Lah Number
In mathematics, Lah numbers, discovered by Ivo Lah in 1955, are coefficients expressing rising factorials in terms of falling factorials.
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:
Signed Lah numbers:
L(n, 1) is always n!; using 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; e.g., partitioning {1, 2, 3} into 3 non-empty subsets results in subsets of length 1.
- {(1), (2), (3)}
Paraphrasing Karamata-Knuth notation for Stirling numbers, it was proposed to use the following alternative notation for Lah numbers:
Read more about Lah Number: Rising and Falling Factorials, Identities and Relations, Table of Values
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