A sure-fire way of answering “How many of these objects are there?” is to generate them all and simply keeping a counter. However, if it so happens that the true count is about 1024, this way of doing it becomes a “sure-fire way of getting nowhere”. This is exactly what happens with so called -regular families of permutations, which are a generalisation of the notion of transitivity in groups. They also measure how Cayley-like a vertex-transitive graph is. Therefore, another approach is required for this (and many other) cases. In this article, we shall describe a general-purpose algorithm that utilises complex cyclotomic numbers to give upper bounds to the counts of these families. Additionally, we present a probabilistic algorithm to give heuristic estimates of the true counts of these families.
just regular family) was introduced much earlier in [2] As defined in [ 1], for any set , an “-regular family” already. In this paper, the author was looking at the well of permutations is a set ℱ ⊆ S , such that for every known issue of graph theory, that there are some vertex , ∈ , there are exactly permutations ∈ ℱ , transitive graphs, which are not Cayley graphs, most such that () = . Sometimes we will take = famously the Petersen graph. Her method involved the {1, 2, 3, . . . , }, in which case we will use the standard construction of “quasi-Cayley graphs”, which are tied to shorthand [] := {1, 2, 3, . . . , }. The -regular family the existence of 1-regular families of automorphisms as as an object is closely related to transitive groups, when follows: “a graph Γ is quasi-Cayley if and only if (Γ) their stabiliser is of size . Indeed, such a group does contains a 1-regular family as a subset”. This is similar follow the property of -regular families and is therefore to what Sabidussi did in [3] with Cayley graphs and their a special case of them. On the other hand, the -regular automorphism groups: “a graph Γ is Cayley if and only families are combinatorial approximations of these tran- if (Γ) contains a subgroup, which acts regularly on sitive groups, as they themselves need not to follow the (Γ) ”. group axioms, only the “ regularity”. With definitions laid out as we have, we can rephrase
The authors of [1] investigated “how close to a Cayley the result of Sabidussi as “a graph Γ is Cayley if and only graph a given vertex-transitive graph is”, which is a ques- if (Γ) contains a subgroup, which is also a 1-regular tion that has been asked many times in prior research in family (with respect to its action on (Γ) )”. diferent variations. As such, there are multiple measures One year after the paper [1], in his bachelor thesis of this “closeness” notion. One such notion is to find [4], the author investigated -regular families from a the smallest transitive subgroup of (Γ) . A diferent computational perspective, specifically, by generating notion relies precisely on the aforementioned -regular these families with the help of a computer. families, and also gets investigated in [1]. There, one In Section 2, we will summarise Kerák’s main result seeks small -regular families as subsets of the (Γ) , from [4], which is relevant to our work, introduce the the group of automorphisms of the given graph Γ . The main problem we will be solving, and describe how genpaper [1] is the first instance we could find which men- erating functions and Fourier transformations can help tions the use of -regular families. us get upper bounds on the number of -regular families. Continuing that, in Section 3, we talk about some of the ITAT’24: Computational Aspects of Large-Scale Problems in Discrete implementation details of these methods to get said upMathematics, September 20–24, 2024, Drienica, Slovakia per bounds. In Section 4, we briefly describe the method * Corresponding author. [4] used and expand on it with a randomised algorithm, †$Thpeasveoal.uktohlloarrs@cofmntprhib.uunteibdae.sqkua(Pll.yK.ollár); which can be used as an estimation heuristic to approxmartin.macaj@fmph.uniba.sk (M. Mačaj) imate the number of -regular families on elements.
© 2024 Copyright for this paper by its authors. Use permitted under Creative Commons License Finally, in Section 5, we summarise our findings. CPWrEooUrckReshdoinpgs IhStpN:/c1e6u1r3-w-0s.o7r3g ACttEribUutRion W4.0oInrtekrnsahtioonpal (PCCroBYce4.0e).dings (CEUR-WS.org)
Our work picks up where [4] left of, where the author was generating the -regular families with the help of a computer. One of his outputs was the Table 1 containing information on how many distinct -regular families on elements are there for some small , . Within this table, “NG” means that for those parameters, the families were only partially generated or not at all.
