Discrete quotients of 3-dimensional N = 4 Coulomb branches via the cycle index

The study of Coulomb branches of 3-dimensional N=4 gauge theories via the associated Hilbert series, the so-called monopole formula, has been proven useful not only for 3-dimensional theories, but also for Higgs branches of 5 and 6-dimensional gauge theories with 8 supercharges. Recently, a conjecture connected different phases of 6-dimensional Higgs branches via gauging of a discrete global $S_n$ symmetry. On the corresponding 3-dimensional Coulomb branch, this amounts to a geometric $S_n$-quotient. In this note, we prove the conjecture on Coulomb branches with unitary nodes and, moreover, extend it to Coulomb branches with other classical groups. The results promote discrete $S_n$-quotients to a versatile tool in the study of Coulomb branches.

The standard lore suggests that Higgs branches of theories with 8 supercharges in dimensions 3, 4, 5, and 6 are classically exact. In 5-dimensional N = 1 theories, the Higgs branch M 5d H g=∞ at infinite gauge coupling, however, grows as new massless degrees of freedom appear in the form of instanton operators. As the Higgs branch is still a hyper-Kähler space, M 5d H g=∞ has a 3dimensional Coulomb branch counterpart, provided the global symmetry is large enough [17,18]. To be precise, this means that a 3-dimensional N = 4 gauge theory exists such that its Coulomb branch agrees with M 5d H g=∞ . Such a quiver is further realised in the study of 5-brane webs and 7-branes.
Similarly, 6-dimensional N = (1, 0) theories exhibit a previously unappreciated rich phase structure of the Higgs branch as recent developments have shown [19,20]. As it turns out, many interesting effects on the 6-dimensional Higgs branches can be described neatly by associated 3dimensional N = 4 theories, whose Coulomb branches M 3d C agree with the 6-dimensional Higgs branches M 6d H as algebraic varieties. In particular, the 3-dimensional quiver gauge theory is understood as a tool that captures the geometry of the moduli space. Besides the small E 8 -instanton transition [19], another interesting phenomenon is discrete gauging [20]. For the latter, it is crucial to realise that the gauging of a discrete global symmetry Γ on M 6d H corresponds to a quotient of M 3d C by Γ. In other words, restriction to the Γ-invariant sector.
In this note we prove earlier conjectures and extend the concept of discrete quotients of 3dimensional Coulomb branches to other scenarios. To begin with, we recall two examples.
Symmetric products of ALE spaces. Consider k D2 branes in the presence of n D6 branes in flat space. The worldvolume theory on the D2 branes is a 3-dimensional N = 4 U(k) gauge theory with one adjoint and n fundamental hypermultiplets. The corresponding quiver theory is the A-type ADHM quiver Adj (1.1) such that the Higgs branch is the moduli space M k,SU(n),C 2 of k SU(n)-instantons on C 2 . Moreover, 3d mirror symmetry predicts that the Coulomb branch is the symmetric product of k copies of the A n−1 singularity [21,22]. In detail, We recall that a 3-dimensional N = 4 U(1) gauge theory with n fundamentals has C 2 /Z n as Coulomb branch; hence, we may write (1.3) These symmetry properties have been conjectured in two complementary studies: firstly, by computing the quantum corrections to the Coulomb branch metric in [21] and, secondly, by computing the Coulomb branch Hilbert series in [1].
Extending the above setting by an orientifold O6 plane changes the resulting 3-dimensional N = 4 worldvolume theory to an USp(2k) gauge theory with one antisymmetric and n fundamental hypermultiplets. The quiver theory is again an ADHM quiver such that the Higgs branch is the moduli space of k SO(2n)-instantons on C 2 . Again, 3d mirror symmetry predicts that the Coulomb branch is the symmetric product of k copies of the D n singularity [21,22]. In other words, Recalling that the Coulomb branch of a SU(2) ∼ = USp(2) gauge theory with n fundamentals is the D n -singularity, one may conclude (1.6) Again, this has been conjectured in [21] and [1] from different approaches.
