Triangles of numbers | Integer sequences | Factorial and binomial topics
Lah number
In mathematics, the Lah numbers, discovered by Ivo Lah in 1954, are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the th derivatives of . Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of n elements can be partitioned into k nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers. Unsigned Lah numbers (sequence in the OEIS): Signed Lah numbers (sequence in the OEIS): L(n, 1) is always n!; in the interpretation above, the only partition of {1, 2, 3} into 1 set can have its set ordered in 6 ways: {(1, 2, 3)}, {(1, 3, 2)}, {(2, 1, 3)}, {(2, 3, 1)}, {(3, 1, 2)} or {(3, 2, 1)} L(3, 2) corresponds to the 6 partitions with two ordered parts: {(1), (2, 3)}, {(1), (3, 2)}, {(2), (1, 3)}, {(2), (3, 1)}, {(3), (1, 2)} or {(3), (2, 1)} L(n, n) is always 1 since, e.g., partitioning {1, 2, 3} into 3 non-empty subsets results in subsets of length 1. {(1), (2), (3)} Adapting the Karamata–Knuth notation for Stirling numbers, it has been proposed to use the following alternative notation for Lah numbers: (Wikipedia).
