Implement the unsigned Lah numbers #17161
Description
Implement the unsigned Lah numbers which for n>=1, k>=1 are
the number of ways a set of n elements can be partitioned into
k nonempty linearly ordered subsets.
Lah numbers are defined for all integers n, k by the following
formula if the Stirling numbers are extended to negative integers
n, k in the way it is recommended by Graham/Knuth/Patashnik
in 'Concrete Mathematics' Section 6.1 (see Table 253).
(This extension is implemented in GAP’s Stirling functions.)
def Lah(n,k):
return sum(stirling_number1(n,j)*stirling_number2(j,k,"gap") for j in (k..n))
(Note how inconsistently at present this formula has to be
written to get consistent results (because stirling_number1
and stirling_number2 are based on different, non-compatible
algorithms! See also ticket #17159)
for n in (-6..6):
print [Lah(n,k) for k in (-6..6)]
1, ., ., ., ., ., ., ., ., ., ., ., .,
30, 1, ., ., ., ., ., ., ., ., ., ., .,
300, 20, 1, ., ., ., ., ., ., ., ., ., .,
1200, 120, 12, 1, ., ., ., ., ., ., ., ., .,
1800, 240, 36, 6, 1, ., ., ., ., ., ., ., .,
720, 120, 24, 6, 2, 1, ., ., ., ., ., ., .,
., ., ., ., ., ., 1, ., ., ., ., ., .,
., ., ., ., ., ., ., 1, ., ., ., ., .,
., ., ., ., ., ., ., 2, 1, ., ., ., .,
., ., ., ., ., ., ., 6, 6, 1, ., ., .,
., ., ., ., ., ., ., 24, 36, 12, 1, ., .,
., ., ., ., ., ., ., 120, 240, 120, 20, 1, .,
., ., ., ., ., ., ., 720, 1800, 1200, 300, 30, 1,
Component: combinatorics
Keywords: Lah numbers
Issue created by migration from https://trac.sagemath.org/ticket/17161