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A209797 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. 0
2, 6, 18, 56, 188, 695, 2838, 12726, 62140, 327760, 1854488, 11189273, 71627546, 484332314, 3446042310, 25712613664, 200599911596, 1632055365951, 13814906940846, 121414108567114, 1105838412755384, 10420517690466168, 101439025287805552, 1018689421191417393 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
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.
LINKS
FORMULA
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)
EXAMPLE
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.
CROSSREFS
Sequence in context: A071721 A125306 A352076 * A064310 A126983 A104629
KEYWORD
nonn
AUTHOR
Adam Goyt, Mar 13 2012
STATUS
approved

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Last modified July 22 06:32 EDT 2024. Contains 374481 sequences. (Running on oeis4.)