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

0,3

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.

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

Example

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.

Mathematica

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

Prog

(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 */

Crossrefs

Column k=1 of A144088.

Cf. A144085.

Sequence in context: A052833 A277490 A296975 * A358064 A348767 A335786

Adjacent sequences: A144083 A144084 A144085 * A144087 A144088 A144089

Keyword

nonn

Author

Abdullahi Umar, Sep 10 2008, Sep 15 2008

Status

approved