( 1-\frac{1+r}{h^2}\right) =\gamma _{W_S}(n)=\widehat{F}_n\big (\gamma _{W_S}(1),\dots ,\gamma _{W_S}(n-1)\big )-\frac{r}{n}\sigma _1(n). \end{aligned}$$

Although for \(r\in \mathbb {C}\setminus \mathbb {Z}^+\) we can no longer observe the relationship between the number of conjugacy classes of \(H_S\) and the coefficients of the conjugacy growth series of \(H_S\wr _X\text {Sym}(X)\), we do obtain a simple recursion for the Nekrasov-Okounkov hook length formula which is independent of complex analysis and hook lengths.

Proof of Theorem 3

This proof closely follows the proof of Theorem 1. Define the q-series identity

$$\begin{aligned} F_{M_A}(q):=\sum \limits _{n=0}^\infty P_n(M_A)q^n:=\left( \frac{1}{2}\prod _{n=1}^\infty \frac{1}{\left( 1-q^n\right) ^2}+\frac{1}{2}\prod _{n=1}^\infty \frac{1}{1-q^{2n}}\right) ^{M_A} \end{aligned}$$

so that \(P_n(M_A)=\gamma _{W_A}(n)\). Then, by the binomial theorem, we have

$$\begin{aligned} \sum \limits _{n=0}^\infty P_n(M_A)q^n=\frac{1}{2^{M_A}}\sum _{k=1}^{M_A}\left( {\begin{array}{c}M_A\\ k\end{array}}\right) \prod _{n=1}^\infty \frac{1}{\left( 1-q^n\right) ^{2k}\left( 1-q^{2n}\right) ^{M_A-k}}. \end{aligned}$$

It suffices to find recurrence relations for each summand. Define

$$\begin{aligned} F_{M_A,k}(q):=\sum _{n=0}^\infty a_k(n)q^n:=\prod _{n=1}^\infty \frac{1}{\left( 1-q^n\right) ^{2k}\left( 1-q^{2n}\right) ^{M_A-k}}. \end{aligned}$$

We take the logarithmic derivative of both sides as in the proof of Theorem 1. First, we take logarithms of both sides to obtain

$$\begin{aligned} \log \left( 1+\sum \limits _{n=1}^\infty a_k(n)q^n\right)= & {} -2k\sum _{n=1}^\infty \log \left( 1-q^n\right) +(k-M_A)\sum _{n=1}^\infty \log \left( 1-q^{2n}\right) \\= & {} -(k+M_A)\sum _{n=1}^\infty \log \left( 1-q^n\right) +(k-M_A)\sum _{n=1}^\infty \log \left( 1+q^n\right) \\= & {} (k+M_A)\sum _{n=1}^\infty \sum _{m=1}^\infty \frac{q^{mn}}{m}+(M_A-k)\sum _{n=1}^\infty \sum _{m=1}^\infty \frac{(-1)^mq^{mn}}{m}, \end{aligned}$$

by the Taylor expansions for \(\log (1-x)\) and \(\log (1+x)\). Then we take the derivatives of both sides to obtain

$$\begin{aligned} \frac{\sum \nolimits _{n=1}^\infty na_k(n)q^{n-1}}{1+\sum \nolimits _{n=1}^\infty a_k(n)q^n}= & {} -\sum _{n=1}^\infty \sum _{d\mid n}d\cdot \left[ (-1)^{\frac{n}{d}}(k-M_A)-(k+M_A)\right] q^{n-1}, \end{aligned}$$

so we have

$$\begin{aligned} \sum \limits _{n=1}^\infty na_k(n)q^n=\left( -\sum _{n=1}^\infty \sum _{d\mid n}d\cdot \left[ (-1)^{\frac{n}{d}}(k-M_A)-(k+M_A)\right] q^n\right) \left( 1+\sum \limits _{n=1}^\infty a_k(n)q^n\right) . \end{aligned}$$

For convenience, define \(b_k(n):=\sum _{d\mid n}d\cdot \left[ (-1)^{\frac{n}{d}}(k-M_A)-(k+M_A)\right] \). Expanding the right hand side and equating coefficients, we now have

$$\begin{aligned} 0=b_k(n)+b_k(n-1)a_k(1)+b_k(n-2)a_k(2)+\cdots +b_k(1)a_k(n-1)+na_k(n). \end{aligned}$$

Using the same identity between the symmetric power functions and the elementary symmetric functions as in the proof of Theorem 1, we arrive at the recursion

$$\begin{aligned} a_k(n)= & {} \widehat{F}_n\big (a_k(1),\dots ,a_k(n-1)\big )-\frac{1}{n}\sum _{d\mid n}d\cdot \left[ (-1)^{\frac{n}{d}}(k-M_A)-(k+M_A)\right] \\= & {} \widehat{F}_n\big (a_k(1),\dots ,a_k(n-1)\big )-\sum _{\delta \mid n}\delta \cdot \left[ (-1)^\delta (k-M_A)-(k+M_A)\right] . \end{aligned}$$

Thus, we have

$$\begin{aligned} \gamma _{W_A}(n)= & {} \frac{1}{2^{M_A}}\sum _{k=0}^{M_A}\left( {\begin{array}{c}M_A\\ k\end{array}}\right) \left( \phantom {\sum _{\delta \mid n}}\widehat{F}_n\big (a_k(1),\dots ,a_k(n-1)\big )\right. \\&\quad \quad \left. -\sum _{\delta \mid n}\delta \cdot \left[ (-1)^\delta (k-M_A)-(k+M_A)\right] \right) . \end{aligned}$$

\(\square \)

Remark

This recurrence relation gives the coefficients \(\gamma _{W_A}(n)\) in terms of the coefficients \(a_k(1)\), \(\dots \), \(a_k(n-1)\) of each summand. Since the linear combination of infinite products is raised to the \((M_A)\)th power in the conjugacy growth series, presumably there is no simple way to obtain a recurrence relation for \(\gamma _{W_A}(n)\) in terms of \(\gamma _{W_A}(1),\dots ,\gamma _{W_A}(n-1)\) as in the symmetric case.

Acknowledgements

The author would like to thank Ken Ono for his invaluable advice and guidance on this project, and also Pierre de la Harpe and the referee for their helpful comments.