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# Lah number

In mathematics, the Lah numbers, discovered by Ivo Lah in 1955,[1] are coefficients expressing rising factorials in terms of falling factorials.

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. Lah numbers are related to Stirling numbers.

Unsigned Lah numbers (sequence A105278 in OEIS):

\( L(n,k) = {n-1 \choose k-1} \frac{n!}{k!}. \)

Signed Lah numbers (sequence A008297 in OEIS):

\( L'(n,k) = (-1)^n {n-1 \choose k-1} \frac{n!}{k!}. \)

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:

{(1, 2, 3)}, {(1, 3, 2)}, {(2, 1, 3)}, {(2, 3, 1)}, {(3, 1, 2)} or {(3, 2, 1)}

L(3, 2) corresponds to the 6 partitions with two ordered parts:

{(1), (2, 3)}, {(1), (3, 2)}, {(2), (1, 3)}, {(2), (3, 1)}, {(3), (1, 2)} or {(3), (2, 1)}

L(n, n) is always 1 since, e.g., partitioning {1, 2, 3} into 3 non-empty subsets results in subsets of length 1.

{(1), (2), (3)}

Adapting the Karamata-Knuth notation for Stirling numbers, it has been proposed to use the following alternative notation for Lah numbers:

\( L(n,k)=\left\lfloor\begin{matrix} n \\ k \end{matrix}\right\rfloor \)

Rising and falling factorials

Let \( x^{(n)} \) represent the rising factorial \( x(x+1)(x+2) \cdots (x+n-1) \) and let \( (x)_n \) represent the falling factorial \( x(x-1)(x-2) \cdots (x-n+1). \)

Then \( x^{(n)} = \sum_{k=1}^n L(n,k) (x)_k and (x)_n = \sum_{k=1}^n (-1)^{n-k} L(n,k)x^{(k)}. \)

For example, \( x(x+1)(x+2) = {\color{Red}6}x + {\color{Red}6}x(x-1) + {\color{Red}1}x(x-1)(x-2) \).

Compare the third row of the table of values.

Identities and relations

\( L(n,k) = {n-1 \choose k-1} \frac{n!}{k!} = {n \choose k} \frac{(n-1)!}{(k-1)!} \)

\( L(n,k) = \frac{n!(n-1)!}{k!(k-1)!}\cdot\frac{1}{(n-k)!} = \left (\frac{n!}{k!} \right )^2\frac{k}{n(n-k)!} \)

\( L(n,k+1) = \frac{n-k}{k(k+1)} L(n,k). \)

\( L(n,k) = \sum_{j} \left[{n\atop j}\right] \left\{{j\atop k}\right\} \), where s(n,j) are the Stirling numbers of the first kind and S(j,k) are the Stirling numbers of the second kind, and with the conventions L(0,0)=1 and L(n , k )=0 if k>n.

\( L(n,1) = n! \)

\( L(n,2) = (n-1)n!/2 \)

\( L(n,3) = (n-2)(n-1)n!/12 \)

\( L(n,n-1) = n(n-1) \)

\( L(n,n) = 1 \)

Table of values

Below is a table of values for the Lah numbers:

\( _n\!\!\diagdown\!\!^k \) | 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 |

See also

Stirling numbers

Pascal matrix

References

John Riordan, Introduction to Combinatorial Analysis, Princeton University Press (1958, reissue 1980) ISBN 978-0-691-02365-6 (reprinted again in 2002 by Dover Publications).

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