Exact Lah Number Calculator

Evaluate the unsigned Lah closed form L(n,k) = C(n-1,k-1)n!/k!, with the empty identity L(0,0) = 1. The empty case is defined separately, while positive n requires at least one block. Inputs must be whole numbers inside the displayed domain; the page never rounds a decimal into an accepted index. This ties the answer to one explicit convention instead of silently mixing sequence offsets or combinatorial interpretations.

A Lah number orders elements inside each block; an ordinary set-partition count does not. The displayed binomial and factorial factors provide a direct independent check. Recompute the displayed recurrence or closed form with the same inputs and compare its previous terms or counting factors. That check supports this bounded result, but it does not transfer the interpretation to a different sequence or counting object.

n is capped at 17 so all accepted Lah values fit exact safe-integer storage. Lah values grow faster than their small indices suggest. Combinatorial and recurrence values can grow rapidly even when the inputs look small. The conservative cap prevents browser stalls and avoids presenting an unsafe floating-point integer as exact; larger work needs arbitrary-precision software and independent resource controls.