Lah number

The unsigned Lah number counts partitions of an n-element set into k nonempty linearly ordered lists. It equals \(\binom{n-1}{k-1}\frac{n!}{k!}\).

Connect

Set partition: A division of a set into disjoint, nonempty subsets whose union is the whole set. Lah numbers refine set partitions by ordering the elements within each block.
Lah number recurrence: The recurrence \(L(n,k)=L(n-1,k-1)+(n+k-1)L(n-1,k)\) for unsigned Lah numbers. A new element either forms a singleton list or enters an existing list.
Stirling number of the first kind: A signed or unsigned number counting permutations by their number of cycles. Cycles are unordered as blocks and cyclically ordered, unlike Lah lists.
Combinatorial species: A framework that describes finite labeled structures through their behavior under relabeling. The list-partition construction can be expressed as a labeled combinatorial species.
Permutation: An arrangement of distinct objects in a specified order. Concatenating the lists gives a permutation, with cuts marking the blocks.
Exponential generating function: A power series whose coefficient of \(x^n/n!\) records the n-th term of a sequence. The Lah numbers have a compact bivariate generating function encoding both indices.
Stirling number of the second kind: The number of ways to partition an n-element set into k nonempty, unordered subsets. Removing the order inside each Lah block yields this less restrictive partition count.
Normal ordering: The rearrangement of creation and annihilation operators into a standard order. Lah-number coefficients arise in normal-ordering identities for powers of bosonic operators.
Composition: An ordered sequence of positive integers whose sum is a specified integer. The list lengths form a composition of n into k parts.
Rising factorial: The product \(x(x+1)\cdots(x+n-1)\), with the empty product equal to 1. Lah numbers convert rising factorials into falling factorials.