A178979 Triangular array read by rows: T(n,k) is the number of set partitions of {1,2,...,n} in which the shortest block has length k (1 <= k <= n).

1, 1, 1, 4, 0, 1, 11, 3, 0, 1, 41, 10, 0, 0, 1, 162, 30, 10, 0, 0, 1, 715, 126, 35, 0, 0, 0, 1, 3425, 623, 56, 35, 0, 0, 0, 1, 17722, 2934, 364, 126, 0, 0, 0, 0, 1, 98253, 15165, 2220, 210, 126, 0, 0, 0, 0, 1, 580317, 86900, 10560, 330, 462, 0, 0, 0, 0, 0, 1

Offset

1,4

Comments

Row sums are Bell numbers A000110.

Column 1 is A000296 (shifted).

From Peter Luschny, Apr 05 2011: (Start)

Sum_{k>1} T(n,k) = A000296(n) count the set partitions with blocks of size > 1.

T(n,1) = A000296(n-1) count the set partitions with blocks of size = 1. Thus for the Bell numbers A000110(n) = Sum_{k>=1} T(n,k) = A000296(n-1) + A000296(n). (End)

Links

Alois P. Heinz, Rows n = 1..141, flattened

Peter Luschny, Set partitions

Formula

E.g.f. for column k: exp((exp(x) - Sum_{i=0..k-1} x^i/i!)) - exp((exp(x) - Sum_{i=0..k} x^i/i!)).

From Ludovic Schwob, Jan 15 2022: (Start)

T(2n,n) = A001700(n) = C(2n-1,n) for n>0.

T(2n-1,n-1) = A001700(n) = C(2n-1,n) for n>1. (End)

Example

T(4,2) = card ({12|34, 13|24, 14|23}) = 3. - Peter Luschny, Apr 05 2011
Triangle begins:
    1;
    1,   1;
    4,   0,  1;
   11,   3,  0,  1;
   41,  10,  0,  0,  1;
  162,  30, 10,  0,  0,  1;
  715, 126, 35,  0,  0,  0,  1;
  ...

Maple

g := k-> exp(x)*(1-(GAMMA(k, x)/GAMMA(k))); egf := k-> exp(g(k))-exp(g(k+1));
T := (n, k)-> n!*coeff(series(egf(k), x, n+1), x, n):
seq(seq(T(n, k), k=1..n), n=1..9); # Peter Luschny, Apr 05 2011
# second Maple program:
b:= proc(n, i) option remember; `if`(n=0, 1, `if`(i>n, 0,
       add(b(n-i*j, i+1) *n!/i!^j/(n-i*j)!/j!, j=0..n/i)))
    end:
T:= (n, k)-> b(n, k) -b(n, k+1):
seq(seq(T(n, k), k=1..n), n=1..12);  # Alois P. Heinz, Mar 25 2016

Mathematica

a[k_]:= Exp[x]-Sum[x^i/i!, {i, 0, k}]; Transpose[Table[Range[20]! Rest[CoefficientList[Series[Exp[a[k-1]]-Exp[a[k]], {x, 0, 20}], x]], {k, 1, 9}]]//Grid

Crossrefs

Cf. A080510, A145877, A000110, A000296.

Sequence in context: A059056 A344393 A127153 * A228270 A335748 A266488

Adjacent sequences: A178976 A178977 A178978 * A178980 A178981 A178982

Keyword

nonn,tabl

Author

Geoffrey Critzer, Jan 02 2011

Status

approved