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A379712 Triangle read by rows: T(n,k) is the number of nonempty labeled antichains of subsets of [n] such that the largest subset is of size k.

%I A379712 #14 Jan 06 2025 06:31:49
%S A379712 1,1,1,1,3,1,1,7,10,1,1,15,97,53,1,1,31,1418,5443,686,1,1,63,40005,
%T A379712 3701128,4043864,43291,1
%N A379712 Triangle read by rows: T(n,k) is the number of nonempty labeled antichains of subsets of [n] such that the largest subset is of size k.
%H A379712 John Tyler Rascoe, <a href="/A379712/a379712.py.txt">Python program</a>.
%e A379712 Triangle begins:
%e A379712    k=0   1     2     3    4  5
%e A379712  n=0 1;
%e A379712  n=1 1,  1;
%e A379712  n=2 1,  3,    1;
%e A379712  n=3 1,  7,   10,    1;
%e A379712  n=4 1, 15,   97,   53,   1;
%e A379712  n=5 1, 31, 1418, 5443, 686, 1;
%e A379712  ...
%e A379712 T(3,0) =  1: {{}}.
%e A379712 T(3,1) =  7: {{1}}, {{2}}, {{3}}, {{1},{2}}, {{1},{3}}, {{2},{3}}, {{1},{2},{3}}.
%e A379712 T(3,2) = 10: {{1,2}}, {{1,3}}, {{2,3}}, {{1},{23}}, {{2},{13}}, {{3},{12}}, {{12},{13}}, {{12},{23}}, {{13},{23}}, {{12},{13},{23}}.
%e A379712 T(3,3) =  1: {{1,2,3}}.
%o A379712 (Python) # see links
%Y A379712 Cf. (column k=1) A000225, A000372, (row sums) A014466, A327806, (column k=2) A379706.
%K A379712 nonn,tabl,more,hard
%O A379712 0,5
%A A379712 _John Tyler Rascoe_, Dec 30 2024