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A144088
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T(n,k) is the number of partial bijections (or subpermutations) of an n-element set with exactly k fixed points.
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3
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1, 1, 1, 4, 2, 1, 18, 12, 3, 1, 108, 72, 24, 4, 1, 780, 540, 180, 40, 5, 1, 6600, 4680, 1620, 360, 60, 6, 1, 63840, 46200, 16380, 3780, 630, 84, 7, 1
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,4
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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
| T(n,k)=C(n,k)(n-k)!sum(m=0,n-k,(-1^m/m!)sum(j=0,n-m,C(n-m,j)/j!));
(n-k)T(n,k)=n(2n-2k-1)T(n-1,k)-n(n-1)(n-k-3)T(n-2,k)-n(n-1)(n-2)T(n-3,k), T(k,k)=1 and T(n,k)=0 if n<k.
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EXAMPLE
| T(3,1) = 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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CROSSREFS
| T(n, 0) = A144085, T(n, 1) = A144086, T(n, 2) = A144087
Sequence in context: A171650 A143777 A152391 * A039948 A111536 A111559
Adjacent sequences: A144085 A144086 A144087 * A144089 A144090 A144091
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KEYWORD
| nice,nonn
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AUTHOR
| A. Umar (aumarh(AT)squ.edu.om), Sep 11 2008, Sep 16 2008
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