A082963 -id:A082963 - cp's OEIS frontend

A286447 Number of inequivalent n X n matrices over GF(3) under action of dihedral group of the square D_4, with a third of 1's, 2's and 3's (ordered occurrences rounded up/down if n^2 != 0 mod 3).

1, 1, 2, 228, 252642, 3286762710, 423091508279496, 488322998306377824150, 5405955851967092442258037800, 561273297862912365721571649672300480, 524055990531978935668322776302483856990581000

Offset: 0

María Merino, Imanol Unanue, May 11 2017

nonn

			For n=3 the a(3)=228 solutions are colorings of 3 X 3 matrices in 3 colors inequivalent under the action of D_4 with exactly 3 occurrences of each color (coefficient of x1^3 x2^3 x3^3).

María Merino, Table of n, a(n) for n = 0..45
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A054739, A082963.

G.f.: g(x1,x2,x3) = 1/8*(y1^(n^2) + 2*y1^n*y2^((n^2 - n)/2) + 3*y2^(n^2/2) + 2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2) + 4*y1^n*y2^((n^2 - n)/2) + y1*y2^((n^2 - 1)/2) + 2*y1*y4^((n^2 - 1)/4)) if n odd, where coefficient correspond to y1 = x1 + x2 + x3, y2 = x1^2 + x2^2 + x3^2, y4 = x1^4 + x2^4 + x3^4 and occurrences of numbers are ceiling(n^2/3) for 1's and floor(n^2/3) for 2's and 3's.

A286525 Number of inequivalent n X n matrices over GF(4) under action of dihedral group of the square D_4, with a fourth of 1's, 2's, 3's and 4's (ordered occurrences rounded up/down if n^2 != 0 mod 4).

Original entry on oeis.org

1, 1, 3, 978, 7885536, 1030690752000, 2681594035175055000, 111102459342780333711432912, 82765346051371433995689422809152600, 984929152509556378339959477248973638627262816, 201525938526971993585665495909682003042353826154218776128

Offset: 0

María Merino, Imanol Unanue, May 11 2017

nonn

			For n=2 the a(2)=3 solutions are the colorings of 2 X 2 matrices in 4 colors inequivalent under the action of D_4 with exactly 1 occurrence of each color (coefficient of x1^1 x2^1 x3^1 x4^1).

María Merino, Table of n, a(n) for n = 0..39
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A054751, A082963, A286447.

G.f.: g(x1,x2,x3,x4) = 1/8*(y1^(n^2) + 2*y1^n*y2^((n^2 - n)/2) + 3*y2^(n^2/2) + 2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2) + 4*y1^n*y2^((n^2 - n)/2) + y1*y2^((n^2 - 1)/2) + 2*y1*y4^((n^2 - 1)/4)) if n odd, where coefficient correspond to y1 = x1 + x2 + x3 + x4, y2 = x1^2 + x2^2 + x3^2 + x4^2, y4 = x1^4 + x2^4 + x3^4 + x4^2 and occurrences of numbers are ceiling(n^2/4) for 1's and floor(n^2/4) for 2's, 3's and 4's.

A286526 Number of inequivalent n X n matrices over GF(5) under action of dihedral group of the square D_4, with a fifth of 1's, 2's, 3's, 4's and 5's (ordered occurrences rounded up/down if n^2 != 0 mod 5).

Original entry on oeis.org

1, 1, 1, 2874, 84086160, 77920099694640, 1787320731699689472000, 1208369393947533515948886636000, 22022604563875220592723146462014970246400, 10631042739086498005729294276105510004209560426195000, 136864426940639977623403211038729959780835360788855628470904385280

Offset: 0

María Merino, Imanol Unanue, May 11 2017

nonn

			For n=3 the a(3)=2874 solutions are colorings of 3 X 3 matrices in 5 colors inequivalent under the action of D_4 with exactly occurrences 2, 2, 2, 2, 1 of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^1).

María Merino, Table of n, a(n) for n = 0..37
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A054752, A082963, A286447, A286525.

G.f.: g(x1,x2,x3,x4,x5) = 1/8*(y1^(n^2) + 2*y1^n*y2^((n^2 - n)/2) + 3*y2^(n^2/2) + 2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2) + 4*y1^n*y2^((n^2 - n)/2) + y1*y2^((n^2 - 1)/2) + 2*y1*y4^((n^2 - 1)/4)) if n odd, where coefficient correspond to y1 = Sum_{i=1..5} x_i, y2 = Sum_{i=1..5} x_i^2, y4 = Sum_{i=1..5} x_i^4 and occurrences of numbers are ceiling(n^2/5) for the first k numbers and floor(n^2/5) for the last (5-k) numbers, if n^2 = k mod 5.

