login
The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^11^2, 1^12^2, and 1^22^1 in the pattern sense.
0

%I #13 Oct 20 2014 17:15:14

%S 2,5,12,33,108,411,1760,8287,42302,231959,1357150,8427205,55288886,

%T 381798657,2765917104,20960284309,165729739624,1364153612335,

%U 11665484410132,103448316470763,949739632313522,9013431476894667,88304011710168714,891917738589610601

%N The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^11^2, 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 2*B(n)+n-1, where B(n) is the n-th Bell number.

%e For n=2 the a(2)=5 solutions are 1^11^1, 1^21^1, 1^21^2, 1^12^1, 1^22^2.

%K nonn

%O 1,1

%A _Adam Goyt_, Mar 13 2012