A209798 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.

2, 5, 12, 33, 108, 411, 1760, 8287, 42302, 231959, 1357150, 8427205, 55288886, 381798657, 2765917104, 20960284309, 165729739624, 1364153612335, 11665484410132, 103448316470763, 949739632313522, 9013431476894667, 88304011710168714, 891917738589610601

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.

Adam M. Goyt and Lara K. Pudwell, Avoiding colored partitions of two elements in the pattern sense, arXiv preprint arXiv:1203.3786, 2012. - From N. J. A. Sloane, Sep 17 2012

Formula

2*B(n)+n-1, where B(n) is the n-th Bell number.

Example

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

Crossrefs

Sequence in context: A209051 A209216 A076864 * A196545 A032292 A151408

Adjacent sequences: A209795 A209796 A209797 * A209799 A209800 A209801

Keyword

nonn

Author

Adam Goyt, Mar 13 2012

Status

approved