OFFSET
1,6
COMMENTS
T(n, k) is the number of ways to distribute k labeled items into n labeled boxes so that there are exactly 3 nonempty boxes.
LINKS
I. V. Statsenko, Generalized layout problem, Innovation science No 4-2, State Ufa, Aeterna Publishing House, 2023, pp. 10-13. In Russian.
FORMULA
T(n, k) = (n!/(n - L)!) * Stirling2(k, L) with L = 3, T(1,1)=T(2,1)=T(2,2) = 0.
EXAMPLE
n\k 1 2 3 4 5 6 7
1: 0
2: 0 0
3: 0 0 6
4: 0 0 24 144
5: 0 0 60 360 1500
6: 0 0 120 720 3000 10800
7: 0 0 210 1260 5250 18900 63210
...
T(4,3) = 24: {1}{2}{3}{} (24 ways).
T(4,4) = 144: {12}{3}{4}{} (144 ways).
MAPLE
L := 3: T := (n, k) -> pochhammer(-n, L)*Stirling2(k, L)*((-1)^L):
seq(seq(T(n, k), k = 1..n), n = 1..10);
CROSSREFS
KEYWORD
nonn,tabl
AUTHOR
Igor Victorovich Statsenko, May 04 2023
STATUS
approved