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%I #17 Apr 02 2016 05:01:16
%S 1,1,1,2,1,1,4,5,4,1,1,8,19,42,61,56,28,8,1,1,16,65,304,1129,3200,
%T 6775,10680,12600,11386,8002,4368,1820,560,120,16,1,1,32,211,1890,
%U 14935,97470
%N T(n, k) is the number of k-element connected subposets of the n-th Boolean lattice, 0 <= k <= 2^n.
%C The n-th Boolean lattice is the set of all subsets of {1,2,...,n}, partially ordered by inclusion.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/BooleanAlgebra.html">Boolean Algebra</a>.
%e The triangle begins:
%e n\k 0 1 2 3 4 5 6 7 8 9 10 11 12 13 14
%e 0 1 1
%e 1 1 2 1
%e 2 1 4 5 4 1
%e 3 1 8 19 42 61 56 28 8 1
%e 4 1 16 65 304 1129 3200 6775 10680 12600 11386 8002 4368 1820 560 120
%e 5 1 32 211 1890 14935 97470 ...
%e For T(2, 2) = 5: [{},{1}], [{},{2}], [{},{1,2}], [{1},{1,2}], [{2},{1,2}].
%o (Sage)
%o def ConnectedSubs(n): # Returns row n of T(n, k).
%o Bn = posets.BooleanLattice(n)
%o counts = [0]*(2^n+1)
%o for X in Subsets(range(2^n)):
%o if Bn.subposet(X).is_connected():
%o counts[len(X)] += 1
%o return counts
%Y Columns: A000012 (k = 0, 2^n), A000079 (k = 1, 2^n - 1), A001047 (k = 3).
%K nonn,more,tabf
%O 0,4
%A _Danny Rorabaugh_, Mar 26 2016
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