A372762 Number T(n,k) of partitions of [n] having exactly k blocks of minimal size; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 5, 9, 0, 1, 0, 31, 10, 10, 0, 1, 0, 82, 70, 35, 15, 0, 1, 0, 344, 336, 140, 35, 21, 0, 1, 0, 1661, 1393, 616, 385, 56, 28, 0, 1, 0, 7942, 6210, 4984, 1386, 504, 84, 36, 0, 1, 0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1
Offset: 0
Examples
T(5,1) = 31: 12345, 1234|5, 1235|4, 123|45, 1245|3, 124|35, 125|34, 12|345, 12|34|5, 12|35|4, 12|3|45, 1345|2, 134|25, 135|24, 13|245, 13|24|5, 13|25|4, 13|2|45, 145|23, 14|235, 14|23|5, 15|234, 1|2345, 15|23|4, 1|23|45, 14|25|3, 14|2|35, 15|24|3, 1|24|35, 15|2|34, 1|25|34. T(5,2) = 10: 123|4|5, 124|3|5, 125|3|4, 134|2|5, 135|2|4, 1|234|5, 1|235|4, 145|2|3, 1|245|3, 1|2|345. T(5,3) = 10: 12|3|4|5, 13|2|4|5, 1|23|4|5, 14|2|3|5, 1|24|3|5, 1|2|34|5, 15|2|3|4, 1|25|3|4, 1|2|35|4, 1|2|3|45. T(5,4) = 0. T(5,5) = 1: 1|2|3|4|5. Triangle T(n,k) begins: 1; 0, 1; 0, 1, 1; 0, 4, 0, 1; 0, 5, 9, 0, 1; 0, 31, 10, 10, 0, 1; 0, 82, 70, 35, 15, 0, 1; 0, 344, 336, 140, 35, 21, 0, 1; 0, 1661, 1393, 616, 385, 56, 28, 0, 1; 0, 7942, 6210, 4984, 1386, 504, 84, 36, 0, 1; 0, 38721, 41331, 22590, 8610, 3717, 840, 120, 45, 0, 1; ...
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- Wikipedia, Partition of a set
Crossrefs
Programs
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Maple
b:= proc(n, m, t) option remember; `if`(n=0, x^t, add(binomial(n-1, j-1)*b(n-j, min(j, m), `if`(j
Formula
Sum_{k=0..n} k * T(n,k) = A372650(n).