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

0,8

Comments

At least one block length occurs exactly k times, and all block lengths occur at least k times.

Links

Alois P. Heinz, Rows n = 0..140, flattened

Wikipedia, Partition of a set

Example

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;

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 *)

Crossrefs

Columns k=0-10 give A000007, A271426, A271762, A271763, A271764, A271765, A271766, A271767, A271768, A271769, A271770.

Row sums give A000110.

Main diagonal gives A000012.

T(2n,n) gives A001147.

T(3n,n) gives A271715.

Cf. A271423.

Sequence in context: A363973 A046781 A244530 * A117435 A282252 A268367

Adjacent sequences: A271421 A271422 A271423 * A271425 A271426 A271427

Keyword

nonn,tabl

Author

Alois P. Heinz, Apr 07 2016

Status

approved