A189232 Triangle read by rows: Number of crossing set partitions of {1,2,...,n} into k blocks.

0, 0, 0, 0, 0, 0, 0, 1, 0, 0, 0, 5, 5, 0, 0, 0, 16, 40, 15, 0, 0, 0, 42, 196, 175, 35, 0, 0, 0, 99, 770, 1211, 560, 70, 0, 0, 0, 219, 2689, 6594, 5187, 1470, 126, 0, 0, 0, 466, 8790, 31585, 37233, 17535, 3360, 210, 0, 0

Offset: 1

Author

Peter Luschny, Apr 28 2011

Keywords

• nonn
• tabl

Examples

			There are 10 crossing set partitions of {1,2,3,4,5}.
T(5,2) = card{13|245, 14|235, 24|135, 25|134, 35|124} = 5.
T(5,3) = card{1|35|24, 2|14|35, 3|14|25, 4|13|25, 5|13|24} = 5.
[1] 0
[2] 0, 0
[3] 0, 0, 0
[4] 0, 1, 0, 0
[5] 0, 5, 5, 0, 0
[6] 0, 16, 40, 15, 0, 0
[7] 0, 42, 196, 175, 35, 0, 0
[8] 0, 99, 770, 1211, 560, 70, 0, 0
		

References

• R. P. Stanley, Enumerative Combinatorics, Vol. 2, Cambridge University Press, 1999 (Exericses 6.19)

Crossrefs

Row sums A016098, A001263.

Programs

Maple

A189232 := (n,k) -> combinat[stirling2](n,k) - binomial(n,k-1)*binomial(n,k)/n:
for n from 1 to 9 do seq(A189232(n,k), k = 1..n) od;

Mathematica

T[n_, k_] := StirlingS2[n, k] - Binomial[n, k-1] Binomial[n, k]/n;
Table[T[n, k], {n, 1, 10}, {k, 1, n}] (* Jean-François Alcover, Jun 24 2019 *)

Formula

T(n,k) = S2(n,k) - C(n,k-1)*C(n,k)/n; S2(n,k) Stirling numbers of the second kind, C(n,k) binomial coefficients.