A144086 Number of partial bijections (or subpermutations) of an n-element set with exactly 1 fixed point.

0, 1, 2, 12, 72, 540, 4680, 46200, 510720, 6244560, 83613600, 1216131840, 19084222080, 321271030080, 5773503415680, 110288062684800, 2231100039168000, 47640952315756800, 1070630750168179200, 25255541547460224000, 623884298434645248000, 16104652019138319436800

Offset: 0

Author

Abdullahi Umar, Sep 10 2008, Sep 15 2008

Keywords

• nonn

Examples

			a(3) = 12 because there are exactly 12 partial bijections (on a 3-element set) with exactly 1 fixed point, namely: (1)->(1), (2)->(2), (3)->(3), (1,2)->(1,3), (1,2)->(3,2), (1,3)->(1,2), (1,3)->(2,3), (2,3)->(2,1), (2,3)->(1,3), (1,2,3)->(1,3,2), (1,2,3)->(3,2,1), (1,2,3)->(2,1,3) - the mappings are coordinate-wise.
		

Links

• Vincenzo Librandi, Table of n, a(n) for n = 0..200
• A. Laradji and A. Umar, Combinatorial results for the symmetric inverse semigroup, Semigroup Forum 75, (2007), 221-236.

Crossrefs

Column k=1 of A144088.

Cf. A144085.

Programs

Mathematica

CoefficientList[Series[x*E^(x^2/(1-x))/(1-x), {x, 0, 20}], x] * Range[0, 20]! (* Vaclav Kotesovec, Feb 24 2014 *)

PARI

x='x+O('x^66); /* that many terms */
k=1; egf=x^k/k!*exp(x^2/(1-x))/(1-x);
Vec(serlaplace(egf)) /* show terms, starting with 1 */
/* Joerg Arndt, Jul 11 2011 */

Formula

a(n) = n*A144085(n-1).

E.g.f.: (x^k/k!)*exp(x^2/(1-x))/(1-x) where k=1. - Joerg Arndt, Jul 11 2011

a(n) = n!*Sum_{m=0..n-1} (-1^m/m!)*Sum_{j=0..n-m} C(n-m)/j!;

(n-1)*a(n) = n*(2*n-3)*a(n-1) - n*(n-1)*(n-4)*a(n-2) - n*(n-1)*(n-2)*a(n-3), a(1)=1 and a(n)=0 if n < 1.

a(n) ~ n^(n+1/4) * exp(2*sqrt(n)-n-3/2) / sqrt(2) * (1 + 31/(48*sqrt(n))). - Vaclav Kotesovec, Feb 24 2014

a(n) = n! * Sum_{k=0..n-1} binomial(k,n-1-k)/(n-1-k)!. - Seiichi Manyama, Aug 06 2024