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%I #20 Oct 20 2014 17:15:14
%S 2,6,16,44,134,468,1880,8534,42804,232972,1359186,8431288,55297064,
%T 381815026,2765949856,20960349828,165729870678,1364153874460,
%U 11665484934400,103448317519318,949739634410652,9013431481088948,88304011718557298,891917738606387792
%N The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^12^2 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.
%H Adam M. Goyt and Lara K. Pudwell, <a href="http://arxiv.org/abs/1203.3786">Avoiding colored partitions of two elements in the pattern sense</a>, arXiv preprint arXiv:1203.3786, 2012. - From _N. J. A. Sloane_, Sep 17 2012
%F a(n) = 2^n + 2*(B(n)-1), where B(n) is the n-th Bell number.
%e For n=2 the a(2)=6 solutions are 1^11^1, 1^11^2, 1^21^1, 1^21^2, 1^12^1, 1^22^2.
%K nonn
%O 1,1
%A _Adam Goyt_, Mar 13 2012
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