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 A144087 a(n) is the number of partial bijections (or subpermutations) of an n-element set with exactly 2 fixed points. 3
 0, 0, 1, 3, 24, 180, 1620, 16380, 184800, 2298240, 31222800, 459874800, 7296791040, 124047443520, 2248897210560, 43301275617600, 882304501478400, 18964350332928000, 428768570841811200, 10170992126597702400, 252555415474602240000, 6550785133563775104000, 177151172210521513804800 (list; graph; refs; listen; history; text; internal format)
 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,changed AUTHOR Abdullahi Umar, Sep 11 2008 STATUS approved

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Last modified August 7 10:36 EDT 2024. Contains 375011 sequences. (Running on oeis4.)