A327869 Sum T(n,k) of multinomials M(n; lambda), where lambda ranges over all partitions of n into distinct parts incorporating k; triangle T(n,k), n>=0, 0<=k<=n, read by rows.

1, 1, 1, 1, 0, 1, 4, 3, 3, 1, 5, 4, 0, 4, 1, 16, 5, 10, 10, 5, 1, 82, 66, 75, 60, 15, 6, 1, 169, 112, 126, 35, 140, 21, 7, 1, 541, 456, 196, 336, 280, 224, 28, 8, 1, 2272, 765, 1548, 1848, 1386, 630, 336, 36, 9, 1, 17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1

Offset: 0

Author

Alois P. Heinz, Sep 28 2019

Keywords

• nonn
• tabl

Comments

Here we assume that every list of parts has at least one 0 because its addition does not change the value of the multinomial.

Number T(n,k) of set partitions of [n] with distinct block sizes and one of the block sizes is k. T(5,3) = 10: 123|45, 124|35, 125|34, 12|345, 134|25, 135|24, 13|245, 145|23, 14|235, 15|234.

Examples

			Triangle T(n,k) begins:
      1;
      1,     1;
      1,     0,     1;
      4,     3,     3,     1;
      5,     4,     0,     4,     1;
     16,     5,    10,    10,     5,    1;
     82,    66,    75,    60,    15,    6,    1;
    169,   112,   126,    35,   140,   21,    7,   1;
    541,   456,   196,   336,   280,  224,   28,   8,  1;
   2272,   765,  1548,  1848,  1386,  630,  336,  36,  9,  1;
  17966, 15070, 15525, 16080, 14070, 3780, 1050, 480, 45, 10, 1;
  ...
		

Links

• Alois P. Heinz, Rows n = 0..140, flattened
• Wikipedia, Multinomial coefficients
• Wikipedia, Partition (number theory)
• Wikipedia, Partition of a set

Crossrefs

Columns k=0-3 give: A007837, A327876, A327881, A328155.

Row sums give A327870.

T(2n,n) gives A328156.

Cf. A327801, A327884.

Programs

Maple

with(combinat):
T:= (n, k)-> add(multinomial(add(i, i=l), l[], 0),
             l=select(x-> nops(x)=nops({x[]}) and
             (k=0 or k in x), partition(n))):
seq(seq(T(n, k), k=0..n), n=0..11);
# second Maple program:
b:= proc(n, i, k) option remember; `if`(i*(i+1)/2 n!*(b(n$2, 0)-`if`(k=0, 0, b(n$2, k))):
seq(seq(T(n, k), k=0..n), n=0..11);

Mathematica

b[n_, i_, k_] := b[n, i, k] = If[i(i+1)/2 < n, 0, If[n==0, 1, If[i<2, 0, b[n, i-1, If[i==k, 0, k]]] + If[i==k, 0, b[n-i, Min[n-i, i-1], k]/i!]]];
T[n_, k_] := n! (b[n, n, 0] - If[k == 0, 0, b[n, n, k]]);
Table[T[n, k], {n, 0, 11}, {k, 0, n}] // Flatten (* Jean-François Alcover, Apr 28 2020, from 2nd Maple program *)