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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.[3] Lah numbers are related to Stirling numbers.[4]
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,[5][6]Karamata–Knuth style notation has taken over. Lah numbers are now often written as
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Expanded Notation of A Number | Mathematics Grade 4 | Periwinkle
Nombres de Lah
Commutative law of addition | Arithmetic properties | Pre-Algebra | Khan Academy
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,
and
For 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 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.[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.
^Ghosal, Sudipta Kr; Mukhopadhyay, Souradeep; Hossain, Sabbir; Sarkar, Ram (2020). "Application of Lah Transform for Security and Privacy of Data through Information Hiding in Telecommunication". Transactions on Emerging Telecommunications Technologies. 32 (2). doi:10.1002/ett.3984. S2CID225866797.