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A194640
Smallest image size for which the number of endofunctions (functions f:{1,2,...,n}->{1,2,...,n}) is a maximum.
1
0, 1, 1, 2, 3, 3, 4, 5, 5, 6, 7, 7, 8, 8, 9, 10, 10, 11, 12, 12, 13, 13, 14, 15, 15, 16, 17, 17, 18, 19, 19, 20, 20, 21, 22, 22, 23, 24, 24, 25, 25, 26, 27, 27, 28, 29, 29, 30, 31, 31, 32, 32, 33, 34, 34, 35, 36, 36, 37, 38, 38, 39, 39, 40, 41, 41, 42, 43, 43
OFFSET
0,4
COMMENTS
a(n) is the smallest number of elements in the image for which the number of functions f:{1,2,...,n}->{1,2,...,n} is a maximum.
LINKS
FORMULA
a(n) = arg max_{k=0..n} Stirling2(n,k) * k! * C(n,k) for n!=2, a(2) = 1.
a(n) = arg max_{k=0..n} A090657(n,k) for n!=2, a(2) = 1.
EXAMPLE
a(3) = 2 because there are 18 functions from {1,2,3} into {1,2,3} that have two elements in their image, 3 functions have one and 6 functions that have three elements in their image.
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:
a:= proc(n) local i, k, m, t;
m, i:= 0, 0;
for k to n do
t:= T(n, k);
if t>m then m, i:= t, k fi
od; i
end:
seq(a(n), n=0..50); # Alois P. Heinz, Sep 08 2011
MATHEMATICA
Prepend[Flatten[Table[Flatten[First[Position[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 1, n}], Max[Table[StirlingS2[n, k] Binomial[n, k] k!, {k, 1, n}]]]]], {n, 1, 50}]], 0]
CROSSREFS
Cf. A000312 (number of endofunctions), A090657.
Sequence in context: A284724 A189579 A278617 * A353086 A189726 A093878
KEYWORD
nonn
AUTHOR
Geoffrey Critzer, Aug 31 2011
STATUS
approved