Lastly, replacing the O6 plane by a hypothetical O6 + plane [23,24] implies that the 3-dimensional N = 4 worldvolume theory turns into a SO(2k + 1) gauge theory with one symmetric and n fundamental hypermultiplets. The quiver is given by and it has been conjectured in [1] that the Coulomb branch is again the k-th symmetric product of a D-type singularity, i.e.
Higgs branches. Following [20], consider n separated M5-branes on a C 2 /Z k singularity. The 6-dimensional N = (1, 0) worldvolume theory has (n − 1) tensor multiplets, a SU(k) n−1 gauge group and bifundamental matter determined by a linear quiver where all nodes are special unitary gauge or flavour nodes. The corresponding 3-dimensional N = 4 quiver gauge theory for n separated M5-branes is equipped with a bouquet of n nodes with 1, i.e. with all nodes being unitary gauge groups. It is important to appreciate the global discrete S n symmetry present in the problem of n identical objects. In particular, the Coulomb branch quiver has an apparent S n symmetry. Following [20], the system admits different phases, in which n i of the n separated M5-branes become coincident at positions x i . These phases are then obtained by gauging a discrete i S n i ⊂ S n global symmetry in the 6-dimensional theory. Let us summarise the main conjectures of [20]: (i) At infinite gauge coupling, the 3-dimensional quiver gauge theory for n coinciding M5-branes on a A k−1 -singularity is given by where all nodes are unitary gauge groups.
(ii) The 6-dimensional Higgs branches and 3-dimensional Coulomb branches then satisfy the following relations: (1.13) (iii) Suppose a partition {n i } of n describes that the n M5-branes coincide in a pattern of n i coinciding branes at different locations. The case of all branes separated corresponds to {1 n }, while all of them coinciding corresponds to {n}, i.e. infinite gauge coupling. Then the associated 3-dimensional quiver is conjectured to be (1.14) with the relations Here, i S n i denotes the product of permutation groups which act on the sets of n i coincident M5-branes.
Similarly, n M5-branes on a C 2 /D k singularity have been considered in [19]. The 6-dimensional N = (1, 0) worldvolume theory is comprised of (2n − 1) tensor multiplets as well as gauge groups and hypermultiplets determined by the quiver . . .
such that there are n USp(2k − 8) and (n − 1) O(2k) gauge nodes in total. The associated 3-dimensional quiver theory for n coincident M5 branes, i.e. infinite gauge coupling in the 6dimensional theory, has been conjectured to be . . .
where Λ 2 denotes the traceless rank-2 antisymmetric representation of USp(2n). The theories are related via One can argue, as shown below, that M 3d C (I D n,k ) is the S n -quotient of the Coulomb branch of . . .
USp (2) . . . n which is a quiver with a bouquet of n nodes of USp (2). Physically, F D n,k captures the phase in which all n M5-branes are separated. The discrete gauging on the Higgs branch is reflected by the relation of the Coulomb branches As shown below, one can generalise the setting to the analogue of (1.15).
Outline. From the above examples, there is an apparent action of an S n group on a Coulomb branch quiver, which appears to be a local operation on the quiver. These examples serve as guideline to prove exact statements on the Coulomb branch Hilbert series upon the action of an S n group.
The remainder is organised as follows: the generalisations of the examples discussed in the introduction are the focus of Section 2. In detail, the generalisation to an arbitrary quiver coupled to either a bouquet of U(1), USp(2), or SO(3) nodes is considered and the statements of discrete S n -quotients are proven on the level of the monopole formula. In Section 3, applications to other types of bouquets, composed of (different) USp(2), SO(3), and O(2) nodes, are considered and proven. Thereafter, Section 4 summarises and concludes. Appendix A provides some background on the cycle index and a proof of an auxiliary identity.
As a remark, a complementary perspective of discrete gauging and its manifestation as discrete quotients on Coulomb branches is presented in [25].

