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A144089
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T(n,k) is the number of partial bijections (or subpermutations) of an n-element set of height k (height(alpha) = |Im(alpha)|) and without fixed points.
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1
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1, 1, 0, 1, 2, 1, 1, 6, 9, 2, 1, 12, 42, 44, 9, 1, 20, 130, 320, 265, 44, 1, 30, 315, 1420, 2715, 1854, 265, 1, 42, 651, 4690, 16275, 25494, 14833, 1854, 1, 56, 1204, 12712, 70070, 198184, 263284, 133496, 14833, 1, 72, 2052, 29904, 240534, 1076544, 2573508
(list; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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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)= (n!/(n-k)!)sum(m=0,k,(-1^m/m!)C(n-m,k-m))
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EXAMPLE
| T(3,2) = 9 because there are exactly 9 partial bijections (on a 3-element set) without fixed points and of height 2, namely: (1,2)->(2,1), (1,2)->(2,3), (1,2)->(3,1), (1,3)->(2,1), (1,3)->(3,1), (1,3)->(3,2), (2,3)->(1,2), (2,3)->(3,1), (2,3)->(3,2),- the mappings are coordinate-wise.
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MATHEMATICA
| t[n_, k_] := n!^2*Hypergeometric1F1[-k, -n, -1]/(k!*(n-k)!^2); Flatten[ Table[ t[n, k], {n, 0, 7}, {k, 0, n}]] (* From Jean-François Alcover, Oct 13 2011 *)
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CROSSREFS
| Sum of rows of T(n, k) is A144085, T(n, n-1) = A000166(n+1) and T(n, n)=A000166(n)
Sequence in context: A137376 A039761 A196073 * A172107 A165891 A039763
Adjacent sequences: A144086 A144087 A144088 * A144090 A144091 A144092
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KEYWORD
| nice,nonn
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AUTHOR
| A. Umar (aumarh(AT)squ.edu.om), Sep 11 2008
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