A270952 - OEIS

%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