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A386556 Number of 2-colorings of an 6 X 6 X 6 grid / cube, up to rotational symmetry, by the number of black cells.

1, 11, 1013, 69045, 3677374, 155822419, 5479820520, 164392285865, 4294750355129, 99256405950180, 2054607644379763, 38477196919023712, 657318781413490584, 10314848558604181280, 149565304110190970723, 2014146095209708440612, 25302710321203873065217, 297678944953786351579885, 3291006113657215317985320

Offset: 0

Marko Riedel, Jul 25 2025

This sequence is finite, having 6^3+1 terms. The cycle index Z(C) of the permutation group C of the rotations of the cube acting on the cells is given by 1/24 (a[1]^216 + 8 * a[1]^6 * a[3]^70 + 9 * a[2]^108 + 6 * a[4]^54).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 35.

Marko Riedel, Table of n, a(n) for n = 0..216

Cf. A334616, A386553, A386554, A386555.

a(n) = [z^n] Z(C; 1+z).

A386555 Number of 2-colorings of an 5 X 5 X 5 grid, up to rotational symmetry, by the number of black cells.

Original entry on oeis.org

1, 10, 355, 13375, 404775, 9775852, 195462780, 3322598660, 49007581605, 637095486150, 7390299421967, 77262191948715, 733990748643925, 6380073195502170, 51040584989484940, 377700327297519628, 2596689746386426870, 16649363658225874915, 99896181927621097005

Offset: 0

Marko Riedel, Jul 25 2025

This sequence is finite, having 5^3+1 terms. The cycle index Z(C) of the permutation group C of the rotations of the cube acting on the cells is given by 1/24 ( a[1]^125 + 8 * a[1]^5 * a[3]^40 + 9 * a[1]^5 * a[2]^60 + 6 * a[1]^5 * a[4]^30).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 35.

Sean A. Irvine, Table of n, a(n) for n = 0..125

Cf. A334616, A386553, A386554, A386556.

a(n) = [z^n] Z(C; 1+z).

A386554 Number of 2-colorings of an 4 X 4 X 4 grid, up to rotational symmetry, by the number of black cells.

Original entry on oeis.org

1, 4, 98, 1744, 26691, 317728, 3125882, 25884268, 184437452, 1147524988, 6311461050, 30983159856, 136842622391, 547369124016, 1993988770170, 6646625005908, 20355293012087, 57473757334488, 150070376844312, 363328255291128, 817488598547832, 1712833203324840

Offset: 0

Marko Riedel, Jul 25 2025

This sequence is finite, having 4^3+1 terms. The cycle index Z(C) of the permutation group C of the rotations of the cube acting on the cells is given by 1/24 (a[1]^64 + 8 * a[1]^4 * a[3]^20 + 9 * a[2]^32 + 6 * a[4]^16).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 35.

Sean A. Irvine, Table of n, a(n) for n = 0..64

Cf. A334616, A386553, A386555, A386556.

a(n) = [z^n] Z(C; 1+z).

A386553 Number of 2-colorings of an 3 X 3 X 3 grid, up to rotational symmetry, by the number of black cells.

Original entry on oeis.org

1, 4, 22, 139, 779, 3455, 12507, 37303, 92968, 195963, 352433, 544382, 725612, 837184, 837184, 725612, 544382, 352433, 195963, 92968, 37303, 12507, 3455, 779, 139, 22, 4, 1

Offset: 0

Marko Riedel, Jul 25 2025

This sequence is finite, having 3^3+1 terms. The cycle index Z(C) of the permutation group C of the rotations of the cube acting on the cells is given by 1/24 (a[1]^27 + 8 * a[1]^3 * a[3]^8 + 9 * a[1]^3 * a[2]^12 + 6 * a[1]^3*a[4]^6).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, 1973, page 35.

Cf. A334616, A386554, A386555, A386556.

a(n) = [z^n] Z(C; 1+z).

A380401 Triangle read by rows: T(n,k) is the number of necklace permutations of a multiset whose multiplicities are given by the k-th partition of n in graded reflected lexicographic order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 3, 2, 1, 1, 24, 12, 6, 4, 2, 1, 1, 120, 60, 30, 16, 20, 10, 4, 5, 3, 1, 1, 720, 360, 180, 90, 120, 60, 30, 20, 30, 15, 5, 6, 3, 1, 1, 5040, 2520, 1260, 630, 318, 840, 420, 210, 140, 70, 210, 105, 54, 35, 10, 42, 21, 7, 7, 4, 1, 1, 40320, 20160, 10080, 5040, 2520, 6720, 3360, 1680, 840, 1120, 560, 188, 1680, 840, 420, 280, 140, 70, 336, 168, 84, 56, 14, 56, 28, 10, 8, 4, 1, 1

Offset: 1

Marko Riedel, Jan 23 2025

See A318810 for a definition of necklace permutation.

			The ordering of the partitions used here is graded reflected lexicographic illustrated below with n=5:
  1,1,1,1,1 => 24
  1,1,1,2 => 12
  1,2,2 => 6
  1,1,3 => 4
  2,3 => 2
  1,4 => 1
  5 => 1
Table begins:
  1
  1,1
  2,1,1
  6,3,2,1,1
  24,12,6,4,2,1,1

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, pages 36-37, 42-43.

Math StackExchange, Marko Riedel et. al, Free circular permutations
Marko Riedel, Maple code for sequence by PET and closed form.

Cf. A000041 (row lengths), A072605 (row sums), A080576 (graded reflected lexicographic order), A212359 (similar triangle for Abramowitz-Stegun order), A318810, A334434, A214609 (up to rotations and reflections).

