A144087 a(n) is the number of partial bijections (or subpermutations) of an n-element set with exactly 2 fixed points.

0, 0, 1, 3, 24, 180, 1620, 16380, 184800, 2298240, 31222800, 459874800, 7296791040, 124047443520, 2248897210560, 43301275617600, 882304501478400, 18964350332928000, 428768570841811200, 10170992126597702400, 252555415474602240000, 6550785133563775104000, 177151172210521513804800

Offset

0,4

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..440

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

Formula

a(n) = (n*(n-1)/2)*A144085(n-2).

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

a(n) = (n!/2)*Sum_{m=0..n-2} (-1^m/m!)*Sum_{j=0..n-m} C(n-m,j)/j!;

(n-2)*a(n) = n*(2*n-5)*a(n-1) - n*(n-1)*(n-5)*a(n-2) - n*(n-1)*(n-2)*a(n-3), a(2)=1 and a(n)=0 if n < 2.

a(n) ~ n^(n + 1/4) * exp(2*sqrt(n) - 3/2 - n) / 2^(3/2) * (1 - 17/(48*sqrt(n))). - Vaclav Kotesovec, Dec 01 2021

a(n) = (n!/2) * Sum_{k=0..n-2} binomial(k,n-2-k)/(n-2-k)!. - Seiichi Manyama, Aug 06 2024

Example

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

Prog

(PARI) x='x+O('x^66); /* that many terms */
k=2; 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=2 of A144088.

Cf. A144085.

Sequence in context: A197209 A317527 A181967 * A110347 A213100 A027324

Adjacent sequences: A144084 A144085 A144086 * A144088 A144089 A144090

Keyword

nonn

Author

Abdullahi Umar, Sep 11 2008

Status

approved