For the rest of the paper, we fix ∈ N and also let = {1, 2, . . . , } be a fixed set of positive integers with 1 ≤ 2 ≤ . . . ≤ . Borrowing some notation from [4], instead of writing down permutations of S with the cycle notation as is usual in algebra, e.g. (1, 2, 4, 6) or (2, 7, 3)(5, 11), we will write down all the 2.2. Generating functions numbers in one sequence into a list. In this list, the num- To bound the values of the numbers of -regular families, ber is in the th position in the list precisely when is we will use generating functions and their properties. To mapped to the number by this permutation. Put another be able to use generating functions, we require a weight way, to each permutation ∈ S , we associate the list of function on the subsets of S , namely : (S ) → N. numbers () = [(1), (2), (3), . . . , ()]. Instead of producing a weight for every subset, we assign For = [6] and = (1, 2, 4, 6), this list will thus each permutation a weight with the function : S → be [2, 4, 3, 6, 5, 1] and for = {2, 3, 5, 7, 11, 13} and N and the total weight of a subset of S shall simply be = (2, 7, 3)(5, 11), the list will be [7, 2, 11, 3, 5, 13]. the sum of individual weights.
When we talk about the elements of S from now on, we will mostly talk about the set of lists induced by the previous construction.
Given any ≤ ( − 1)!, we know that ∀ℱ ∈ : |ℱ | = . This facilitates the fact that the only 0-regular set is the empty set and so ⃒⃒ 0 ⃒⃒ = 1 for all regardless of its size. Since we have chosen | | = , we have that |S | = !, which means that for all > ( − 1)!, the set is empty, so ∀ > ( − 1)! : ⃒⃒ ⃒⃒ = 0.
Our main efort for the remainder of this paper is to get upper bounds for the total number of -regular families for = 5. In Section 4 we will also describe a randomised algorithm that provides a heuristic estimate for the number of -regular families on elements are there in that specific case. We are also going to reference the values within the row for = 4 in the Table 1 to check the correctness of our methods.
The weight function of single permutations : S → N will depend on a chosen list w = [1, . . . , ] of weights (and also the chosen set ) and is defined by Observation 1. Within this list association, an -regular () = ∑︀=1 · ()[]. In a sense, we really should family of S is such a collection of the lists ℱ ⊆ S , so call this weight function w, but we will omit it in this that when the lists in ℱ are aligned below one another, section. For example, if = [4] and w = [1, 2, 5, 15], each column contains each ∈ precisely times. for the identity permutation we have () = 1 · 1 + 2 · 2 + 5 · 3 + 15 · 4 = 80. For = (1, 2), we have
Later, we will want to talk about individual en- () = [2, 1, 3, 4] and () = 79, and for = (1, 4, 3), tries of these lists and for that reason, we shall ac- we have () = [4, 2, 1, 3] and () = 62. cess them via indices written with square brackets, i.e. As mentioned before, the weight function for sets ()[] = (). For us, the first entry has index 1, e.g. of permutations : (S ) → N, will be the sum [7, 2, 11, 3, 5, 13][1] = 7, [2, 4, 3, 6, 5, 1][4] = 6. of the individual permutation weights, i.e., () = ∑︀∈ (). For any ⊆ N, we can ask for the set Definition 1. For a given non-negative integer , let = { ⊆ S | ( ) ∈ }. Ideally we want to con ⊆ (S ) be the set of -regular families of S . struct the function (that is, choose the list w as well Also let = ⋃︀≥ 0 . as the set in a clever way) so that there exists an with ℱ ∈ ⇔ ( ) ∈ . However, ensuring both we know this generating function is also equal to implications hold while also ensuring that he entries of () = ∏︀∈S (1 + ()), creating the following es and w are suficiently small is highly non-trivial (for sential equation: , w big, it is not so dificult to come up with examples), so we will relax the requirement and demand that only ∑︁ · = () = ∏︁ (1 + ()) (2) ℱ ∈ ⇒ ( ) ∈ holds. This is why our algo- ∈N ∈S rithms will, in general, produce upper bounds for those Utilising a standard trick of Fourier transformations, counts, and occasionally produce the exact values. let a primitive th root of unity. Then to get the boundonCeocanntinseueinthgatthfeorsiamnpyle-erxegamulparlefaamb oilvyeℱw,iethachw=eig[4h]t, ing sum ∑︀∈N , we can average the values of () at is multiplied times by each ∈ , so (ℱ ) = all the ℎ roots of unity, that is: (1 + 2 + 5 + 15) · · (1 + 2 + 3 + 4) = 230. In general we have the following lemma. ∑︁ = ∑︁ ( ) (3) Lemma 1. Fix w = [1, . . . , ], and let ℱ ⊆ S be an -regular family. Then (ℱ ) = · ∑︀∈ · ∑∑︀︀∈w . Therefore, with = ∑︀∈ ·
∈w and = { · | ∈ N}, the condition ℱ ∈ ⇒ (ℱ ) ∈ is satisfied for all , in particular ℱ ∈ ⇒ (ℱ ) ∈ .
