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

0,3

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.

Links

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

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

Example

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).

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]

Crossrefs

Cf. A065440, A086331, A350134.

Row sums of A349454.

Sequence in context: A003580 A052580 A134095 * A302702 A189981 A326000

Adjacent sequences: A204039 A204040 A204041 * A204043 A204044 A204045

Keyword

nonn

Author

Geoffrey Critzer, Jan 09 2012

Status

approved