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A271206 Number T(n,k) of set partitions of [n] having exactly k triples (t,t+1,t+2) such that t+i is in block b+i for some b; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.

%I A271206 #15 Feb 05 2017 06:43:50
%S A271206 1,1,2,4,1,10,4,1,28,18,5,1,89,77,30,6,1,315,345,164,45,7,1,1233,1617,
%T A271206 919,299,63,8,1,5285,8003,5262,2011,492,84,9,1,24583,41871,31180,
%U A271206 13611,3857,754,108,10,1,123062,231474,191889,94020,30128,6755,1095,135,11,1
%N A271206 Number T(n,k) of set partitions of [n] having exactly k triples (t,t+1,t+2) such that t+i is in block b+i for some b; triangle T(n,k), n>=0, 0<=k<=max(0,n-2), read by rows.
%H A271206 Alois P. Heinz, <a href="/A271206/b271206.txt">Rows n = 0..100, flattened</a>
%H A271206 Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e A271206 T(3,1) = 1: 1|2|3.
%e A271206 T(4,1) = 4: 12|3|4, 14|2|3, 1|24|3, 1|2|34.
%e A271206 T(5,1) = 18: 123|4|5, 125|3|4, 12|35|4, 12|3|45, 13|24|5, 1|23|4|5, 145|2|3, 14|25|3, 14|2|35, 14|2|3|5, 15|24|3, 1|245|3, 1|24|35, 1|24|3|5, 15|2|34, 1|25|34, 1|2|345, 1|2|34|5.
%e A271206 T(5,2) = 5: 12|3|4|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45.
%e A271206 T(5,3) = 1: 1|2|3|4|5.
%e A271206 Triangle T(n,k) begins:
%e A271206 :  0 :     1;
%e A271206 :  1 :     1;
%e A271206 :  2 :     2;
%e A271206 :  3 :     4,     1;
%e A271206 :  4 :    10,     4,     1;
%e A271206 :  5 :    28,    18,     5,     1;
%e A271206 :  6 :    89,    77,    30,     6,    1;
%e A271206 :  7 :   315,   345,   164,    45,    7,   1;
%e A271206 :  8 :  1233,  1617,   919,   299,   63,   8,   1;
%e A271206 :  9 :  5285,  8003,  5262,  2011,  492,  84,   9,  1;
%e A271206 : 10 : 24583, 41871, 31180, 13611, 3857, 754, 108, 10, 1;
%p A271206 b:= proc(n, i, t, m) option remember; expand(`if`(n=0, 1, add((v->
%p A271206      `if`(t and v, x, 1)*b(n-1, j, v, max(m, j)))(j=i+1), j=1..m+1)))
%p A271206     end:
%p A271206 T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1, false, 0)):
%p A271206 seq(T(n), n=0..14);
%t A271206 b[n_, i_, t_, m_] := b[n, i, t, m] = Expand[If[n==0, 1, Sum[Function[v, If[t && v, x, 1]*b[n-1, j, v, Max[m, j]]][j==i+1], {j, 1, m+1}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, 1, False, 0]]; Table[T[n], {n, 0, 14}] // Flatten (* _Jean-François Alcover_, Feb 05 2017, translated from Maple *)
%Y A271206 Column k=0 gives A271207.
%Y A271206 Row sums give A000110.
%Y A271206 Cf. A185982.
%K A271206 nonn,tabf
%O A271206 0,3
%A A271206 _Alois P. Heinz_, Apr 01 2016