Lah number

In mathematics, the (signed and unsigned) Lah numbers are coefficients expressing rising factorials in terms of falling factorials and vice versa.

They were discovered by Ivo Lah in 1954.

[1][2] Explicitly, the unsigned Lah numbers

are given by the formula involving the binomial coefficient

Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of

nonempty linearly ordered subsets.

[3] Lah numbers are related to Stirling numbers.

, the Lah number

non-empty subsets results in subsets of size 1, that can only be permuted in one way.

In the more recent literature,[5][6] Karamata–Knuth style notation has taken over.

Lah numbers are now often written as

Below is a table of values for the Lah numbers: The row sums are

represent the rising factorial

represent the falling factorial

The Lah numbers are the coefficients that express each of these families of polynomials in terms of the other.

where the coefficients 6, 6, and 1 are exactly the Lah numbers

The Lah numbers satisfy a variety of identities and relations.

In Karamata–Knuth notation for Stirling numbers

are the unsigned Stirling numbers of the first kind and

are the Stirling numbers of the second kind.

The Lah numbers satisfy the recurrence relations

The n-th derivative of the function

can be expressed with the Lah numbers, as follows[7]

Generalized Laguerre polynomials

are linked to Lah numbers upon setting

This formula is the default Laguerre polynomial in Umbral calculus convention.

[8] In recent years, Lah numbers have been used in steganography for hiding data in images.

Compared to alternatives such as DCT, DFT and DWT, it has lower complexity of calculation—

[9][10] The Lah and Laguerre transforms naturally arise in the perturbative description of the chromatic dispersion.

[11][12] In Lah-Laguerre optics, such an approach tremendously speeds up optimization problems.