A287239 Number of inequivalent n X n matrices over an alphabet of size 6 under action of dihedral group of the square D_4, with one-sixth each of 1s, 2s, 3s, 4s, 5s and 6s (ordered occurrences rounded up/down if n^2 != 0 mod 6).

Original entry on oeis.org

1, 1, 1, 5688, 504508320, 2029169127793680, 333772217080092664473600, 1966297518276227170017585421188600, 474436367892839446541884570454351985506872320, 4529567636413022031420100639004131328550592354551163392000, 1664947024157601976065851576560401128416782438266187161307818265349050000

Offset: 0

María Merino, Imanol Unanue, May 22 2017

nonn

Computed using Polya's enumeration theorem for coloring.

			For n = 3 the a(3) = 5688 solutions are colorings of 3 X 3 matrices in 6 colors inequivalent under the action of D_4 with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2 x6^2).

María Merino, Table of n, a(n) for n = 0..34
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A286392, A082963, A286447, A286525, A286526.

G.f.: g(x1,x2,x3,x4,x5,x6)=1/8*(y1^(n^2)+2*y1^n*y2^((n^2-n)/2)+3*y2^(n^2/2)+2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2)+4*y1^n*y2^((n^2-n)/2)+y1*y2^((n^2-1)/2)+2*y1*y4^((n^2-1)/4)) if n odd, where coefficient correspond to y1=Sum_{i=1..6} x_i, y2=Sum_{i=1..6} x_i^2, y4=Sum_{i=1..6} x_i^4 and occurrences of numbers are ceiling(n^2/6) for the first k numbers and floor(n^2/6) for the last (6-k) numbers, if n^2 = k mod 6.

A287245 Number of inequivalent n X n matrices over GF(7) under action of dihedral group of the square D_4, with one-seventh each of 1's, 2's, 3's, 4's, 5's, 6's and 7's (ordered occurrences rounded up/down if n^2 != 0 mod 7).

Original entry on oeis.org

1, 1, 1, 11340, 2270280240, 27055587870486000, 21628439666761521875561280, 920451958269648700957746787694592000, 1914192808178753950843058828570207003149548000000, 216425158352284448578663515683744576588775769063470820304640000

Offset: 0

María Merino, Imanol Unanue, May 22 2017

nonn

Computed using Polya's enumeration theorem for coloring.

			For n = 3 the a(3) = 11340 solutions are colorings of 3 X 3 matrices in 7 colors inequivalent under the action of D_4 with exactly occurrences 2, 2, 1, 1, 1, 1, 1 of each color (coefficient of x1^2 x2^2 x3^1 x4^1 x5^1 x6^1 x7^1).

María Merino, Table of n, a(n) for n = 0..33
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A286393, A082963, A286447, A286525, A286526, A287239.

G.f.: g(x1,x2,x3,x4,x5,x6,x7)=1/8*(y1^(n^2)+2*y1^n*y2^((n^2-n)/2)+3*y2^(n^2/2)+2*y4^(n^2/4)) if n even and 1/8*(y1^(n^2)+4*y1^n*y2^((n^2-n)/2)+y1*y2^((n^2-1)/2)+2*y1*y4^((n^2-1)/4)) if n odd, where coefficient correspond to y1=Sum_{i=1..7} x_i, y2=Sum_{i=1..7} x_i^2, y4=Sum_{i=1..7} x_i^4 and occurrences of numbers are ceiling(n^2/7) for the first k numbers and floor(n^2/7) for the last (7-k) numbers, if n^2 = k mod 7.

A287249 Number of inequivalent n X n matrices over GF(8) under action of dihedral group of the square D_4, with one-eighth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's and 8's (ordered occurrences rounded up/down if n^2 != 0 mod 8).

Original entry on oeis.org

1, 1, 1, 22680, 10216251360, 288592936632000000, 675888739586283307003920000, 150403128386758194407881602780164966400, 2270715491453850844620503532869818724155487772912000, 2190916399747036514334089808617857198357442887303702763561256837120

Offset: 0

María Merino, Imanol Unanue, May 22 2017

nonn

Computed using Polya's enumeration theorem for coloring.

