A376748 -id:A376748 - cp's OEIS frontend

A376823 Number of colorings of a toroidal n X n grid using exactly three colors under translational symmetry.

0, 9, 2022, 2679246, 33887517990, 4169289730628814, 4883659745750360600262, 53651309691205903304049168186, 5474401089420129832016444491921748358, 5153775207320113272335114827748860107542139918, 44553974378043749018442678682265335735181851572329684070

Offset: 1

Andrew Howroyd, Oct 05 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..40

Main diagonal of A294685.

Cf. A179043, A184278, A376748 (colors permutable), A376822, A376824, A376825.

a(n) = A184278(n) - 3*A179043(n) + 3.

A376747 Number of non-isomorphic colorings of a toroidal n X n grid using exactly two swappable colors.

Original entry on oeis.org

0, 4, 31, 2107, 671103, 954459519, 5744387279871, 144115188277194943, 14925010118699132241919, 6338253001141180784480847871, 10985355337065420437221545952731135, 77433143050453552574875182200691073835007, 2213872302702432822841084717014014514981767643135, 256208234097415541381052629523530965709132732687965552639

Offset: 1

Marko Riedel, Oct 03 2024

nonn

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequence.

Main diagonal of A294791.

Cf. A376748, A376749, A376808, A376822.

a(n) = (1/(n^2*2!))*(Sum_{sigma in S_2} 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..2} (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.

A376749 Number of non-isomorphic colorings of a toroidal n X n grid using exactly four swappable colors.

Original entry on oeis.org

0, 1, 874, 10741819, 1870851589562, 5465007068038102643, 269482732023591671431784330, 221537990355601030571170905795094315, 3007205014171762201565124875608675533096268906, 669557518440386985607930852942771727146772232484581602227, 2433673642945425535196140161775877796522974318753784273286700783313050

Offset: 1

Marko Riedel, Oct 03 2024

nonn

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973.

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequence.

Main diagonal of A294793.

Cf. A376747, A376748, A376808, A376824.

a(n) = (1/(n^2*4!))*(Sum_{sigma in S_4} 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..4} (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.

cp's OEIS Frontend

A376823 Number of colorings of a toroidal n X n grid using exactly three colors under translational symmetry.

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A376747 Number of non-isomorphic colorings of a toroidal n X n grid using exactly two swappable colors.

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A376749 Number of non-isomorphic colorings of a toroidal n X n grid using exactly four swappable colors.

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