A271424 Number T(n,k) of set partitions of [n] with minimal block length multiplicity equal to k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.
1, 0, 1, 0, 1, 1, 0, 4, 0, 1, 0, 11, 3, 0, 1, 0, 51, 0, 0, 0, 1, 0, 132, 55, 15, 0, 0, 1, 0, 771, 105, 0, 0, 0, 0, 1, 0, 3089, 945, 0, 105, 0, 0, 0, 1, 0, 18388, 1218, 1540, 0, 0, 0, 0, 0, 1, 0, 96423, 15456, 3150, 0, 945, 0, 0, 0, 0, 1, 0, 627529, 26785, 24255, 0, 0, 0, 0, 0, 0, 0, 1
Offset: 0
Examples
T(4,1) = 11: 1234, 123|4, 124|3, 12|3|4, 134|2, 13|2|4, 1|234, 1|23|4, 14|2|3, 1|24|3, 1|2|34. T(4,2) = 3: 12|34, 13|24, 14|23. T(4,4) = 1: 1|2|3|4. T(6,3) = 15: 12|34|56, 12|35|46, 12|36|45, 13|24|56, 13|25|46, 13|26|45, 14|23|56, 15|23|46, 16|23|45, 14|25|36, 14|26|35, 15|24|36, 16|24|35, 15|26|34, 16|25|34. Triangle T(n,k) begins: 1; 0, 1; 0, 1, 1; 0, 4, 0, 1; 0, 11, 3, 0, 1; 0, 51, 0, 0, 0, 1; 0, 132, 55, 15, 0, 0, 1; 0, 771, 105, 0, 0, 0, 0, 1; 0, 3089, 945, 0, 105, 0, 0, 0, 1; 0, 18388, 1218, 1540, 0, 0, 0, 0, 0, 1; 0, 96423, 15456, 3150, 0, 945, 0, 0, 0, 0, 1;
Links
- Alois P. Heinz, Rows n = 0..140, flattened
- Wikipedia, Partition of a set
Crossrefs
Programs
-
Maple
with(combinat): b:= proc(n, i, k) option remember; `if`(n=0, 1, `if`(i<1, 0, add(multinomial(n, n-i*j, i$j) *b(n-i*j, i-1, k)/j!, j={0, $k..n/i}))) end: T:= (n, k)-> b(n$2, k)-`if`(n=k,0,b(n$2, k+1)): seq(seq(T(n, k), k=0..n), n=0..12); -
Mathematica
multinomial[n_, k_List] := n!/Times @@ (k!); b[n_, i_, k_] := b[n, i, k] = If[n == 0, 1, If[i<1, 0, Sum[multinomial[n, Join[{n-i*j}, Array[i&, j]]]* b[n-i*j, i-1, k]/j!, {j, Join[{0}, Range[k, n/i]] // Union}]]]; T[n_, k_] := b[n, n, k] - If[n == k, 0, b[n, n, k + 1]]; Table[T[n, k], {n, 0, 12}, {k, 0, n}] // Flatten (* Jean-François Alcover, Jan 16 2017, adapted from Maple *)
Comments