			For n = 3 the a(4) = 10216251360 solutions are colorings of 4 X 4 matrices in 8 colors inequivalent under the action of D_4 with exactly 2 occurrences of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2 x6^2 x7^2 x8^2).

María Merino, Table of n, a(n) for n = 0..32
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A286394, A082963, A286447, A286525, A286526, A287239, A287245.

G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8) = (1/8)*(y1^(n^2)+2*y1^n*y2^((n^2-n)/2)+3*y2^(n^2/2)+2*y4^(n^2/4)) if n even and (1/8)*(y1^(n^2)+4*y1^n*y2^((n^2-n)/2)+y1*y2^((n^2-1)/2)+2*y1*y4^((n^2-1)/4)) if n odd, where coefficient correspond to y1=Sum_{i=1..8} x_i, y2=Sum_{i=1..8} x_i^2, y4=Sum_{i=1..8} x_i^4 and occurrences of numbers are ceiling(n^2/8) for the first k numbers and floor(n^2/8) for the last (8-k) numbers, if n^2 = k mod 8.

A287250 Number of inequivalent n X n matrices over GF(9) under action of dihedral group of the square D_4, with one-ninth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's and 9's (ordered occurrences rounded up/down if n^2 != 0 mod 9).

Original entry on oeis.org

1, 1, 1, 45360, 20432427120, 1731557619792000000, 17601269260059379482191694720, 11370476506038919496334983007474778275840, 944848320304251231447932170156537415535539635814400000, 6641336088298446224006555306105706090482482272285249518936232000000000

Offset: 0

María Merino, Imanol Unanue, May 22 2017

nonn

Computed using Polya's enumeration theorem for coloring.

			For n = 3 the a(3) = 45360 solutions are colorings of 3 X 3 matrices in 9 colors inequivalent under the action of D_4 with exactly 1 occurrence of each color (coefficient of x1^1 x2^1 x3^1 x4^1 x5^1 x6^1 x7^1 x8^1 x9^1).

María Merino, Table of n, a(n) for n = 0..31
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A286396, A082963, A286447, A286525, A286526, A287239, A287245, A287249, A287261.

G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8,x9) = (1/8)*(y1^(n^2)+2*y1^n*y2^((n^2-n)/2)+3*y2^(n^2/2)+2*y4^(n^2/4)) if n even and (1/8)*(y1^(n^2)+4*y1^n*y2^((n^2-n)/2)+y1*y2^((n^2-1)/2)+2*y1*y4^((n^2-1)/4)) if n odd, where coefficient correspond to y1=Sum_{i=1..9} x_i, y2=Sum_{i=1..9} x_i^2, y4=Sum_{i=1..9} x_i^4 and occurrences of numbers are ceiling(n^2/9) for the first k numbers and floor(n^2/9) for the last (9-k) numbers, if n^2 = k mod 9.

A287261 Number of inequivalent n X n matrices over an alphabet of size 10 under action of dihedral group of the square D_4, with one-tenth of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's and 0's (ordered occurrences rounded up/down if n^2 != 0 mod 10).

Original entry on oeis.org

1, 1, 1, 1, 40864828320, 7792009289281728000, 187746872107299580970294400000, 614005731326101652800803825889630961295360, 176445174659483893854948844253232539237396497554309120000, 7090469783239448892319287907564531885316857076509137838529329991091840000

Offset: 0

María Merino, Imanol Unanue, May 22 2017

nonn

Computed using Polya's enumeration theorem for coloring.

			For n = 3 the a(4) = 40864828320 solutions are colorings of 4 X 4 matrices in 10 colors inequivalent under the action of D_4 with exactly occurrences 2, 2, 2, 2, 2, 2, 1, 1, 1, 1 of each color (coefficient of x1^2 x2^2 x3^2 x4^2 x5^2 x6^2 x7^1 x8^1 x9^1 x10^1).

María Merino, Table of n, a(n) for n = 0..29
M. Merino and I. Unanue, Counting squared grid patterns with Pólya Theory, EKAIA, 34 (2018), 289-316 (in Basque).

Cf. A286397, A082963, A286447, A286525, A286526, A287239, A287245, A287249, A287250.

