cp's OEIS Frontend

A376808 Number of non-isomorphic colorings of a toroidal n X n grid using any number of swappable colors.

%I A376808 #22 Oct 12 2024 15:37:38
%S A376808 1,9,2387,655089857,185543613289205809,106103186941524316132396201360,
%T A376808 218900758256599151027392153440612298654753249,
%U A376808 2689595989958732045849530682270318547733917269644639109073775285
%N A376808 Number of non-isomorphic colorings of a toroidal n X n grid using any number of swappable colors.
%C A376808 Two colorings are equivalent if there is a permutation of the colors that takes one to the other in addition to translational symmetries on the torus (Power Group Enumeration). The maximum number of colors is n * n.
%D A376808 F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.
%H A376808 Andrew Howroyd, <a href="/A376808/b376808.txt">Table of n, a(n) for n = 1..24</a>
%H A376808 Marko Riedel et al., <a href="https://math.stackexchange.com/questions/2506511/">Burnside lemma and translational symmetries of the torus</a>.
%H A376808 Marko Riedel, <a href="/A376808/a376808.maple.txt">Maple code for sequence by PGE.</a>
%F A376808 a(n) = Sum_{Q=1..n^2} (1/(n^2*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|n} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(n/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma). The notation j_l(sigma) is from the Harary text and gives the number of cycles of length l in the permutation sigma. [[.]] is an Iverson bracket.
%e A376808 For the 2x2 we find
%e A376808   +-+-+   +-+-+   +-+-+   +-+-+   +-+-+
%e A376808   |X|X|   |X|X|   |X|X|   |X| |   |X| |
%e A376808   +-+-+   +-+-+   +-+-+   +-+-+   +-+-+
%e A376808   |X|X|   |X| |   | | |   |X| |   | |X|
%e A376808   +-+-+   +-+-+   +-+-+   +-+-+   +-+-+
%e A376808   +-+-+   +-+-+   +-+-+   +-+-+
%e A376808   |X|Y|   |X| |   |X| |   |X|Y|
%e A376808   +-+-+   +-+-+   +-+-+   +-+-+
%e A376808   | | |   |Y| |   | |Y|   |Z| |
%e A376808   +-+-+   +-+-+   +-+-+   +-+-+
%e A376808 so a(2) = 9.
%Y A376808 Main diagonal of A295197.
%Y A376808 Cf. A294791, A294792, A294793, A294794.
%K A376808 nonn
%O A376808 1,2
%A A376808 _Marko Riedel_, Oct 04 2024