Lah numbers L(n,k) = C(n-1,k-1)·n!/k! = n!/(k!)·C(n-1,k-1). They count partitions of [n] into k ordered lists. L(3,2)=6: {(1),(2,3)}, {(1),(3,2)}, {(2),(1,3)}, {(2),(3,1)}, {(3),(1,2)}, {(3),(2,1)}.
How to Use
Enter n and k:
- L(n,k): Value
- Triangle: All values
- Row sum: Fubini connection
Factorial Connection
x^{(n)} = Σ L(n,k)·(x)_k converts rising factorial to falling factorial. L(n,k) is the 'change of basis' matrix between these two polynomial bases. This is why Lah numbers are also called 'transition coefficients'.
Stirling Relation
L(n,k) = Σ_j s(n,j)·S(j,k) where s=Stirling first kind, S=Stirling second kind. Equivalently, the Lah matrix = (unsigned Stirling 1st)×(Stirling 2nd). Beautiful matrix factorization!
Step-by-Step Instructions
- 1Enter n, k.
- 2Compute L(n,k).
- 3View triangle.
- 4See factorial link.
- 5Check row sum.