G.f.: g(x1,x2,x3,x4,x5,x6,x7,x8,x9,x10) = (1/8)*(y1^(n^2)+2*y1^n*y2^((n^2-n)/2)+3*y2^(n^2/2)+2*y4^(n^2/4)) if n even and (1/8)*(y1^(n^2)+4*y1^n*y2^((n^2-n)/2)+y1*y2^((n^2-1)/2)+2*y1*y4^((n^2-1)/4)) if n odd, where coefficient correspond to y1=Sum_{i=1..10} x_i, y2=Sum_{i=1..10} x_i^2, y4=Sum_{i=1..10} x_i^4 and occurrences of numbers are ceiling(n^2/10) for the first k numbers and floor(n^2/10) for the last (10-k) numbers, if n^2 = k mod 10.

A081623 Number of ways in which the points on an n X n square lattice can be equally occupied with spin "up" and spin "down" particles. If n is odd, we arbitrarily take the lattice to contain one more spin "up" particle than the number of spin "down" particles.

Original entry on oeis.org

1, 1, 6, 126, 12870, 5200300, 9075135300, 63205303218876, 1832624140942590534, 212392290424395860814420, 100891344545564193334812497256, 191645966716130525165099506263706416, 1480212998448786189993816895482588794876100

Offset: 0

A. Timothy Royappa, Apr 22 2003

			a(2) = C(4,2) = 6.
a(3) = C(9,5) = 126.

Alois P. Heinz, Table of n, a(n) for n = 0..57
Brian Hayes, The World in a Spin, American Scientist 88:5 (September-October 2000), pp. 384-388. [alternate link]
James Grime, Maths Problem: Complete Noughts and Crosses (Burnside's Lemma)
Noah Lordi, Maedee Trank-Greene, Akira Kyle, and Joshua Combes, Quantum permutation puzzles with indistinguishable particles, arXiv:2410.22287 [quant-ph], 2024. See p. 8.

A082963 is the equivalent sequence up to reflection and rotation.

a:= n-> (s-> binomial(s, floor(s/2)))(n^2):
seq(a(n), n=0..15);  # Alois P. Heinz, Jul 21 2017

a(n)=binomial(n^2,n^2\2) \\ Charles R Greathouse IV, May 09 2013

a(n) = C(n^2, (n^2+1)/2) if n is odd and C(n^2, n^2/2) if n is even.
a(n) = binomial(n^2,floor(n^2/2)). - Alois P. Heinz, Jul 21 2017
Largest coefficient of (1 + x)^(n^2). - Ilya Gutkovskiy, Apr 24 2025

a(0)=1 prepended by Alois P. Heinz, Jul 21 2017

cp's OEIS Frontend

A286447 Number of inequivalent n X n matrices over GF(3) under action of dihedral group of the square D_4, with a third of 1's, 2's and 3's (ordered occurrences rounded up/down if n^2 != 0 mod 3).

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A286525 Number of inequivalent n X n matrices over GF(4) under action of dihedral group of the square D_4, with a fourth of 1's, 2's, 3's and 4's (ordered occurrences rounded up/down if n^2 != 0 mod 4).

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A286526 Number of inequivalent n X n matrices over GF(5) under action of dihedral group of the square D_4, with a fifth of 1's, 2's, 3's, 4's and 5's (ordered occurrences rounded up/down if n^2 != 0 mod 5).

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A287239 Number of inequivalent n X n matrices over an alphabet of size 6 under action of dihedral group of the square D_4, with one-sixth each of 1s, 2s, 3s, 4s, 5s and 6s (ordered occurrences rounded up/down if n^2 != 0 mod 6).

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A287245 Number of inequivalent n X n matrices over GF(7) under action of dihedral group of the square D_4, with one-seventh each of 1's, 2's, 3's, 4's, 5's, 6's and 7's (ordered occurrences rounded up/down if n^2 != 0 mod 7).

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A287249 Number of inequivalent n X n matrices over GF(8) under action of dihedral group of the square D_4, with one-eighth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's and 8's (ordered occurrences rounded up/down if n^2 != 0 mod 8).

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A287250 Number of inequivalent n X n matrices over GF(9) under action of dihedral group of the square D_4, with one-ninth each of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's and 9's (ordered occurrences rounded up/down if n^2 != 0 mod 9).

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A287261 Number of inequivalent n X n matrices over an alphabet of size 10 under action of dihedral group of the square D_4, with one-tenth of 1's, 2's, 3's, 4's, 5's, 6's, 7's, 8's, 9's and 0's (ordered occurrences rounded up/down if n^2 != 0 mod 10).

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A081623 Number of ways in which the points on an n X n square lattice can be equally occupied with spin "up" and spin "down" particles. If n is odd, we arbitrarily take the lattice to contain one more spin "up" particle than the number of spin "down" particles.

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