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A144086
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Number of partial bijections (or subpermutations) of an n-element set with exactly 1 fixed point.
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2
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0, 1, 2, 12, 72, 540, 4680, 46200, 510720, 6244560, 83613600, 1216131840, 19084222080, 321271030080, 5773503415680, 110288062684800, 2231100039168000, 47640952315756800, 1070630750168179200, 25255541547460224000, 623884298434645248000, 16104652019138319436800
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,3
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REFERENCES
| Laradji, A. and Umar, A. Combinatorial results for the symmetric inverse semigroup. Semigroup Forum 75, (2007), 221-236.
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FORMULA
| 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
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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.
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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 */
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CROSSREFS
| a(n) = A144088(n, 1) and a(n) = n*A144085(n-1)
Sequence in context: A062119 A052556 A052833 * A005443 A002867 A130426
Adjacent sequences: A144083 A144084 A144085 * A144087 A144088 A144089
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KEYWORD
| nonn
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AUTHOR
| A. Umar (aumarh(AT)squ.edu.om), Sep 10 2008, Sep 15 2008
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