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A111516
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Triangle read by rows: number of order-preserving partial transformations (of an n-element totally ordered set) of waist k (waist(alpha) = max(Im(alpha)).
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0
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1, 1, 1, 1, 3, 4, 1, 7, 12, 18, 1, 15, 32, 56, 88, 1, 31, 80, 160, 280, 450, 1, 63, 192, 432, 832, 1452, 2364, 1, 127, 448, 1120, 2352, 4244, 7700, 12642
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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COMMENTS
| G(n; n) is A050146 and sum(k=1,n,G(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
| G(n,k)=sum(j=0,k,C(n,j)*C(k+j-2,j-1)); G(n,k)=2*G(n-1,k)-G(n-1,k-1)+G(n,k-1), G(n,0)=1 (n>=0), G(0,k)=0 (k>0)
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EXAMPLE
| G(2,2) = 4 because there are exactly 4 order-preserving partial transformations (on a 2-element chain) of waist 2, namely: (1)->(2), (2)->(2),(1,2)->(1,2),(1,2)->(2,2) - the mappings are coordinate-wise
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CROSSREFS
| Cf. A050146, A123164.
Sequence in context: A076412 A053707 A075052 * A116392 A174607 A105578
Adjacent sequences: A111513 A111514 A111515 * A111517 A111518 A111519
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KEYWORD
| nonn,tabl
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AUTHOR
| A. Umar (aumarh(AT)squ.edu.om), Aug 25 2008
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