A and D-type
This section focuses on generic good 3-dimensional N = 4 quiver gauge theories that are coupled to bouquets of either U(1), USp(2), or SO(3) nodes. Upon S n -quotient, the bouquet is replaced by a single U(n), USp(2n), or SO(2n + 1) node supplemented by an additional hypermultiplet that transforms as in the corresponding ADHM quiver.

A-type -U(1)-bouquet
Taking (1.3) as well as (1.13) and (1.15) as motivation, one can generalise the statement to a generic quiver with one (or many) bouquet(s) attached and provide a proof on the level of the monopole formula.
Consider an arbitrary quiver, denoted by •, coupled to either a U(n) gauge node with one additional adjoint hypermultiplet or a bouquet of n U(1) nodes. I.e. define the two quiver theories (2.1) To be precise, the U(n) as well as all of the U(1) nodes couple to the same single node in • via bifundamental matter. Viewed from the U(n) or U(1) nodes, the quiver • is considered as providing background charges k = (k 1 , . . . , k s ) for some s ∈ N, i.e. the magnetic charges from the single node they couple to. To see this, consider this single node in • as flavour node with background charges k as, for example, in [4,5,16]. Thus, there are two Hilbert series to compute: (i) the monopole formula H(t, k) of • with the single node turned into a flavour node with fluxes k, and (ii) the monopole formula of T {n}, (or T {1 n }, ) where the flavour node provides fluxes k. The Hilbert series of T {n},• (or T {1 n },• ) can be obtained via gluing the Hilbert series H(t, k) with that of T {n}, (or T {1 n }, ) along the common flavour node, which turns it into a gauge node. Since the Hilbert series with background charges for • is same in both cases and is not affected by the S n -quotient, it will henceforth be ignored. Now, the above conjecture (1.13) is generalised by Proposition 1. Let the quiver gauge theories T {n},• and T {1 n },• be as defined in (2.1), then their Coulomb branches satisfy Preliminaries. In order to prove Proposition 1, one defines as Hilbert series of the 3-dimensional N = 4 U(1) gauge theory with background charges k. In detail, the conformal dimension and dressing factor read for q ∈ Z. Then (2.2) can be expressed via the following generating series: such that Proposition 1 becomes In order to compute HS n (t, z) explicitly from the symmetrisation of f (t, z), one employs the cycle index (A.1).
To compare the result, recall the ingredients for the monopole formula of an U(n) gauge node with one adjoint hypermultiplet and background charges. The conformal dimension reads wherein the contributions from the adjoint hypermultiplet cancel the vector multiplet contributions entirely. The magnetic charges appearing in the monopole formula are ordered q 1 ≥ q 2 ≥ . . . ≥ q n . The U(n) dressing factors have been defined in [1]. The shorthand notation |q i − k| ≡ s l=1 |q i − k l | summarises the contributions from the magnetic charges k l of the single node in • the U(n) couples to via bifundamental matter.
Proof. The recursive formula (A.2) suggests to prove (2.2) by induction. One explicitly verifies the claim for n = 1, 2; thereafter one proceeds to HS n (t, z) with general n, i.e.
assuming the validity for all HS k (t, z) with k < n. To begin with, one verifies the base case: (i) n = 1: trivial. Returns the U(1) case.
(ii) n = 2: The two contributions read (2.10) Combining both, one obtains which coincides with the monopole formula for the quiver • coupled to a U(2) gauge node with an adjoint hypermultiplet.
Thereafter, one proceeds with the inductive step (n − 1) → n for (2.8). The strategy of the proof is to consider the different contributions for the distinct summation regions of the magnetic charges q i in detail and show that these agree with the monopole formula of T {n},• .
(i) q 1 > q 2 > . . . > q n can only originate from one term: a 1 HS n−1 , in which one denotes the magnetic charge in a 1 by q and those of HS n−1 by q i , i = 1, . . . , n − 1. Then there are exactly n contributing cases: . . . , but these can all be relabelled to a single case. Then one finds and observes the dressing factor for a residual U(1) n gauge symmetry, which is in fact the correct stabiliser of q 1 > q 2 > . . . > q n in U(n).
The relevant contributions can only arise from a 1 HS n−1 to a l HS n−l . Then a 1 HS n−1 has two different contributions: firstly, but recall that the q i in HS n−1 are already ordered. Then there are exactly (n − l) possible ways to arrange q in between the q i . However, a simple relabelling makes them all identical and one obtains: Secondly, there is the contribution where (l − 1) q i are equal in HS n−1 and one has to align the q from a 1 with those (l − 1) equal magnetic charges. This yields precisely one case Similarly, there exists exactly one matching contribution for a j HS n−j , where the q from a j has to match the (l − j) equal q i from HS n−j . One obtains The total contribution becomes Such that one obtains the contribution: and one recognises the correct dressing factor for the residual S(U(l) × U(1) n−l ) gauge symmetry of l equal magnetic charges. (iii) Now, one can easily generalise to any partition (l 1 , . . . , l p ), i j=1 l i = n (not necessarily ordered) that describes the set-up of The total contribution becomes which is the correct contribution with a dressing factor reflecting the residual S( p j=1 U(l j )) gauge symmetry. Again, the factor n is cancelled by the 1 n pre-factor in the cycle index. Consequently, one has addressed all possible {q i }, i = 1, . . . , n, summation regions that appear in (2.8) and, most importantly, one has proven that these correspond exactly to the definition of the fully refined monopole formula for a U(n) gauge node with one adjoint hypermultiplet and background charges. This concludes the proof of Proposition 1.
Comments. The proof shows that given the Coulomb branch of an arbitrary quiver • with a U(1)-bouquet of size n, one may quotient by S n . The result is the same as the Coulomb branch of • coupled to a U(n)-node with one additional adjoint hypermultiplet. From the nature of the proof, i.e. the S n -quotient is a local operation on the Coulomb branch, there exist two immediate corollaries: (i) Similarly to (1.15), one can consider a generic partition {n i } of n which corresponds to the quotient by i S n i on T {1 n },• . Since • has been arbitrary, one can repeat the proof by subdividing the size n bouquet, focusing on the sub-bouquet of size n i , while treating the remaining U(1)-nodes as part of the background quiver. In other word, the quiver has a Coulomb branch which satisfies (ii) Furthermore, one may consider quivers where multiple bouquets are attached to different nodes. Then one can repeat the process of discrete quotients to any of the bouquets individually, as the operation is entirely local.

