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A144066
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T(n, k) is the number of order-preserving partial transformations (of an n-element chain) of height k (height(alpha) = |Im(alpha)|).
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0
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1, 1, 1, 1, 6, 1, 1, 21, 15, 1, 1, 60, 102, 28, 1, 1, 155, 490, 310, 45, 1, 1, 378, 1935, 2220, 735, 66, 1, 1, 889, 6741, 12285, 7315, 1491, 91, 1
(list; table; 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-classes of the monoid of order-preserving partial transformations (of an n-element chain). Sum of rows of T(n, k) is A123164.
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LINKS
| Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations, Journal of Algebra 278, (2004), 342-359.
Laradji, A. and Umar, A. Combinatorial results for semigroups of order-decreasing partial transformations, J. Integer Seq. 7 (2004), 04.3.8
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FORMULA
| J(n,k)=C(n,k)*A112857(n,k); C(n-1,k-1)*J(n,k)=2((n-k+1)/(n-k))J(n-1,k)
+ C(n,k)J(n-1,k-1)
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EXAMPLE
| J(2,1) = 6 because there are exactly 6 order-preserving partial transformations (on a 2-element chain)of height 1, namely: (1)->(1), (1)->(2), (2)->(1), (2)->(2), (1,2)->(1,1),(1,2)->(2,2)- the mappings are coordinate-wise.
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CROSSREFS
| A123164, A112857
Sequence in context: A146988 A203954 A060972 * A056941 A157638 A142596
Adjacent sequences: A144063 A144064 A144065 * A144067 A144068 A144069
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KEYWORD
| nonn,tabl
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AUTHOR
| A. Umar (aumarh(AT)squ.edu.om), Sep 09 2008
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