Proof. Let ℱ be an -regular family. By definition, (ℱ ) = ∑︁ () = ∑︁ ∑︁ · ()[], ∈ℱ
∈ℱ ∈w where the ()[] are elements of . This is a finite sum and hence we can swap the order of summations: (ℱ ) = . . . = ∑︁ · ∑︁ ()[]. ∈w ∈ℱ ∈N 1 ·
=1 Combining Equation 1 and Equation 3, we get The evaluation of this average requires the evaluations of on the right hand side. Those values depend on the weight function , which in turn depends on the particular choice of and w. This constitutes the first algorithm, which accepts inputs , w and outputs the upper bound from Equation 4, see Section 3.1 for implementation.
Because ℱ is -regular, together over all and all , In the previous section, we transformed a permutation ()[] equals each ∈ precisely times: into its weight () = ∑︀∈w · ()[] and we transformed a permutation set ⊆ S into ( ) = (ℱ ) = . . . = ∑︁ · ∑︁ · . ∑︀∈ (). Enconding each set of permutations into ∈w ∈ a single number caused us to lose a lot of information about said set of permutations. Instead, let us assign To finish the proof of the claim, pull out of both sums, the “weight” of the permutation to be the permutation since it is independent of the sums, and substitute in the itself in the list notation: () = (). Treating these defined : lists of integers as vectors for the purposes of addition (ℱ ) = . . . = · ∑∈︁w · ∑ ∈︁ = · . a∑n︀d∈scala(rm)ufoltripalnicyatio⊆n, wSe m.Tahyisnwowayd,ewfineeretain(m)o=re information about .
This means that divides (ℱ ), so (ℱ ) ∈ .
Lemma 2. Let ℱ ⊆ S be an -regular family. Then
Said another way that is more usual in the context of (ℱ ) = · ∑︀∈ · [1, 1, . . . , 1], where the vector has generating functions, if = |{ ⊆ S : ( ) = }|, | | = ones. With d = ∑︀∈ · [1, 1, . . . , 1] and then the following bound is a direct consequence of the = { · d| ∈ N}, the condition ℱ ∈ ⇒ (ℱ ) ∈ previous lemma: is satisfied for all , in particular ℱ ∈ ⇒ (ℱ ) ∈ .
∈N dimensional case from the previous subsection. Each We take () to be the generating function for this se- position within the lists gets each number ∈ prequence of numbers, that is, () = ∑︀∈N · . By cisely times in the expression for (ℱ ), when ℱ is an standard theory of generating functions (see e.g. [5]), -regular family. ∑︁ I · 11 · 22 · . . . · = I∈N = (1, 2, . . . , ) =
= ∏︁ (1 + ∏︁ ()[]) ∈S =1
(6) Let = ∑︀∈ , which means that d can be written as [, , . . . , ]. Here too, we want to use the trick of Fourier transformations. In this case we can start with the observation that: Here, the second sum is over all vector-indices, whose each coordinate is a multiple of . Clearly, the sum on the right-hand-side includes all coeficients with indices · d, therefore we have the following upper bound as a consequence:
Testing this algorithm on various inputs, we recognised that to get a meaningful decrease of the computed upper bound, we needed to substantially increase the ∑︁ · d ≤ 1 · ∑︁ ( 1 , . . . , ) (8) value of . Because ( ) = ∏︀∈S (1 + · ()), the ∈N I∈Z computation of each ( ) is done in linear time with respect to the order of (which is ), and therefore the
The algorithm implementing this approach and its runtime complexity is (! · 2). Coupled with the fact optimisations is described in Subsection 3.3. that we had to substantially increase to get a significant improvement, we stopped using this exact approach, 3. Algorithmic Upper Bounds even if the further algorithms share the same core idea. Still, we include the description of the algorithm here 3.1. The basic algorithm to illustrate this core idea. Later, in the results, we will highlight where the particular approaches had their comLet us return to the ideas from Section 2.2 and create an putational limits. algorithm based on them. For the Subsection 3.1, fix w = [1, . . . , ] and also . Furthermore, as before, let 3.2. Long lists of coeficients = ∑︀∈ · ∑︀∈w . At the end of Subsection 2.2, we ended up with the bounding inequality described To make matters for the previous algorithm worse, even in 4. Our algorithm will thus, for every , evaluate the if we could reduce the time complexity of the algorithm polynomial at that value and average these results. We down, we still have a memory overflow waiting to hapremark that by fixing , w (which will be the inputs of pen. In fact, one of our intermediate algorithm iterations the algorithm), we are also fixing the value as well as had time