A209797 - OEIS

%I #7 Mar 31 2012 10:33:22

%S 2,6,18,56,188,695,2838,12726,62140,327760,1854488,11189273,71627546,

%T 484332314,3446042310,25712613664,200599911596,1632055365951,

%U 13814906940846,121414108567114,1105838412755384,10420517690466168,101439025287805552,1018689421191417393

%N The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^11^2 and 1^22^1 in the pattern sense.

%C A partition of the set [n] is a family nonempty disjoint sets whose union is [n].  The blocks are written in order of increasing minima.  A partition of the set [n] can be written as a word p=p_1p_2...p_n where p_i=j if element i is in block j.  A partition q=q_1q_2...q_n contains partition p=p_1p_2...p_k if there is a subword q_{i_1}q_{i_2}...q_{i_k} such that q_{i_a}<q_{i_b} whenever p_a<p_b, these words are called order isomorphic.  A colored partition q contains the colored partition p in the pattern sense if there is a copy of the uncolored partition p in the uncolored partition q, and the colors on this copy of p are order isomorphic to the colors on p, otherwise we say q avoids p in the pattern sense.

%F For n >=2, 2*B(n)+B(n-1)+sum(sum(B(n-j-k), k = 0 .. n-j), j = 2 .. n)+sum(B(j-1)*(B(n-j)+sum((k+binomial(n-j, k))*B(n-j-k), k = 1 .. n-j)), j = 2 .. n-1)

%e For n=2 the a(2)=6 solutions are 1^11^1, 1^21^1, 1^21^2, 1^12^1, 1^12^2, 1^22^2.

%K nonn

%O 1,1

%A _Adam Goyt_, Mar 13 2012