Stirling Numbers of the Second Kind Table

Enter a maximum order N to generate the triangular table of Stirling numbers of the second kind S(n,k) from n=0 to N, with the row sums (Bell numbers) shown alongside.

Input

Enter a maximum order N to generate the triangular table of Stirling numbers of the second kind S(n,k) from n=0 to N. The sum of each row (the Bell number) is shown too.

Maximum order N

Enter an integer from 0 to 30.

Result

Bell number B(8) (sum of row n=8)

4,140

Triangle of Stirling numbers of the second kind S(n,k)

n / k	k=0	k=1	k=2	k=3	k=4	k=5	k=6	k=7	k=8	Bell
0	1									1
1	0	1								1
2	0	1	1							2
3	0	1	3	1						5
4	0	1	7	6	1					15
5	0	1	15	25	10	1				52
6	0	1	31	90	65	15	1			203
7	0	1	63	301	350	140	21	1		877
8	0	1	127	966	1,701	1,050	266	28	1	4,140

The rightmost cell of each row is the Bell number, equal to that row sum. Cells where k exceeds n are blank because no such partition exists.

How it works

The Stirling number of the second kind S(n,k) counts the number of ways to partition a set of n distinguishable elements into k non-empty, unlabeled subsets. It is a fundamental quantity in combinatorics and set partition theory.
This tool fills the lower-triangular table using the recurrence S(n,k) = k * S(n-1,k) + S(n-1,k-1), starting from S(0,0)=1 and setting S(n,0)=0 for n greater than or equal to 1.
Every value is computed with BigInt (arbitrary-precision integers), so even at higher orders there is no overflow or rounding error and the results are exact integers.
The sum of each row, S(n,0)+S(n,1)+…+S(n,n), equals the Bell number B(n), which is the total number of partitions of an n-element set. The Bell number appears in the rightmost column of the table.
The maximum order N must be between 0 and 30 inclusive. Columns are indexed by k and rows by n; cells where k exceeds n are left blank because no such partition exists.

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