A085483 Triangle read by rows: S_B(n,k) = "Type B" Stirling numbers of the second kind.

2, 2, 5, 2, 15, 14, 2, 35, 84, 43, 2, 75, 350, 430, 142, 2, 155, 1260, 2795, 2130, 499, 2, 315, 4214, 15050, 19880, 10479, 1850, 2, 635, 13524, 73143, 149100, 132734, 51800, 7193, 2, 1275, 42350, 334110, 987042, 1320354, 854700, 258948, 29186, 2, 2555, 130620, 1466515, 6038550, 11390673, 10878000, 5394750, 1313370, 123109

Offset: 1

Author

James East, Aug 15 2003

Keywords

• nonn
• tabl

Examples

			S_B(2,2)=5 because the relevant partitions of {-2,-1,1,2} are: {-2|-1|1|2}, {-1,1|-2|2}, {-1|1|-2,2}, {-1,1|-2,2}, {1,-2|-1,2}.
Triangle begins:
  2;
  2,   5;
  2,  15,   14;
  2,  35,   84,   43;
  2,  75,  350,  430,  142;
  2, 155, 1260, 2795, 2130, 499;
  ...
		

Links

• Michael De Vlieger, Table of n, a(n) for n = 1..11325 (rows n = 1..150, flattened)
• Eli Bagno and David Garber, Combinatorics of q,r-analogues of Stirling numbers of type B, arXiv:2401.08365 [math.CO], 2024. See page 4.
• Takao Komatsu, Eli Bagno, and David Garber, A q,r-analogue of poly-Stirling numbers of second kind with combinatorial applications, arXiv:2209.06674 [math.CO], 2022.

Crossrefs

Cf. A008277, A005425.

S_B(n, 1) + ... + S_B(n, n) = A002872(n).

Programs

Mathematica

nn = 10; f[n_] := Sum[2^(n - 3 k) n!/((n - 2 k)! k!), {k, 0, n}]; Do[f[n], {n, 0, nn}]; Table[f[k]*StirlingS2[n, k], {n, nn}, {k, n}] (* Michael De Vlieger, Sep 21 2022, after Robert G. Wilson v at A005425 *)

Formula

A partition of {-n, ..., -1, 1, ..., n} into nonempty subsets X_1, ..., X_r is called "symmetric" if for each i -X_i = X_j for some j. S_B(n, k) is the number of such symmetric partitions whose induced partition on {1, ..., n} involves k nonempty subsets. S_B(n, k) = S(n, k) * a(k), where S(n, k) is A008277 and a(k) is A005425.