A330759 Number T(n,k) of set partitions into k blocks of strict integer partitions of n; triangle T(n,k), n>=0, 0<=k<=A003056(n), read by rows.
1, 0, 1, 0, 1, 0, 2, 1, 0, 2, 1, 0, 3, 2, 0, 4, 5, 1, 0, 5, 6, 1, 0, 6, 9, 2, 0, 8, 13, 3, 0, 10, 23, 10, 1, 0, 12, 27, 11, 1, 0, 15, 40, 19, 2, 0, 18, 51, 26, 3, 0, 22, 71, 40, 5, 0, 27, 100, 73, 16, 1, 0, 32, 127, 93, 19, 1, 0, 38, 163, 132, 31, 2, 0, 46, 215, 184, 45, 3
Offset: 0
Examples
T(10,1) = 10: (10), 1234, 127, 136, 145, 19, 235, 28, 37, 46. T(10,2) = 23: 123|4, 124|3, 12|34, 12|7, 134|2, 13|24, 13|6, 14|23, 14|5, 15|4, 16|3, 17|2, 1|234, 1|27, 1|36, 1|45, 1|9, 23|5, 25|3, 2|35, 2|8, 3|7, 4|6. T(10,3) = 10: 12|3|4, 13|2|4, 14|2|3, 1|23|4, 1|24|3, 1|2|34, 1|2|7, 1|3|6, 1|4|5, 2|3|5. T(10,4) = 1: 1|2|3|4. Triangle T(n,k) begins: 1; 0, 1; 0, 1; 0, 2, 1; 0, 2, 1; 0, 3, 2; 0, 4, 5, 1; 0, 5, 6, 1; 0, 6, 9, 2; 0, 8, 13, 3; 0, 10, 23, 10, 1; 0, 12, 27, 11, 1; 0, 15, 40, 19, 2; 0, 18, 51, 26, 3; 0, 22, 71, 40, 5; 0, 27, 100, 73, 16, 1; ...
Links
- Alois P. Heinz, Rows n = 0..1000, flattened
- Wikipedia, Partition of a set
Crossrefs
Programs
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Maple
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2
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Mathematica
b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n == 0, x^k, b[n, i-1, k] + With[{t = Min[n-i, i-1]}, b[n-i, t, k]*k + b[n-i, t, k+1]]]]; T[n_] := CoefficientList[b[n, n, 0], x]; T /@ Range[0, 20] // Flatten (* Jean-François Alcover, Mar 12 2021, after Alois P. Heinz *)