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%I #15 Feb 05 2017 06:43:50
%S 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 919,299,63,8,1,5285,8003,5262,2011,492,84,9,1,24583,41871,31180,
%U 13611,3857,754,108,10,1,123062,231474,191889,94020,30128,6755,1095,135,11,1
%N 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 Alois P. Heinz, <a href="/A271206/b271206.txt">Rows n = 0..100, flattened</a>
%H Wikipedia, <a href="https://en.wikipedia.org/wiki/Partition_of_a_set">Partition of a set</a>
%e T(3,1) = 1: 1|2|3.
%e T(4,1) = 4: 12|3|4, 14|2|3, 1|24|3, 1|2|34.
%e 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 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 T(5,3) = 1: 1|2|3|4|5.
%e Triangle T(n,k) begins:
%e : 0 : 1;
%e : 1 : 1;
%e : 2 : 2;
%e : 3 : 4, 1;
%e : 4 : 10, 4, 1;
%e : 5 : 28, 18, 5, 1;
%e : 6 : 89, 77, 30, 6, 1;
%e : 7 : 315, 345, 164, 45, 7, 1;
%e : 8 : 1233, 1617, 919, 299, 63, 8, 1;
%e : 9 : 5285, 8003, 5262, 2011, 492, 84, 9, 1;
%e : 10 : 24583, 41871, 31180, 13611, 3857, 754, 108, 10, 1;
%p b:= proc(n, i, t, m) option remember; expand(`if`(n=0, 1, add((v->
%p `if`(t and v, x, 1)*b(n-1, j, v, max(m, j)))(j=i+1), j=1..m+1)))
%p end:
%p T:= n-> (p-> seq(coeff(p, x, i), i=0..degree(p)))(b(n, 1, false, 0)):
%p seq(T(n), n=0..14);
%t 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 Column k=0 gives A271207.
%Y Row sums give A000110.
%Y Cf. A185982.
%K nonn,tabf
%O 0,3
%A _Alois P. Heinz_, Apr 01 2016
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