D-type -USp(2)-bouquet
Next, one can generalise (1.6) by considering an arbitrary quiver coupled to one (or many) bouquet(s) of USp(2) ∼ = SU(2) gauge nodes and provide a proof at the level of the monopole formula.
Again, consider an arbitrary quiver, labelled by •, coupled to either an USp(2n) gauge node with one additional anti-symmetric hypermultiplet or a USp(2)-bouquet of size n. Again, the following notation is employed: (2.26) As above, the USp(2n) as well as all of the USp(2) nodes couple to the same single node in • via bifundamental matter. From the USp(2n) or USp(2) point of view, the quiver • contributes background charges k = (k 1 , . . . , k s ) for some s ∈ N, i.e. the magnetic charges from the single node they couple to. All other contributions from • could be summarised in a function of the fugacity, which is not affected by the S n -quotient and is henceforth ignored, cf. the discussion below (2.1).
With this preliminaries at hand, the statement of Proposition 2 becomes Proof. As before, the cycle index (A.1) can be employed to realise the symmetrisation in (2.32) such that the proof proceeds by induction in n HS n (t) = 1 n n k=1 a k · HS n−k (t) , a k = f (t k ) . where the two contributions are treated as follows: Adding them up yields t ∆(q 1 )+∆(0) (2.37) , q 1 = q 2 = 0 .
Next, one proceeds as in the A-type case of Section 2.1, i.e. the inductive step (n − 1) → n. Since there is a slight complication when considering q 1 ≥ . . . ≥ q n ≥ 0, the details of the proof need to be elaborated.
(i) q 1 > . . . > q n > 0 can only originate from a 1 HS n−1 via where the n different possibilities to place q between the (n − 1) q i eliminated the pre-factor 1 n . Moreover, the dressing factor correctly reproduces the stabiliser of q 1 > . . . > q n > 0 inside USp(2n), i.e. U(1) n . (ii) q 1 > . . . > q n−l > 0 = q n−l+1 = . . . = q n for which contributions arise from a 1 HS n−1 to a l HS n−l . To start with, a 1 HS n−1 provides two contributions and arranging q between the non-vanishing q i gives a multiplicity of (n − l). The other term is which has multiplicity one. Similarly, the contribution form a j HS n−j is Summing up all contributions, one obtains and one recognises the dressing factor of U(1) n−l × USp(2l). Note in particular the use of the results of Appendix A.2, but this time for Q l (t 2 ) = l. (iii) q 1 = . . . = q l > q l+1 > . . . > q n > 0 for which contributions arise from a 1 HS n−1 to a l HS n−l .
Comments. The proof establishes that the Coulomb branch of an arbitrary quiver • with a USp(2)-bouquet of size n coincides upon quotient by S n with the Coulomb branch of the quiver • where the bouquet is replaced by a USp(2n) gauge node with an additional anti-symmetric hypermultiplet.
The nature of the proof allows to draw two immediate corollaries, as in the A-type case: (i) One may consider an arbitrary partition {n i } of n such that one quotients T {1 n },• by i S n i . (ii) In addition, one may consider quivers with more multiple bouquets, as the operation is local on the Coulomb branch. To clarify, the SO(2n + 1) as well as all of the SO(3) nodes couple to the same single node in • via bifundamental matter. From the view point of the SO(2n + 1) or SO(3) nodes, the quiver • contributes background charges k = (k 1 , . . . , k s ) for some s ∈ N, i.e. the magnetic charges from the single node they couple to. All other contributions from • could be summarised in a function of the fugacity, which is not affected by the S n -quotient and is henceforth ignored, cf. the discussion below (2.1).

