A376822 -id:A376822 - cp's OEIS frontend

A294684 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry, 1 <= k <= n.

0, 1, 5, 2, 12, 62, 4, 38, 350, 4154, 6, 106, 2190, 52486, 1342206, 12, 360, 14622, 699598, 35792566, 1908897150, 18, 1180, 99878, 9587578, 981706830, 104715443850, 11488774559742, 34, 4148, 699250, 134223974, 27487816990, 5864063066498, 1286742755471398, 288230376353050814

Offset: 1

Marko Riedel, Nov 06 2017

Colors are not being permuted, i.e., Power Group Enumeration does not apply here.

			Triangle begins:
   0;
   1,    5;
   2,   12,    62;
   4,   38,   350,    4154;
   6,  106,  2190,   52486,   1342206;
  12,  360, 14622,  699598,  35792566,   1908897150;
  18, 1180, 99878, 9587578, 981706830, 104715443850, 11488774559742;
  ...
For the 2 X 2 and two colors we find
  +---+  +---+  +---+  +---+  +---+
  |X| |  | |X|  |X| |  |X|X|  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+
  | | |  |X|X|  | |X|  | | |  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+  +-+-+

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

Andrew Howroyd, Table of n, a(n) for n = 1..1275 (first 50 rows)
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequences A294684, A294685, A294686, A294687.

Main diagonal is A376822.

Cf. A184271, A294685, A294686, A294687, A294791, A294792, A294793, A294794. T(n,1) is A052823.

With[{Q = 2}, Table[(Q!/n/k) Sum[Sum[EulerPhi[d] EulerPhi[f] StirlingS2[GCD[d, f] (n/d) (k/f), Q], {f, Divisors@ k}], {d, Divisors@ n}], {n, 8}, {k, n}]] // Flatten (* Michael De Vlieger, Nov 08 2017 *)

T(n,m) = {2*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), 2, 2) ))/(n*m)} \\ Andrew Howroyd, Oct 05 2024

T(n,k) = (Q!/(n*k))*(Sum_{d|n} Sum_{f|k} phi(d) phi(f) S(gcd(d,f)*(n/d)*(k/f), Q)) with Q=2 and S(n,k) Stirling numbers of the second kind.

T(n,k) = A184271(n,k) - 2. - Andrew Howroyd, Oct 05 2024

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

Original entry on oeis.org

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.

A376824 Number of colorings of a toroidal n X n grid using exactly four colors under translational symmetry.

Original entry on oeis.org

0, 6, 20720, 257706024, 44900438149488, 131160169581733489616, 6467585568566200114362823920, 5316911768534424725926923896066891424, 72172920340122292837562997014593985220649867760, 16069380442569287654590340470284256047904187412954757496784

Offset: 1

Andrew Howroyd, Oct 05 2024

nonn

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

Main diagonal of A294686.

Cf. A179043, A184272, A184278, A376749 (colors permutable), A376822, A376823, A376825.

a(n) = A184272(n) - 4*A184278(n) + 6*A179043(n) - 4.

A376825 Number of colorings of a toroidal n X n grid using exactly five colors under translational symmetry.

Original entry on oeis.org

0, 0, 92680, 8221452750, 11696087875731720, 403564024914127655401650, 362489465982555360136794113733480, 8470302887983624205463771824482291388274750, 5106052803042976484591492152983188808422646355702792360, 78886090441754278328274880503253722147584506163456748572863233329010

Offset: 1

Andrew Howroyd, Oct 05 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..25

Main diagonal of A294687.

Cf. A376822, A376823, A376824.

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.

cp's OEIS Frontend

A294684 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry, 1 <= k <= n.

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

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A376824 Number of colorings of a toroidal n X n grid using exactly four colors under translational symmetry.

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A376825 Number of colorings of a toroidal n X n grid using exactly five 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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