OFFSET
3,5
COMMENTS
Equivalently, the number of ordered set partitions of the set [n] into k blocks of size at least three. When the boxes are unlabeled we obtain A059022.
FORMULA
E.g.f. with additional constant 1: 1/(1 - t*(exp(x) - 1 - x - x^2/2!)) = 1 + t*x^3/3! + t*x^4/4! + t*x^5/5! + (t+20*t^2)*x^6/6! + ....
Recurrence relation: T(n+1,k) = k*(T(n,k) + n*(n-1)/2*T(n-2,k-1)). T(n,k) = k!*A059022(n,k).
EXAMPLE
Table begins
n\k | 1 2 3
----+-----------------
3 | 1
4 | 1
5 | 1
6 | 1 20
7 | 1 70
8 | 1 182
9 | 1 420 1680
10 | 1 912 12600
11 | 1 1914 62370
...
T(6,2) = 20: The arrangements of 6 objects into 2 boxes { } and [ ] so that each box contains at least 3 items are {1,2,3}[4,5,6], {1,2,4}[3,5,6], {1,2,5}[3,4,6], {1,2,6}[3,4,5], {1,3,4}[2,5,6], {1,3,5}[2,4,6], {1,3,6}[2,4,5], {1,4,5}[2,3,6], {1,4,6}[2,3,5], {1,5,6}[2,3,4] and the 10 other possibilities where the contents of a pair of boxes are swapped.
CROSSREFS
KEYWORD
nonn,easy,tabf
AUTHOR
Peter Bala, Dec 04 2011
STATUS
approved