D-type -SO(3)-bouquet
Proposition 3. Let T {n},• and T {1 n },• be as defined in (2.50) then their Coulomb branches satisfy Preliminaries. For the proof below, one defines the basic ingredient: which is the Coulomb branch Hilbert series. Here, the conformal dimension reads for the magnetic charge q ∈ N and background fluxes k. The dressing factors associated to SO(3) are those of USp(2), i.e. (2.54) The SO(2n + 1) gauge node with one hypermultiplet transforming in Sym 2 ([1, 0, . . . , 0]) and background charges k has conformal dimension ∆(q 1 , . . . , q n ; k) = 1 2 In the monopole formula, the magnetic charges q i are restricted to q 1 ≥ q 2 ≥ . . . ≥ q n ≥ 0. Moreover, the dressing factors of SO(2n + 1) are those of USp(2n), see [1]. The shorthand notation |q i ± k| ≡ s l=1 |q i ± k l | summarises the contributions from the magnetic charges k l of the single node in • the SO(2n + 1) couples to via bifundamental matter.
Proof. To prove Proposition 3, one needs to verify the Hilbert series relations (2.6) or (2.32) for the case of a SO(3)-bouquet. As before, the proof relies on the recursive formula (A.2) of the cycle index and proceeds by induction as in (2.33). As a first step, one considers the base case.
(ii) n = 2: The proposal reads where the two contributions are treated as follows: (2.58) Adding them up, yields t ∆(q 1 )+∆(0) (2.59) Comparing this to the monopole formula of SO(5) with one Sym 2 [1, 0] hypermultiplet and background charges, the conformal dimension follows from (2.55) and the dressing factors read Consequently, Proposition 3 is true for n = 2.
The argument proceeds as in the D-type case of Section 2.2, one proves the inductive step (n−1) → n. As the dressing factors as well as the lattice of magnetic charges are identical to the USp(2n) case, it is unnecessary to spell out the details of the proof. The only point to appreciate is that the conformal dimension is the sum of the individual SO(3) conformal dimensions.
Comments. Again, the same corollaries are in order: (i) one can generalise to arbitrary partitions, and (ii) one can consider multiple bouquets. Moreover, note that the Coulomb branch of is the D n+3 -singularity. Hence, Proposition 3 provides the missing analogue of (1.3), (1.6) for (1.7), i.e. (2.62)

Other applications
After establishing the generalisations underlying the A and D-type singularities, one may wonder if there are other types of bouquets that can be considered. One could ask what are sufficient conditions such that a G n gauge node, which may be supplemented by additional matter, can be obtained from an S n -quotient of a certain G 1 -bouquet. To be precise, the G n node as well as all nodes of the G 1 -bouquet are coupled to the same single node in a given quiver gauge theory via bifundamental matter. From the aforementioned cases one formulates three conditions: While the first statement is concise, the second and third are less precise. However, by recalling [7,8] the interpretation of the dressing factors P (t, q i ) as Hilbert series of C[g] G ∼ = C[t] W G , with t a Cartan sub-algebra of g and W G the Weyl group, one can identify all classical groups that allow for a S n factor in W G .
Suppose W Gn = S n (Γ) n and denote the chosen Cartan sub-algebra as t ∼ = V n , for some 1-dimensional vector space V , then and a similar argument is valid for the summation ranges in the monopole formula of G n and G 1 . By inspecting classical Weyl groups in Table 1 one concludes that choosing G n to be either U(n), SO(2n + 1), USp(2n), or O(2n) together with one adjoint hypermultiplet leads to possible S n -quotients on the Coulomb branch. All the different cases are elaborated on in the subsequent sections. The only exception is U(n) as it agrees with Proposition 1.

SO(3)-bouquet
Specifying the above to a SO(2n + 1) gauge node with one adjoint hypermultiplet coupled to an arbitrary quiver •, one finds: then their Coulomb branches satisfy To prove Corollary 1 one follows all the steps of the proof of Proposition 3. The only point the take care of is that the conformal dimensions add up, which is not difficult to see.

