A294686 -id:A294686 - cp's OEIS frontend

A294791 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry and swappable colors.

0, 1, 4, 1, 7, 31, 3, 23, 179, 2107, 3, 55, 1095, 26271, 671103, 7, 189, 7327, 350063, 17896831, 954459519, 9, 595, 49939, 4794087, 490853415, 52357746895, 5744387279871, 19, 2101, 349715, 67115111, 13743921631, 2932032057731, 643371380132743, 144115188277194943, 29, 7315, 2485591, 954444607, 390937468407, 166799988703927, 73201365371896619

Offset: 1

Marko Riedel, Nov 08 2017

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.)

			For the 2 X 2 grid and two colors we find T(2,2) = 4:
  +---+  +---+  +---+  +---+
  |X| |  |X| |  |X|X|  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+
  | | |  | |X|  | | |  |X| |
  +-+-+  +-+-+  +-+-+  +-+-+

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 sequences A294791, A294792, A294793, A294794.

Cf. A152175, A294684, A294685, A294686, A294687, A294792, A294793, A294794, A295197. T(n,1) is A056295.

T(n,k) = (1/(n*k*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|k} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(k/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma) with Q=2. 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.

A294792 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry and swappable colors.

Original entry on oeis.org

0, 0, 3, 1, 18, 345, 2, 136, 7254, 447156, 5, 946, 158355, 29032254, 5647919665, 18, 7324, 3580802, 1961010826, 1143822046786, 694881637942816, 43, 56450, 82968843, 136166703562, 238244961999013, 434202285631866206, 813943290958393433377, 126, 447138, 1960981598, 9651082393912, 50656925726930746, 276966813318877426118, 1557582240509759704455566

Offset: 1

Marko Riedel, Nov 08 2017

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.)

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

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Cf. A294684, A294685, A294686, A294687, A294791, A294793, A294794, A295197. T(n,1) is A056296.

T(n,k) = (1/(n*k*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|k} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(k/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma) with Q=3. 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.

A294793 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry and swappable colors.

Original entry on oeis.org

0, 0, 1, 0, 13, 874, 1, 235, 51075, 10741819, 2, 3437, 2823766, 2261625725, 1870851589562, 13, 51275, 155495153, 486711524815, 1600136051453135, 5465007068038102643, 50, 742651, 8643289534, 107092397450897, 1405227969932349726, 19188864521773558375127, 269482732023591671431784330, 221, 10741763, 486710971595, 24009547064476683

Offset: 1

Marko Riedel, Nov 08 2017

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.)

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

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Cf. A294684, A294685, A294686, A294687, A294791, A294792, A294794, A295197. T(n,1) is A056297.

T(n,k) = (1/(n*k*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|k} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(k/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma) with Q=4. 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.

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.

Original entry on oeis.org

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

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.

Original entry on oeis.org

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

A294687 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry, 1 <= k <= n.

Original entry on oeis.org

0, 0, 0, 0, 300, 92680, 0, 15750, 13794150, 8221452750, 24, 510312, 1686135376, 4495236798162, 11696087875731720, 300, 13794450, 193054017440, 2425003938178050, 30852000867277668428, 403564024914127655401650, 2400, 343501500, 21664357535320, 1317601563731383350, 82985159653854019928352, 5411356249329837891442095560

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,      0;
   0,    300,      92680;
   0,  15750,   13794150,    8221452750;
  24, 510312, 1686135376, 4495236798162, 11696087875731720;
  ...

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

Andrew Howroyd, Table of n, a(n) for n = 1..325 (first 25 rows).
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Main diagonal is A376825.

Cf. A294684, A294685, A294686, A294791, A294792, A294793, A294794. T(n,1) is A056285.

T(n,m)=my(k=5); k!*sumdiv(n, d, sumdiv(m, e, eulerphi(d) * eulerphi(e) * stirling(n*m/lcm(d,e), k, 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=5 and S(n,k) Stirling numbers of the second kind.

A294794 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry and swappable colors.

Original entry on oeis.org

0, 0, 0, 0, 3, 775, 0, 145, 115100, 68522769, 1, 4281, 14051164, 37460388596, 97467398965031, 3, 115381, 1608801153, 20208371722051, 257100007425866689, 3363033541015148835823, 20, 2863227, 180536313547, 10980013072900632, 691542997115450167856, 45094635411084308447578413, 3020745549854628001139950947779, 136, 68522707

Offset: 1

Marko Riedel, Nov 08 2017

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.)

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

Marko Riedel et al., Burnside lemma and translational symmetries of the torus.

Cf. A294684, A294685, A294686, A294687, A294791, A294792, A294793, A295197. T(n,1) is A056298.

T(n,k) = (1/(n*k*Q!))*(Sum_{sigma in S_Q} Sum_{d|n} Sum_{f|k} phi(d) phi(f) [[forall j_l(sigma) > 0 : l|lcm(d,f) ]] P(gcd(d,f)*(n/d)*(k/f), sigma)) where P(F, sigma) = F! [z^F] Product_{l=1..Q} (exp(lz)-1)^j_l(sigma) with Q=5. 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.

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.

cp's OEIS Frontend

A294791 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly two colors under translational symmetry and swappable colors.

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A294792 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly three colors under translational symmetry and swappable colors.

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A294793 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly four colors under translational symmetry and swappable colors.

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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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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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A294687 Triangle read by rows: T(n,k) is the number of non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry, 1 <= k <= n.

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A294794 Triangle read by rows, 1 <= k <= n: T(n,k) = non-isomorphic colorings of a toroidal n X k grid using exactly five colors under translational symmetry and swappable colors.

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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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