In mathematics, the Lah numbers, discovered by Ivo Lah in 1954,[1][2] are coefficients expressing rising factorials in terms of falling factorials. They are also the coefficients of the th derivatives of .[3]
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.[4] Lah numbers are related to Stirling numbers.[5]
Unsigned Lah numbers (sequence A105278 in the OEIS):
Signed Lah numbers (sequence A008297 in the OEIS):
L(n, 1) is always n!; in the interpretation above, the only partition of {1, 2, 3} into 1 set can have its set ordered in 6 ways:
L(3, 2) corresponds to the 6 partitions with two ordered parts:
L(n, n) is always 1 since, e.g., partitioning {1, 2, 3} into 3 non-empty subsets results in subsets of length 1.
Adapting the Karamata–Knuth notation for Stirling numbers, it has been proposed to use the following alternative notation for Lah numbers:
Let represent the rising factorial and let represent the falling factorial .
Then and
For example,
Compare the third row of the table of values.
Below is a table of values for the Lah numbers:
k n |
1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | 11 | 12 |
---|---|---|---|---|---|---|---|---|---|---|---|---|
1 | 1 | |||||||||||
2 | 2 | 1 | ||||||||||
3 | 6 | 6 | 1 | |||||||||
4 | 24 | 36 | 12 | 1 | ||||||||
5 | 120 | 240 | 120 | 20 | 1 | |||||||
6 | 720 | 1800 | 1200 | 300 | 30 | 1 | ||||||
7 | 5040 | 15120 | 12600 | 4200 | 630 | 42 | 1 | |||||
8 | 40320 | 141120 | 141120 | 58800 | 11760 | 1176 | 56 | 1 | ||||
9 | 362880 | 1451520 | 1693440 | 846720 | 211680 | 28224 | 2016 | 72 | 1 | |||
10 | 3628800 | 16329600 | 21772800 | 12700800 | 3810240 | 635040 | 60480 | 3240 | 90 | 1 | ||
11 | 39916800 | 199584000 | 299376000 | 199584000 | 69854400 | 13970880 | 1663200 | 11880 | 4950 | 110 | 1 | |
12 | 479001600 | 2634508800 | 4390848000 | 3293136000 | 1317254400 | 307359360 | 43908480 | 3920400 | 217800 | 7260 | 132 | 1 |
In recent years, Lah numbers have been used in steganography for hiding data in images. A few researchers,[6] [7] such as Dr Sudipta Kumar Ghosal, have exploited them in this domain as an alternative to DCT, DFT and DWT because of the low complexity——of calculation of their integer coefficients.