← Back to the home page

Restricted r-Lah numbers

ℓm(r)(n,k)\ell_m^{(r)}(n, k) is the number of partitions of an (n+r)(n + r) element set into (k+r)(k + r) lists, where a list means a non-empty, linearly ordered subset, such that rr distinguished elements have to be in distinct ordered blocks, and each block has at most mm elements.

Formula

ℓm(r)(n,k)=n!k!∑i=0k+r∑t=0r(−1)i(rt)(k+r−ti−t)(n+2r−mi−t−1k+2r−t−1)mt\ell_m^{(r)}(n, k) = \frac{n!}{k!} \sum_{i=0}^{k+r} \sum_{t=0}^r (-1)^i \binom{r}{t} \binom{k + r - t}{i - t} \binom{n + 2r - mi - t - 1}{k + 2r - t - 1} m^t

Generating Functions

see source

References

Mark Shattuck: Some formulas for the restricted r-Lah numbers

Comments

Loading comments...