Concept
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. Explicitly, the unsigned Lah numbers are given by the formula involving the binomial coefficient for . 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 litterature, 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 . 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 . The Lah numbers satisfy a variety of identities and relations. In Karamata–Knuth notation for Stirling numberswhere are the Stirling numbers of the first kind and are the Stirling numbers of the second kind. for . The Lah numbers satisfy the recurrence relationswhere , the Kronecker delta, and for all . The n-th derivative of the function can be expressed with the Lah numbers, as followsFor example, Generalized Laguerre polynomials are linked to Lah numbers upon setting This formula is the default Laguerre polynomial in Umbral calculus convention. 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.
