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%I #28 Oct 15 2018 11:36:57
%S 1,0,1,0,1,1,0,1,2,2,0,1,4,5,5,0,1,7,12,17,15,0,1,12,29,45,64,52,0,1,
%T 20,66,121,201,265,203,0,1,33,145,336,585,966,1197,877,0,1,54,315,901,
%U 1741,3172,4971,5852,4140,0,1,88,676,2347,5375,10100,18223,27267,30751,21147
%N Number T(n,k) of set partitions of [n] where k is minimal such that for each block b the smallest integer interval containing b has at most k elements; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
%H Alois P. Heinz, <a href="/A276727/b276727.txt">Rows n = 0..20, flattened</a>
%F T(n,k) = A276719(n,k) - A276719(n,k-1) for k>0, T(n,0) = A000007(n).
%e T(4,1) = 1: 1|2|3|4.
%e T(4,2) = 4: 12|34, 12|3|4, 1|23|4, 1|2|34.
%e T(4,3) = 5: 123|4, 13|24, 13|2|4, 1|234, 1|24|3.
%e T(4,4) = 5: 1234, 124|3, 134|2, 14|23, 14|2|3.
%e T(5,4) = 17: 1234|5, 124|35, 124|3|5, 134|25, 134|2|5, 13|245, 13|25|4, 14|235, 14|23|5, 1|2345, 1|235|4, 14|25|3, 14|2|35, 14|2|3|5, 1|245|3, 1|25|34, 1|25|3|4.
%e Triangle T(n,k) begins:
%e 1;
%e 0, 1;
%e 0, 1, 1;
%e 0, 1, 2, 2;
%e 0, 1, 4, 5, 5;
%e 0, 1, 7, 12, 17, 15;
%e 0, 1, 12, 29, 45, 64, 52;
%e 0, 1, 20, 66, 121, 201, 265, 203;
%e 0, 1, 33, 145, 336, 585, 966, 1197, 877;
%p b:= proc(n, m, l) option remember; `if`(n=0, 1,
%p add(b(n-1, max(m, j), [subsop(1=NULL, l)[],
%p `if`(j<=m, 0, j)]), j={l[], m+1} minus {0}))
%p end:
%p A:= (n, k)-> `if`(n=0, 1, `if`(k<2, k, b(n, 0, [0$(k-1)]))):
%p T:= (n, k)-> A(n, k) -`if`(k=0, 0, A(n, k-1)):
%p seq(seq(T(n, k), k=0..n), n=0..12);
%t b[n_, m_, l_List] := b[n, m, l] = If[n == 0, 1, Sum[b[n - 1, Max[m, j], Append[ReplacePart[l, 1 -> Nothing], If[j <= m, 0, j]]], {j, Append[l, m + 1] ~Complement~ {0}}]]; A[n_, k_] := If[n == 0, 1, If[k < 2, k, b[n, 0, Array[0&, k - 1]]]]; T [n_, k_] := A[n, k] - If[k == 0, 0, A[n, k - 1]]; Table[T[n, k], {n, 0, 12}, { k, 0, n}] // Flatten (* _Jean-François Alcover_, Feb 04 2017, translated from Maple *)
%Y Columns k=0-10 give: A000007, A057427, A000071(n+1), A320553, A320554, A320555, A320556, A320557, A320558, A320559, A320560.
%Y Row sums give A000110.
%Y Main diagonal gives A000110(n-1) for n>0.
%Y T(2n,n) gives A276728.
%Y Cf. A263757, A276719, A276891.
%K nonn,tabl
%O 0,9
%A _Alois P. Heinz_, Sep 16 2016
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