USp(2)-bouquet
Next, consider a USp(2n) gauge node with one adjoint hypermultiplet coupled to an arbitrary quiver • via bifundamental hypermultiplets. The proof of Corollary 2 follows from the proof of Proposition 2, by verifying that the conformal dimensions add up appropriately.

O(2)-bouquet
Finally, let an arbitrary quiver • be coupled either to an O(2n) gauge node with one additional anti-symmetric hypermultiplet or to a O(2)-bouquet of size n. Since this is the first time O(2n) gauge nodes appear, some remarks on the proof are in order. Analogous to Section 2, define the basic ingredient: which is the Coulomb branch Hilbert series. Here, the conformal dimension reads for the magnetic charge q ∈ N and background fluxes k. Following [3], the dressing factors associated to O(2) are those of SO (3), i.e. (3.10) Note that there is no extra topological fugacity for O (2).
As the dressing factors and GNO lattice for O(2n) originate from SO(2n + 1), which are the same as for USp(2n), the proof of Corollary 3 is consequence of the proofs of Propositions 2 and 3.

Remarks and example
With Corollaries 1-3 at ones disposal, one can immediately generalise to the following: Corollary 4. Let G n be either U(n), SO(2n + 1), USp(2n), or O(2n) and {n i } be a partition of n. The Coulomb branch of the quiver gauge theory Likewise, one could consider quiver gauge theories coupled to various bouquets at different nodes.

Discussion and conclusions
In this note we have shown that discrete S n -quotients on Coulomb branches of quivers with various bouquets are entirely local operations. By this we mean that the geometric S n -quotient on M C is realised on the quiver (and the monopole formula) as an operation on the bouquet alone; the remainder of the quiver is untouched by the S n action.
We provided the A and D-type Propositions 1-3 in Section 2 and proved them via the cycle index for S n . Subsequently, we explore various other possibilities in Section 3 and derived Corollaries 1-4. In comparison, the gauge nodes in Section 2 are supplemented by loops corresponding to matter as in the ADHM quivers, whereas the gauge nodes in Section 3 are equipped with one additional adjoint hypermultiplet. The A-type case of U(n) nodes is the only scenario for which both notions coincide.
The results are important for a number of reasons: firstly, it allows to deduce if certain 3dimensional N = 4 Coulomb branches are S n orbifolds of one another. For instance, the sub-regular nilpotent orbit of G 2 is an S 3 quotient of the minimal nilpotent orbit of SO(8), cf. [26]. Due to the discrete quotient proposition, the statement follows immediately by inspecting the 3-dimensional N = 4 quivers Secondly, the propositions allow to systematically study the different phases of 6-dimensional Higgs branches as put forward by [20]. For instance, Proposition 2 allows to conclude a similar statement to (1.15) on the different phases of the Higgs branches of multiple M5-branes on a C 2 /D k singularity [19]. The conjecture becomes that for a partition {n i } of n, such that the M5-branes coincide in a pattern of n i , the 3-dimensional quiver reads . . .
. . . and its Coulomb branch satisfies Thirdly, the discrete quotient procedure establishes another operation on quiver gauge theories solely through their associated Hilbert series. This highlights the diverse applicability of the Hilbert series and adds to the catalogue of quiver operations such as the ideas of quiver subtraction [27] and Kraft-Procesi small instanton transition [19].
In view of other approaches to 3-dimensional N = 4 Coulomb branches, like the abelianisation method [28,29] or the attempt to define the Coulomb branch mathematically [30][31][32], it would be interesting to understand whether these can reproduce the discrete quotients. The cycle index of a permutation group Γ of degree n is defined as average of the cycle index monomials of all permutations g ∈ Γ. Every g ∈ Γ can be decomposed into disjoint cycles c 1 c 2 c 3 · · · . Let j k (g) be the number of cycles in g of length k, then one recalls the following definitions from q-theory: For the q-binomial coefficient exists a q-version of the Pascal identities; for instance k j q = q j k − 1 j q + k − 1 j − 1 q , (A.5d) for j = 1, 2, . . . , k − 1. Then, one can rewrite Having expressed (A.4) as in (A.6) has the benefit that one can follow an argument of [33]. The proof proceeds by induction over l employing (A.5d). Firstly, the base case is verified easily Secondly, the inductive step is shown via where the induction hypothesis has been used in the last step.