Calculators
Lah Number Calculator
Calculate exact unsigned Lah numbers for up to 200 distinct items and ordered lists privately in your browser.
Count partitions into ordered lists.
The unsigned Lah number L(n, k) counts ways to partition n distinct items into k nonempty ordered lists. Order within each list matters; the collection of lists is unordered. For three items and two lists, the six possibilities are {AB, C}, {BA, C}, {AC, B}, {CA, B}, {BC, A} and {CB, A}. For 1 ≤ k ≤ n, L(n, k) = C(n − 1, k − 1) × n! / k!. For example L(4, 2) = 3 × 24 / 2 = 36. By convention L(0, 0) = 1; k = 0 for positive n or k > n gives zero. Unlike Stirling numbers of the second kind, ordering inside each group matters. These are unsigned Lah numbers, not signed Lah numbers. Enter n and k from 0 to 200 using one to three digits. Leading zeros and surrounding whitespace are accepted; signs, decimals, separators and scientific notation are rejected. Browser BigInt arithmetic preserves every digit, and zero counts as one digit. Inputs are not uploaded or stored.
Frequently Asked Questions
Everything you need to know about this tool, how it works, and privacy.
What does a Lah number count?
Unsigned Lah numbers count partitions of distinct items into nonempty lists. Items within each list are ordered, while the lists themselves are unordered. L(3, 2) = 6 and L(4, 2) = 36.
How is the result calculated?
For 1 ≤ k ≤ n, use L(n, k) = C(n − 1, k − 1) × n! / k!. Exact BigInt products and divisions avoid floating-point rounding.
What happens with zero or too many lists?
L(0, 0) = 1 represents the empty arrangement. Positive n with zero lists, or k greater than n, gives zero.
Which inputs are supported?
Enter n and k from 0 to 200 using digits only. Leading zeros and surrounding whitespace are accepted. Results are exact and processing stays in your browser.