0,3

Laradji, A. and Umar, A. Combinatorial results for the symmetric inverse semigroup. Semigroup Forum 75, (2007), 221-236.

a(n)=n!*sum(m=0,n,(-1^m/m!)*sum(j=0,n-m,C(n-m,j)/j!))

a(n)=(2n-1)a(n-1)-(n-1)(n-3)a(n-2)-(n-1)(n-2)a(n-3), a(0)=1, a(n)=0 if n<0

E.g.f. for number of partial bijections of an n-element set with exactly k fixed points is x^k/k!*exp(x^2/(1-x))/(1-x). [From Vladeta Jovovic (vladeta(AT)eunet.yu), Nov 09 2008]

a(3) = 18 because there are exactly 18 partial derangements (on a 3-element set), namely: the empty map, (1)->(2), (1)->(3), (2)->(1), (2)->(3), (3)->(1), (3)->(2), (1,2)->(2,1), (1,2)->(2,3), (1,2)->(3,1), (1,3)->(2,1), (1,3)->(3,1), (1,3)->(3,2), (2,3)->(1,2), (2,3)->(3,1), (2,3)->(3,2), (1,2,3)->(2,3,1), (1,2,3)->(3,1,2) - the mappings are coordinate-wise.

a(n) is A144088(n, 0)

Sequence in context: A020114 A009597 A060223 this_sequence A003708 A000986 A143920

Adjacent sequences: A144082 A144083 A144084 this_sequence A144086 A144087 A144088

nonn

A. Umar (aumarh(AT)squ.edu.om), Sep 10 2008, Sep 15 2008