A362763 -id:A362763 - cp's OEIS frontend

A362764 Number of nonisomorphic 2-sets of permutations of an n-set.

0, 1, 5, 23, 89, 484, 2904, 22002, 190555, 1876337, 20445337, 244087420, 3161870208, 44155439706, 661065427533, 10561205778825, 179324080364960, 3224650449785185, 61218223893368714, 1223523447160283853, 25679025453032962132, 564657001202726477313

Offset: 1

Andrew Howroyd, May 03 2023

nonn

Isomorphism is up to permutation of the elements of the n-set.

			The a(3)=5 sets with permutations shown in cycle notation are:
  {(1)(2)(3), (1)(23)},
  {(1)(2)(3), (123)},
  {(1)(23), (12)(3)},
  {(1)(23), (123)},
  {(123), (132)}.

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

Column k=2 of A362763.

Cf. A000041, A362645, A362765.

a(n) = A362645(n) - A000041(n).

A362765 Number of nonisomorphic 3-sets of permutations of an n-set.

Original entry on oeis.org

0, 0, 6, 116, 2494, 87984, 4250015, 271412031, 21965480315, 2195837248568, 265649147125826, 38249422194113490, 6463715127098722285, 1266831272477388372744, 285028258253204630333567, 72965650731125156284328720, 21086743012582217859035501699

Offset: 1

Andrew Howroyd, May 03 2023

nonn

Isomorphism is up to permutation of the elements of the n-set.

			The a(3)=6 sets with permutations shown in cycle notation are:
  {(1)(2)(3), (1)(23), (12)(3)},
  {(1)(2)(3), (1)(23), (123)},
  {(1)(2)(3), (123), (132)},
  {(1)(23), (12)(3), (123)},
  {(1)(23), (12)(3), (13)(2)},
  {(1)(23), (123), (132)}.

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

Column k=3 of A362763.

Cf. A362646, A362764.

A362766 Number of nonisomorphic sets of permutations of an n-set.

Original entry on oeis.org

2, 2, 4, 24, 711936, 11076899964874307395625695676727296

Offset: 0

Andrew Howroyd, May 03 2023

nonn

Isomorphism is up to permutation of the elements of the n-set.

a(6) has 214 decimal digits.

			The a(2)=4 sets with permutations shown in cycle notation are:
  {},
  {(1)(2)},
  {(12)},
  {(1)(2), (12)}.

Andrew Howroyd, Table of n, a(n) for n = 0..6

Row sums of A362763.

Cf. A050923.

a(n) = Sum_{k=0..n!} A362763(n,k).

A050923(n) / n! <= a(n) <= A050923(n).

A381842 Triangle read by rows: T(n,k) is the number of non-equivalent subsets of size k in S_n, 0 <= k <= n!.

Original entry on oeis.org

1, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 2, 1, 1, 1, 1, 4, 10, 41, 103, 309, 691, 1458, 2448, 3703, 4587, 5050, 4587, 3703, 2448, 1458, 691, 309, 103, 41, 10, 4, 1, 1, 1, 1, 6, 37, 715, 13710, 256751, 4140666, 58402198, 726296995, 8060937770, 80604620206, 732149722382

Offset: 0

Raghavendra Tripathi, Mar 09 2025

We say two subsets A, B of size k are equivalent if there are permutations p, q in S_n such that pAq=B.

The n-th row contains n! + 1 entries corresponding to subsets of S_n of size 0 to n!.

			Triangle begins:
  [0] 1, 1;
  [1] 1, 1;
  [2] 1, 1, 1;
  [3] 1, 1, 2, 2, 2, 1, 1;
  [4] 1, 1, 4, 10, 41, 103, 309, 691, 1458, 2448, 3703, 4587, 5050, ...;

Andrew Howroyd, Table of n, a(n) for n = 0..880 (rows 0..6)
Aman Kushwaha and Raghavendra Tripathi, A note on Erdős matrices and Marcus-Ree inequality, arXiv:2503.09542 [math.MG], 2025. See p. 12.

Cf. A000041, A362763 (up to conjugation).

T(n, 1) = 1.
T(n, 2) = A000041(n) - 1.
T(n, k) = T(n, n!-k).

a(39) onwards from Andrew Howroyd, Mar 09 2025

cp's OEIS Frontend

A362764 Number of nonisomorphic 2-sets of permutations of an n-set.

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A362765 Number of nonisomorphic 3-sets of permutations of an n-set.

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A362766 Number of nonisomorphic sets of permutations of an n-set.

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A381842 Triangle read by rows: T(n,k) is the number of non-equivalent subsets of size k in S_n, 0 <= k <= n!.

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