OFFSET
1,3
COMMENTS
T(n, k) is the number of ways to distribute k labeled items into n labeled boxes so that there are exactly 2 nonempty boxes.
LINKS
Igor Victorovich 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 = 2, T(1, 1) = 0.
EXAMPLE
n\k 1 2 3 4 5 6 7
1: 0
2: 0 2
3: 0 6 18
4: 0 12 36 84
5: 0 20 60 140 300
6: 0 30 90 210 450 930
7: 0 42 126 294 630 1302 2646
...
T(4,2) = 12: {1}{2}{}{} (12 ways).
T(4,3) = 36: {12}{3}{}{} (36 ways).
T(4,4) = 84: {123}{4}{}{} (84 ways).
MAPLE
L := 2: 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 01 2023
STATUS
approved