A204042 The number of functions f:{1,2,...,n}->{1,2,...,n} (endofunctions) such that all of the fixed points in f are isolated.

1, 1, 2, 12, 120, 1520, 23160, 413952, 8505280, 197631072, 5125527360, 146787894440, 4601174623584, 156693888150384, 5761055539858528, 227438694372072120, 9596077520725211520, 430920897407809702208, 20520683482765477749120, 1032920864149903149579336, 54797532208320308334631840

Offset: 0

Author

Geoffrey Critzer, Jan 09 2012

Keywords

• nonn

Comments

Note this sequence counts the functions enumerated by A065440 for which the statement is vacuously true.

a(n) is also the number of partial endofunctions on {1,2,...,n} without fixed points.

Examples

			a(2)=2 because there are two functions f:{1,2}->{1,2} in which all the fixed points are isolated: 1->1,2->2  and 1->2,2->1 (which has no fixed points).
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..386

Crossrefs

Cf. A065440, A086331, A350134.

Row sums of A349454.

Programs

Maple

a:= n-> add((j-1)^j*binomial(n, j), j=0..n):
seq(a(n), n=0..20);  # Alois P. Heinz, Dec 16 2021

Mathematica

t = Sum[n^(n-1) x^n/n!, {n,1,20}]; Range[0,20]! CoefficientList[Series[Exp[x] Exp[Log[1/(1-t)]-t], {x,0,20}], x]

Formula

E.g.f.: exp(x)*A(x) where A(x) is the e.g.f. for A065440.

a(n) ~ exp(exp(-1)-1)*n^n. - Vaclav Kotesovec, Sep 24 2013

a(n) = Sum_{j=0..n} (j-1)^j * binomial(n,j). - Alois P. Heinz, Dec 16 2021