|
|
A090657
|
|
Triangle read by rows: T(n,k) = number of functions from [1,2,...,n] to [1,2,...,n] such that the image contains exactly k elements (0<=k<=n).
|
|
12
|
|
|
1, 0, 1, 0, 2, 2, 0, 3, 18, 6, 0, 4, 84, 144, 24, 0, 5, 300, 1500, 1200, 120, 0, 6, 930, 10800, 23400, 10800, 720, 0, 7, 2646, 63210, 294000, 352800, 105840, 5040, 0, 8, 7112, 324576, 2857680, 7056000, 5362560, 1128960, 40320
(list;
table;
graph;
refs;
listen;
history;
text;
internal format)
|
|
|
OFFSET
|
0,5
|
|
COMMENTS
|
|
|
LINKS
|
|
|
FORMULA
|
T(n,k) = C(n,k) * k! * A048993(n,k).
T(n,k) = C(n,k) * Sum_{j=0..k} (-1)^(k-j) * C(k,j) * j^n.
T(n,k) = n * (T(n-1,k-1) + k/(n-k) * T(n-1,k)) with T(n,n) = n! and T(n,0) = 0 for n>0.
|
|
EXAMPLE
|
Triangle begins:
1;
0, 1;
0, 2, 2;
0, 3, 18, 6;
0, 4, 84, 144, 24;
...
|
|
MAPLE
|
T:= proc(n, k) option remember;
if k=n then n!
elif k=0 or k>n then 0
else n * (T(n-1, k-1) + k/(n-k) * T(n-1, k))
fi
end:
seq(seq(T(n, k), k=0..n), n=0..10);
|
|
MATHEMATICA
|
Table[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 0, n}], {n, 0, 10}] // Flatten (* Geoffrey Critzer, Sep 09 2011 *)
|
|
CROSSREFS
|
|
|
KEYWORD
|
|
|
AUTHOR
|
|
|
EXTENSIONS
|
Revised description from Jan Maciak, Apr 25 2004
|
|
STATUS
|
approved
|
|
|
|