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A144084
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T(n, k) is the number of partial bijections of height k (height(alpha) = |Im(alpha)|) of an n-element set.
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1
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1, 1, 1, 1, 4, 2, 1, 9, 18, 6, 1, 16, 72, 96, 24, 1, 25, 200, 600, 600, 120, 1, 36, 450, 2400, 5400, 4320, 720, 1, 49, 882, 7350, 29400, 52920, 35280, 5040
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| T(n,k) is also the number of elements in the Green's J equivalence classes in the symmetric inverse monoid, I sub n.
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REFERENCES
| Howie, J. M., Fundamentals of semigroup theory. Oxford: Clarendon Press, (1995).
Munn, W. D., The characters of the symmetric inverse semigroup. Proc. Cambridge Philos. Soc. 53 (1957), 13-18.
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FORMULA
| T(n,k)= (C(n,k)^2)*k!
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EXAMPLE
| T(3,1) = 9 because there are exactly 9 partial bijections (on a 3-element set) of height 1, namely: (1)->(1), (1)->(2), (1)->(3), (2)->(1), (2)->(2), (2)->(3), (3)->(1), (3)->(2), (3)->(3)
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CROSSREFS
| T(n, k) = |A021010|. Sum of rows of T(n, k) is A002720. T(n, n) is the order of the symmetric group on an n-element set, n!
Sequence in context: A160905 A183157 A063983 * A021010 A193607 A075397
Adjacent sequences: A144081 A144082 A144083 * A144085 A144086 A144087
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KEYWORD
| nonn
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AUTHOR
| A. Umar (aumarh(AT)squ.edu.om), Sep 10 2008, Sep 30 2008
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