complexity just (! · ), because instead of evaluating diferent values ( ), this algorithm ex- in Section 3.1, the evaluations themselves are linear in panded the product form of symbolically. This is be- in terms of time, so the total time complexity is roughly cause what the computation actually needs is the sum of (! · +1). For just = 5 and = 127, that causes certain polynomial coeficients of - specifically those the algorithm to run for about 15 hours, while giving a whose index is ≡ 0 (mod ). To perform this expansion weak bound. of (), whose product form we established in Equation Since the evaluations of themselves cannot be easily (2), we represent each intermediate step as a long list sped up, let us try to decrease the number of evaluations. of coeficients, in which each position is the sum of all We will choose some properties of the set that will coeficients, whose indices have the same remainder lead to optimisation of our algorithm’s speed. With the (mod ). At the end, the result we seek is then simply claims that follow, we will prove that we can reduce the the 0th coeficient of this list. complexity down to only (− 1). Also, to simplify
While this provided a significant time-improvement, the further expressions, we will write [1, . . . , ] ingiven that we only “evaluated” the polynomial once, the stead of ( 1 , . . . , ). Because of the equality in (6), memory requirement was roughly (), i.e., an approxi- it easily follows that: mately linear amount of memory is required for this computation. On our laptop, even with tricks of the “meet in Lemma 3. For any [1, . . . , ] ∈ Z and any ∈ S the middle” character, choosing ≈ 1.2· 108 was roughly we have the limit of its memory capabilities. A stronger machine [1, . . . , ] = [ (1), . . . , ()]. could be employed for the task, but that too does not solve the problem. This is because the memory problem For fixed , the next lemma will gain us a much more is, in a sense, unavoidable. Once we start to dabble with significant speed-up: cyclotomic numbers that we need to represent exactly, and we need to have (), where is the Euler totient Lemma 4. Adding 1 to all the exponents of the roots of function, coeficients to represent those. Choosing to unity does not change the value of , i.e.: have many distinct prime factors seemed to yield worse [1, . . . , ] = [1 + 1, . . . , + 1]. upper bounds than when had only a few of those and then () is still (), so we cannot escape the issue, at least not in this way.
Proof. Focus on the specific product 11 · 22 · . . . · within . Evaluating it at the powers of from the lefthand-side, we of course get another power of . The value of that power is
ory requirements, some with bad runtimes in relation to the results we were getting from them. Our third and most recently used method for this problem seeks to improve this via generating polynomials of many variables, as was discussed and introduced in Section 2.3.
In what follows, is still the set {1, 2, . . . , } and let = ∑︀∈ . To simplify the optimisation arguments that follow in the later part of the section, assume that is a prime number bigger than , hence and are coprime. Finally, let be a primitive th root of unity, just as in the Section 2.3. In that section, we arrived at the inequality in (8), which bounded the total number of all -regular families on elements by the expression:
This means that all our algorithm needs to do is to evaluate the multinomial on all possible vectors of roots of unity and average the result. However, this is outrageously slow. Notice that for that to happen, we require evaluations of . Moreover, as we discussed ∑︁ · .
=1
hand-side gives us (again omitting the itself and writing down just the power): ∑︁ ( + 1) · = ∑︁ · + .
=1 =1
and both of these expressions represent powers of . Since is a primitive th root of unity, the + makes no diference to the product.
As such, this product within has the same value on the inputs we started with and, analogously, so will every other product in . This proves our claim.
We can iteratively repeat this argument to get the following equality as an immediate corollary [1, . . . , ] = [1 + , . . . , + ] (9) that holds for all 1 ≤ ≤ , with = being the trivial case, as exponents of operate (mod ). tion a bit about how GAP internally stores cyclotomic numbers. We touched on this earlier in Section 3.2, and I = · !.