C(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, eulerphi(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)]))} \\ Andrew Howroyd, Jan 23 2025

For a distribution of colors n1+n2+...+nm = n the number of necklaces is (1/n)*Sum_{d|gcd(n1,n2,...,nm)} phi(d) (n/d)!/Prod_{q=1..m} (nq/d)!

T(n,k) = A318810(A334434(n,k)).

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

Original entry on oeis.org

1, 9, 2387, 655089857, 185543613289205809, 106103186941524316132396201360, 218900758256599151027392153440612298654753249, 2689595989958732045849530682270318547733917269644639109073775285

Offset: 1

Marko Riedel, Oct 04 2024

nonn

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.

			For the 2x2 we find
  +-+-+   +-+-+   +-+-+   +-+-+   +-+-+
  |X|X|   |X|X|   |X|X|   |X| |   |X| |
  +-+-+   +-+-+   +-+-+   +-+-+   +-+-+
  |X|X|   |X| |   | | |   |X| |   | |X|
  +-+-+   +-+-+   +-+-+   +-+-+   +-+-+
  +-+-+   +-+-+   +-+-+   +-+-+
  |X|Y|   |X| |   |X| |   |X|Y|
  +-+-+   +-+-+   +-+-+   +-+-+
  | | |   |Y| |   | |Y|   |Z| |
  +-+-+   +-+-+   +-+-+   +-+-+
so a(2) = 9.

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

Andrew Howroyd, Table of n, a(n) for n = 1..24
Marko Riedel et al., Burnside lemma and translational symmetries of the torus.
Marko Riedel, Maple code for sequence by PGE.

Main diagonal of A295197.

Cf. A294791, A294792, A294793, A294794.

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.

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.

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.

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.

A368213 Triangular array read by rows: Number of permutations of [n] that factor into exactly k-cycles, ordered by n (rows) and divisors k of n (columns).

Original entry on oeis.org

1, 1, 1, 1, 0, 2, 1, 3, 0, 6, 1, 0, 0, 0, 24, 1, 15, 40, 0, 0, 120, 1, 0, 0, 0, 0, 0, 720, 1, 105, 0, 1260, 0, 0, 0, 5040, 1, 0, 2240, 0, 0, 0, 0, 0, 40320, 1, 945, 0, 0, 72576, 0, 0, 0, 0, 362880, 1, 0, 0, 0, 0, 0, 0, 0, 0, 0, 3628800, 1, 10395, 246400, 1247400, 0, 6652800, 0, 0, 0, 0, 0, 39916800

Offset: 1

Marko Riedel, Dec 17 2023

			Row n=6 is 1, 15, 40, 120 because there is one permutation of [6] consisting of six fixed points, there are 15 permutations consisting of three transpositions, there are forty permutations consisting of two three-cycles and there are one hundred and twenty permutations consisting of just one six-cycle (6!/6).
Triangular array starts:
[ 1] 1;
[ 2] 1,   1;
[ 3] 1,   0,    2;
[ 4] 1,   3,    0,    6;
[ 5] 1,   0,    0,    0,    24;
[ 6] 1,  15,   40,    0,     0, 120;
[ 7] 1,   0,    0,    0,     0,   0, 720;
[ 8] 1, 105,    0, 1260,     0,   0,   0, 5040;
[ 9] 1,   0, 2240,    0,     0,   0,   0,    0, 40320;
[10] 1, 945,    0,    0, 72576,   0,   0,    0,     0, 362880;

P. Flajolet and R. Sedgewick, Analytic Combinatorics, Cambridge University Press, 2009, pages 120-122.

Marko Riedel, math.stackexchange.com, Number of permutations in the symmetric group.
Marko Riedel, Computing the number of permutations of [n] with k m-cycles with analytic combinatorics.
Marko Riedel, Computing the number of permutations of [n] with k m-cycles using the principle of inclusion-exclusion.

Cf. A005225 (row sums), A008290.

Cf. A123023 (column 2), A052502 (column 3), A060706 (column 4).

T:= (n, m)-> `if`(irem(n,m)=0, n!/m^(n/m)/(n/m)!, 0):
seq(seq(T(n, m), m = 1..n), n=1..15);

A368213[n_,k_]:=If[Divisible[n,k],n!/(k^(n/k)(n/k)!),0];
Table[A368213[n,k],{n,15},{k,n}] (* Paolo Xausa, Dec 18 2023 *)

def T(n, d): return factorial(n) // (d ** (n//d) * factorial(n//d))
for n in range(1, 19):
    print([T(n, d) if n % d == 0 else 0 for d in range(1, n+1)])
# Peter Luschny, Dec 17 2023

T(n, k) = n! / ( k^(n/k) * (n/k)! ) if k divides n otherwise 0.

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User: Marko Riedel

A386556 Number of 2-colorings of an 6 X 6 X 6 grid / cube, up to rotational symmetry, by the number of black cells.

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A386555 Number of 2-colorings of an 5 X 5 X 5 grid, up to rotational symmetry, by the number of black cells.

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A386554 Number of 2-colorings of an 4 X 4 X 4 grid, up to rotational symmetry, by the number of black cells.

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A386553 Number of 2-colorings of an 3 X 3 X 3 grid, up to rotational symmetry, by the number of black cells.

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A380401 Triangle read by rows: T(n,k) is the number of necklace permutations of a multiset whose multiplicities are given by the k-th partition of n in graded reflected lexicographic order.

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

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

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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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