Determining the basis for strings with cyclic and permutation symmetry

I'd like to determine a basis set of strings for strings of length m composed of n letters that are compatible with cyclic symmetry of the string and permutation symmetry of the letters.

It's relatively straightforward to do this by hand in individual cases. For example $$m=5, n=3$$ with the letters labelled $$1,2,3$$, then the minimal number of strings I believe are

$${00000}, {00001},{00011},{00101},{00012},{00112},{00102}, {00121}, {01012}$$

where I can generated all the other strings by cyclically permuting the indices of the strings above, or by permuting the letters {1,2,3} amongst themselves.

Is there are a way of determining the number of basis strings for each $$m,n$$? Or at least the number of basis strings constructed in this way.

• Does the first string actually represent a pair of two strings of length five namely $00000$ and $00001$? – Marko Riedel Feb 22 at 14:19
• Indeed, thank you for pointing out the typo. – Aran Feb 22 at 15:07

The problem of counting the basis strings is an instance of Power Group Enumeration as defined by Harary and Palmer in the text Graphical Enumeration. We have the cyclic group acting on the slots with cycle index

$$Z(C_m) = \frac{1}{m} \sum_{d|m} \varphi(d) a_d^{m/d}.$$

The group acting on the colors is the symmetric group $$S_n$$ with recurrence by Lovasz for the cycle index $$Z(S_n):$$

$$Z(S_n) = \frac{1}{n} \sum_{l=1}^n a_l Z(S_{n-l}) \quad\text{where}\quad Z(S_0) = 1.$$

With these two cycle indices it suffices to run the PGE algorithm as documented e.g. at the following MSE link.

This yields e.g. for four colors and $$m$$ slots the sequence

$$1, 2, 3, 7, 11, 39, 103, 367, 1235, 4439, \\ 15935, 58509, 215251, 799697, \ldots$$

which points us to OEIS A056292 where we find confirmation of these data. Similarly for six colors and $$m$$ slots we obtain

$$1, 2, 3, 7, 12, 43, 126, 539, 2304, 11023, \\ 54682, 284071, 1509852, 8195029, \ldots$$

which points to OEIS A056294, again for confirmation.

The algorithm is shown below.

with(numtheory);

pet_cycleind_cyclic :=
proc(n)
option remember;

1/n*add(phi(d)*a[d]^(n/d), d in divisors(n));
end;

pet_cycleind_symm :=
proc(n)
option remember;

if n=0 then return 1; fi;

end;

neckl_pg :=
proc(m, n)
option remember;
local idx_slots, idx_colors, res, term_a, term_b,
v_a, v_b, inst_a, inst_b, len_a, len_b, p, q;

if m = 1 or n = 1 then return 1 fi;

idx_slots := pet_cycleind_cyclic(m);
idx_colors := pet_cycleind_symm(n);

res := 0;

for term_a in idx_slots do
for term_b in idx_colors do
p := 1;

for v_a in indets(term_a) do
len_a := op(1, v_a);
inst_a := degree(term_a, v_a);

q := 0;

for v_b in indets(term_b) do
len_b := op(1, v_b);
inst_b := degree(term_b, v_b);

if len_a mod len_b = 0 then
q := q + len_b*inst_b;
fi;
od;

p := p*q^inst_a;
od;

res := res +
lcoeff(term_a)*lcoeff(term_b)*p;
od;
od;

res;
end;

The problem described here seems to be identical to the problem of finding the number of equivalence classes of base-m necklaces of length n, where necklaces are considered equivalent under both rotations as well as permutations of the symbols.

The solution is given in N. J. Fine, Classes of periodic sequences, Illinois J. Math., 2 (1958), 285-302.