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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!.

%I A381842 #65 Apr 09 2025 11:21:13
%S A381842 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,
%T A381842 4587,5050,4587,3703,2448,1458,691,309,103,41,10,4,1,1,1,1,6,37,715,
%U A381842 13710,256751,4140666,58402198,726296995,8060937770,80604620206,732149722382
%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!.
%C A381842 We say two subsets A, B of size k are equivalent if there are permutations p, q in S_n such that pAq=B.
%C A381842 The n-th row contains n! + 1 entries corresponding to subsets of S_n of size 0 to n!.
%H A381842 Andrew Howroyd, <a href="/A381842/b381842.txt">Table of n, a(n) for n = 0..880</a> (rows 0..6)
%H A381842 Aman Kushwaha and Raghavendra Tripathi, <a href="https://arxiv.org/abs/2503.09542">A note on Erdős matrices and Marcus-Ree inequality</a>, arXiv:2503.09542 [math.MG], 2025. See p. 12.
%F A381842 T(n, 1) = 1.
%F A381842 T(n, 2) = A000041(n) - 1.
%F A381842 T(n, k) = T(n, n!-k).
%e A381842 Triangle begins:
%e A381842   [0] 1, 1;
%e A381842   [1] 1, 1;
%e A381842   [2] 1, 1, 1;
%e A381842   [3] 1, 1, 2, 2, 2, 1, 1;
%e A381842   [4] 1, 1, 4, 10, 41, 103, 309, 691, 1458, 2448, 3703, 4587, 5050, ...;
%Y A381842 Cf. A000041, A362763 (up to conjugation).
%K A381842 nonn,tabf
%O A381842 0,10
%A A381842 _Raghavendra Tripathi_, Mar 09 2025
%E A381842 a(39) onwards from _Andrew Howroyd_, Mar 09 2025