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.John Riordan, Introduction to Combinatorial Analysis, Princeton University Press (1958, reissue 1980) (reprinted again in 2002 by Dover Publications). Explicitly, the unsigned Lah numbers are given by the formula involving the binomial coefficient

for , and the signed Lah numbers are related to them by .

Signed Lah numbers are only of historical interest as it's how they were defined in Lah's seminal paper, but their sign pattern (, instead of as used for signed Stirling numbers) make them of little to no use in formulas of mathematical interest.

Unsigned Lah numbers have an interesting meaning in combinatorics: they count the number of ways a set of elements can be partitioned into nonempty linearly ordered subsets. Lah numbers are related to Stirling numbers.

For , the Lah number is equal to the factorial in the interpretation above, the only partition of into 1 set can have its set ordered in 6 ways: is equal to 6, because there are six partitions of into two ordered parts: is always 1 because the only way to partition into non-empty subsets results in subsets of size 1, that can only be permuted in one way. In the more recent literature, Karamata–Knuth style notation has taken over. Lah numbers are now often written as

Table of values Below is a table of values for the Lah numbers: 0 1 2 3 4 5 6 7 8 9 1001 10 1 20 2 1 30 6 6 1 40 24 36 12 1 50 120 240 120 20 1 60 720 1800 1200 300 30 1 70 5040 15120 12600 4200 630 42 1 80 40320 141120 141120 58800 11760 1176 56 1 90 362880 1451520 1693440 846720 211680 28224 2016 72 1 10036288001632960021772800127008003810240635040604803240901

The row sums are .

Rising and falling factorials

Let represent the rising factorial and let represent the falling factorial . The Lah numbers are the coefficients that express each of these families of polynomials in terms of the other. Explicitly,andFor example,and

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

Identities and relations The Lah numbers satisfy a variety of identities and relations.

In Karamata–Knuth notation for Stirling numberswhere are the unsigned Stirling numbers of the first kind and are the Stirling numbers of the second kind.

, for . Recurrence relations The Lah numbers satisfy the recurrence relationswhere , the Kronecker delta, and for all .

Exponential generating function

Derivative of exp(1/x)

The n-th derivative of the function can be expressed with the Lah numbers, as followsFor example,

Link to Laguerre polynomials Generalized Laguerre polynomials are linked to Lah numbers upon setting This formula is the default Laguerre polynomial in Umbral calculus convention.

Practical application 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——of their integer coefficients. The Lah and Laguerre transforms naturally arise in the perturbative description of the chromatic dispersion. In Lah-Laguerre optics, such an approach tremendously speeds up optimization problems.

Lah number

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