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Search: id:A144066
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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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1
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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
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OFFSET
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0,5
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COMMENT
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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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REFERENCES
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Laradji, A. and Umar, A. Combinatorial results for semigroups of order-preserving partial transformations. Journal of Algebra 278, (2004), 342-359.
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FORMULA
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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
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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
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A123164, A112857
Sequence in context: A146957 A146988 A060972 this_sequence A056941 A157638 A142596
Adjacent sequences: A144063 A144064 A144065 this_sequence A144067 A144068 A144069
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KEYWORD
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nonn
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AUTHOR
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A. Umar (aumarh(AT)squ.edu.om), Sep 09 2008
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