(in the case that is prime and greater than ) for I = optimisation. Before we get to that optimisation, we men- [0, . . . , 0] we have I = 1 and for Γ I = {Γ} we have
we will be more specific here. For prime the cyclo- canonical representatives of orbits of Γ . Then one needs 1 dimensional linear space to derive eficient ways of determining whether given tomic field
Q( ) is an − over Q. Instead of “the standard basis” 1, , . . . , − 2,
2, . . . , − 1, that is, each element of Q( ) is expressed as 1 + 2 2, . . . + − 1 − 1 and it is internally represented as vector [1, 2 . . . , − 1]. In
represented by the vector [− , − , . . . , − ].
Galois group of Q( ) which consists of automorphisms , which map 1 ↦→
1, ↦→ and act on the roots of unity as a permutation for all 1 ≤ that 1 is the identity map. With all the notation and knowledge of this subsection, we are ready to state the < . Note next optimisation claim: ∑︀=−11 . Then Lemma 5. Let [1, . . . , ] ∈ Z and let [1, . . . , ] = − 1 ∑︁ =1 − 1 =1 ∑︁ [1, . . . , ] = ([1, . . . , ]) = − − 1 ∑︁ =1 (10) vector I is the representative of its orbit as well as how to compute I. One possible way to pick the canonical representative is to choose those I’s, which are lexicographically smallest as vectors. For this choice of representatives, we have the following restrictions: • A canonical representative has non-decreasing elements, because of Lemma 3. • Thanks to Lemma 4, the representative has sum of its coeficients ≡
0 (mod ). • As a consequence of Theorem 1, the first non-zero coordinate of a canonical representative is 1.
their orbits and the algorithm needs to be able to determine which is “the representative”. For example, if = [1, 2, 3, 4, 7] and = 17 both [0, 1, 2, 3, 11] and [0, 1, 9, 10, 14] belong to the same orbit of Γ .
4.
algorithm to generate all -regular families on elements. In what follows, assume = 5 is fixed and, since the specific choice of does not change the number of regular families, let = [5] = {1, 2, 3, 4, 5}. On top of the set contains precisely those permutations ∈ S , such that (1) = . We remark that this makes all have equal size: |1| = . . . = |5| = (5 − 1)! = 24.
-sized subsets ⊆ any -regular family ℱ ⊆ contain each precisely times too.
, such that all the other positions
Proof. Let us note that for any [1, . . . , ] ∈
Z we have (( 1 )1 · . . .· ( ) ) = ( 1 )1 · . . .· ( ) . Let us return to the work of Kerák [4] and his recursive . Because ∑︀− 1 = − 1, rear- that, partition the set S into five sets 1, . . . , 5, so that =1 ∑︀− 1 (︁ ∑︀=−11
︁) =1 {, . . . , − 1}. Therefore ∑︀−=11 ranging the last double-sum yields −
we have ([1, . . . , ]) = [1, . . . , ], which is the first claimed equation.
([1, . . . , ]) = ∑︀=−11 .
of S, Z, and Z* on the set Z modify evaluation of (X). Combining these lemmas we obtain information about the action of group Γ = Z and its behaviour with the evaluations of (X). S ×
Z ×
Z* on the set Theorem 1. Let I [1, . . . , ] = 1 + . . . + − 1 = [1, . . . , ] ∈ − 1. Then there exists a number I which depends only on (the conjugacy class of) the stabilizer Γ I such that ∑︁ [1, . . . , ] = I(− 1 − . . . − − 1) J∈IΓ
complicated to present here. Let us just point out that Z and let sively trying out all subsets ⊆ of size , exiting early if any position overflows , i.e., if any position has ⊆ more than of any number ∈ {1, 2, 3, 4, 5}. Once a has been found to not overflow any position with any number, his algorithm proceeds recursing in a (11) depth-first manner, until all possible combinations of subsets of the ’s have been tested. After all these families have been generated, counting them up is easy.
