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A112857
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Triangle read by rows: number of Green's R-classes in the semigroup of order-preserving partial transformations (of an n-element chain) consisting of elements of height k (height(alpha) = |Im(alpha)|). .
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5
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1, 1, 1, 1, 3, 1, 1, 7, 5, 1, 1, 15, 17, 7, 1, 1, 31, 49, 31, 9, 1, 1, 63, 129, 111, 49, 11, 1, 1, 127, 321, 351, 209, 71, 13, 1, 1, 255, 769, 1023, 769, 351, 97, 15, 1, 1, 511, 1793, 2815, 2561, 1471, 545, 127, 17, 1, 1, 1023, 4097, 7423, 7937, 5503, 2561, 799, 161, 19, 1
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OFFSET
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0,5
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COMMENTS
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Sum of rows of R(n, k) is A007051, R(n,k) = |A118801|.
Row-reversed variant of A119258. - R. J. Mathar, Jun 20 2011
Pairwise sums of row terms starting from the right yields triangle A038207. - Gary W. Adamson, Feb 06 2012
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LINKS
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Table of n, a(n) for n=0..65.
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
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R(n,k)=sum(j=p,n,C(n,j*C(j-1,p-1)).
R(n,k)=2*R(n-1,k)+R(n-1,k-1). R(n,0)= 1 = R(n,n)
n-th row = top row of M^n, deleting the zeros; where M is an infinite square production matrix with (1,1,1,...) as the superdiagonal and (1,2,2,2,...) as the main diagonal. - Gary W. Adamson, Feb 06, 2012
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EXAMPLE
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R(3,2) = 5 because in a regular semigroup of transformations the Green's R-classes coincide with convex partitions of subsets of {1,2,3} with convex classes (modulo the subsets): {1}, {2}/{1}, {3}/{2}, {3}/{1,2}, {3}/{1}, {2,3}
1;
1,1;
1,3,1;
1,7,5,1;
1,15,17,7,1;
1,31,49,31,9,1;
1,63,129,111,49,11,1;
1,127,321,351,209,71,13,1;
1,255,769,1023,769,351,97,15,1;
1,511,1793,2815,2561,1471,545,127,17,1;
1,1023,4097,7423,7937,5503,2561,799,161,19,1;
...
As to matrix M, top row of M^3 = (1, 7, 5, 1, 0, 0, 0,...)
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MAPLE
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A112857 := proc(n, k) if k=0 or k=n then 1; elif k <0 or k>n then 0; else 2*procname(n-1, k)+procname(n-1, k-1) ; end if; end proc: # R. J. Mathar, Jun 20 2011
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CROSSREFS
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Cf. A007051, A118801, A135233, columns: A000225, A000337, A055580, A027608.
Cf. A038207
Sequence in context: A108625 A177992 * A118801 A080936 A094507 A065625
Adjacent sequences: A112854 A112855 A112856 * A112858 A112859 A112860
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KEYWORD
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nonn,tabl
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AUTHOR
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Abdullahi Umar, Aug 25 2008
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STATUS
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approved
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