cp's OEIS Frontend

A208275 The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^11^1 and 1^22^1 in the pattern sense.

%I A208275 #27 Oct 08 2018 08:45:48
%S A208275 2,5,10,21,46,107,262,675,1818,5105,14882,44929,140070,450055,1487294,
%T A208275 5047327,17562546,62578845,228062522,849213293,3227667742,12511072803,
%U A208275 49417391350,198758992859,813460577482,3385607683977,14320923895890,61532392279385
%N A208275 The number of partitions of the set [n] where each element can be colored 1 or 2 avoiding the patterns 1^11^1 and 1^22^1 in the pattern sense.
%C A208275 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 A208275 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 A208275 sum(sum(binomial(i-1, j)*binomial(n-i, j)*j!, j = 0 .. min(i-1, n-i)), i = 1 .. n)+sum(sum((i-1)*binomial(i-2, j)*binomial(n-i, j)*j!, j = 0 .. min(i-2, n-i)), i = 2 .. n)+sum(sum(binomial(i-1, j)*binomial(n-i-1, j)*j!, j = 0 .. min(i-1, n-i-1)), i = 1 .. n-1)+1
%e A208275 For n=2 the a(2)=5 solutions are 1^11^2, 1^21^1, 1^12^1, 1^12^2, 1^22^2.
%t A208275 a[n_] := With[{B = Binomial},
%t A208275   Sum[B[i-1, j] B[n-i, j] j!, {i, 1, n}, {j, 0, Min[i-1, n-i]}] +
%t A208275   Sum[B[i-2, j] B[n-i, j] (i-1) j!, {i, 2, n}, {j, 0, Min[i-2, n-i]}] +
%t A208275   Sum[B[i-1, j] B[n-i-1, j] j!, {i, 1, n-1}, {j, 0, Min[i-1, n-i-1]}] + 1
%t A208275 ];
%t A208275 Array[a, 28] (* _Jean-François Alcover_, Oct 08 2018 *)
%K A208275 nonn
%O A208275 1,1
%A A208275 _Adam Goyt_, Mar 12 2012