Sum of all counts exactly Scientific notation
Input:
Output: Upper bound for the size ⃒⃒ ⃒⃒ ℱ ⊆ S , its complement = S ∖ ℱ is ( − 1)! − - are disivible by 5 constitute a trivial (and very excessive!) regular. This means, that after 12, whatever the true bound for the total number of -regular families: counts were in the first half will also be in the second half, reflected about the column = 12. Hence, the total ⃒⃒ ⃒⃒⃒ ≤ ︃( 120)︃+(︃ 120)︃+. . .+ number of -regular families that exist on 5 elements is ⃒ 0 5 then calculated as double of all the numbers, except for the = 12 case, that should be counted only once. ︃( 120)︃
120 ≈ 2.658· 1035.
produce before running out of memory was 10757440577219567348022770930 ≈ 1.076 · 1028. 5. Conclusion After that, we switched to the algorithm using the multivariate generating function and we include a In this paper, we described methods of both getting up- few of the output results in the table too. The lowest per bounds for the number of -regular families on upper bound we obtained at the time of writing is elements as well as a method to get a heuristic estima- 4263880475370843510800356 ≈ 4.264· 1024. Coupled tion of the true counts. We checked our work for = 4 with the estimation heuristic makes us fairly confident elements, where to get a fast enough version of the al- the upper bounds are approaching the true count. gorithm 2, we did not need to use Theorem 1, so did Let us also contrast our method to the naive approach not need to be prime, just coprime to . The bounds here of “test every subset of S5 and check -regularity”. We quickly converge to the true count of 1200 (see Table 3) know there are 2120 subsets of S5 and let us say we and the estimation heuristic also gets really close to this could check 230 of them in a second (this is approaching number (see the discussion in Section 4). roughly the clock speed of a modern computer and is
For the case = 5, one can immediately see that an probably a vast exaggeration of the machine’s capabili-regular family has size divisible by 5. Therefore, the ties) - this approach would thus take 290 seconds, roughly number of subsets of a set with 120 elements, whose sizes 3 · 109 ages of the universe.
On the other hand, for reasons not mentioned in this 0019 grant. paper, choosing = {1, 25, 625, 15625, 390625} is The second author acknowledges support from the suficient to get the precise value of ⃒⃒ ⃒⃒ . Here, = APVV Research Grant APVV-19-0308 and from the VEGA 406901, and the algorithm evaluates roughly (in fact Research Grants 1/0423/20, 1/0727/22 and 1/0437/23. less than) 3 points of the multinomial , which means roughly 6.737 · 1016 evaluations. Pesimistically, let us say that one evaluation takes a minute, which would References isstil9l.3“oanglyestaokfet”haebuonuitve1r2s8e,. 1A77ll, 2th3a9t, i7s5t1oyseaayrst,hwathtihche [1] tRo. mJaojcrpayh,isGm. sA, .EJuornoepse,an-rJeoguurlnaarlfoafmCiolimesboinfagtroarpichsa7u9competition is not even close.
However, the = 6 case is tricky. For one, because the (2019) 97–110. doi:https://doi.org/10.1016/ upper bounding algorithm has time complexity (− 1), j.ejc.2018.12.002. raising increases the runtime drastically. To get a mean- [2] G. Gauyacq, On quasi-cayley graphs, Discrete Apingful bound, one will probably need to substantially im- plied Mathematics 77 (1997) 43–58. doi:https:// itbpnoirgoot,hvweee.iglptl.hrn(︀oiso1b2taa0wlb)︀gioloirrskittichtomboawfcukertltrlha,cbekre.icnTaguhsteeheetshmteismbelravatenioscnhgeiantlggwofaarycitthtoomros [3] fsdGüeo.mriSMaa.bnoaitrtdihcgues/scms1hia0,ot.Vliak1err0.6ot18erx6g(-/1/tC9Sr6oa04nr1p)s6iu4t62sivI-6De–2:4g113r82a80Xp.3(hU59s9R,78L)M1:00ho.0tn0tap9tss8:h/-/eaXfpt.ei. 60 ≈ 9.661 · 1034. [4] F. Kerák, -regular sets of permutations, Bachelor’s thesis, Comenius University, Bratislava, 2020.
Acknowledgments [5] H. Wilf, generatingfunctionology: Third Edition, CRC Press, 2005. URL: https://books.google.sk/ The first author acknowledges Funding by the EU books?id=bVFuBwAAQBAJ.
NextGenerationEU through the Recovery and Resilience [6] The GAP Group, GAP – Groups, AlgoPlan for Slovakia under the project No. 09I03-03-V02- rithms, and Programming, Version 4.12.2, 00036. The first author also acknowledges support from https://www.gap-system.org, 2022. the Grant of Comenius University No. UK/1114/2024. [7] P. Kollar, R-RegularFamilies, https://github.com/ Thirdly, the first author acknowledges support from the OldAnchovyTopping/R-RegularFamilies, 2024. Acgrants VEGA Research Grant 1/0437/23 and SK-AT-23- cessed: 2024-06-30.