A194640 Smallest image size for which the number of endofunctions (functions f:{1,2,...,n}->{1,2,...,n}) is a maximum.

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

Author

Geoffrey Critzer, Aug 31 2011

Keywords

• nonn

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.

Examples

			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.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000312 (number of endofunctions), A090657.

Programs

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]

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.