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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

Table of n, a(n) for n=1..24.

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: A190861 A071721 A125306 * A064310 A126983 A104629

Adjacent sequences:  A209794 A209795 A209796 * A209798 A209799 A209800

KEYWORD

nonn

AUTHOR

Adam Goyt, Mar 13 2012

STATUS

approved

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Last modified March 20 07:42 EDT 2019. Contains 321345 sequences. (Running on oeis4.)