A376823 -id:A376823 - cp's OEIS frontend

A294685 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry, 1 <= k <= n.

0, 0, 9, 2, 91, 2022, 9, 738, 43315, 2679246, 30, 5613, 950062, 174184755, 33887517990, 91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814, 258, 338259, 497812638, 816999710223, 1429469771994078, 2605213713043722909, 4883659745750360600262, 729, 2679228, 11765822365, 57906482267826, 303941554100145501

Offset: 1

Marko Riedel, Nov 06 2017

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

			Triangle begins:
   0;
   0,     9;
   2,    91,     2022;
   9,   738,    43315,     2679246;
  30,  5613,   950062,   174184755,   33887517990;
  91, 43404, 21480921, 11765865678, 6862930841141, 4169289730628814;
  ...

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

Andrew Howroyd, Table of n, a(n) for n = 1..820 (first 40 rows)
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Main diagonal is A376823.

Cf. A184271, A184284, A294684, A294686, A294687, A294791, A294792, A294793, A294794. T(n,1) is A056283.

T(n,m)=6*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), 3, 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=3 and S(n,k) Stirling numbers of the second kind.

T(n,k) = A184284(n,k) - 3*A184271(n,k) + 3. - Andrew Howroyd, Oct 05 2024

A376822 Number of colorings of a toroidal n X n grid using exactly two colors under translational symmetry.

Original entry on oeis.org

0, 5, 62, 4154, 1342206, 1908897150, 11488774559742, 288230376353050814, 29850020237398264483838, 12676506002282327791964489726, 21970710674130840874443091905462270, 154866286100907105149651981766316633972734, 4427744605404865645682169434028029029963535286270

Offset: 1

Andrew Howroyd, Oct 05 2024

nonn

Andrew Howroyd, Table of n, a(n) for n = 1..50

Main diagonal of A294684.

Cf. A179043, A376747 (colors permutable), A376823, A376824, A376825.

a(n) = A179043(n) - 2.

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.

A376748 Number of non-isomorphic colorings of a toroidal n X n grid using exactly three swappable colors.

Original entry on oeis.org

0, 3, 345, 447156, 5647919665, 694881637942816, 813943290958393433377, 8941884948534360647405572800, 912400181570021638669407666368774097, 858962534553352212055863239761275173880606456, 7425662396340624836407113113710889289196975262054947345, 587417576454184723055270940786413231085263155884260701824558793960

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 A294792.

Cf. A376747, A376749, A376808, A376823.

a(n) = (1/(n^2*3!))*(Sum_{sigma in S_3} 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..3} (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

A294685 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry, 1 <= k <= n.

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A376822 Number of colorings of a toroidal n X n grid using exactly two 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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A376748 Number of non-isomorphic colorings of a toroidal n X n grid using exactly three swappable colors.

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