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A145035
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T(n,k) is the number of order-decreasing and order-preserving partial transformations (of an n-chain) of waist k (waist(alpha) = max(Im(alpha))).
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1
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1, 1, 1, 1, 3, 2, 1, 7, 8, 6, 1, 15, 24, 28, 22, 1, 31, 64, 96, 112, 90, 1, 63, 160, 288, 416, 484, 394, 1, 127, 384, 800, 1344, 1896, 2200, 1806, 1, 255, 896, 2112, 4000, 6448, 8952, 10364, 8558, 1, 511, 2048, 5376, 11264, 20160, 31616, 43392, 50144, 41586, 1, 1023, 4608, 13312, 30464, 59520, 102592, 157760, 214656, 247684, 206098
(list; table; graph; refs; listen; history; internal format)
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OFFSET
| 0,5
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LINKS
| Laradji, A. and Umar, A., Combinatorial Results for Semigroups of Order-Decreasing Partial Transformations , Journal of Integer Sequences, Vol. 7 (2004), Article 04.3.8. [From A. Umar (aumarh(AT)squ.edu.om), Oct 07 2008]
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FORMULA
| T(n,k)=(n-k+1)*sum_{j=1,n} binomial(n,j)*binomial(k+j-2,j-1) /n, k>0.
T(n,k)=2T(n-1,k)-T(n-1,k-1)+T(n,k-1), (n>=k>=1), T(n,0)=1, T(n,1)=-1+2^n.
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EXAMPLE
| T(3,2) = 8 because there are exactly 8 order-decreasing and order-preserving partial transformations (of a 3-chain) of waist 2, namely: 2->2, 3->2, (1,2)->(1,2), (1,3)->(1,2), (2,3)->(1,2), (2,3)->(2,2), (1,2,3)->(1,1,2), (1,2,3)->(1,2,2).
1;
1,1;
1,3,2;
1,7,8,6;
1,15,24,28,22;
1,31,64,96,112,90;
1,63,160,288,416,484,394;
1,127,384,800,1344,1896,2200,1806;
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MAPLE
| A145035 := proc(n, k) if k = 0 then 1; else (n-k+1)*sum(binomial(n, j)*binomial(k+j-2, j-1), j=1..n)/n ; end if; end proc: # R. J. Mathar, Jun 11 2011
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CROSSREFS
| Row sums of T(n, k) is A006318
Sequence in context: A111960 A130462 A059380 * A192020 A171128 A122832
Adjacent sequences: A145032 A145033 A145034 * A145036 A145037 A145038
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KEYWORD
| nonn,easy,tabl
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AUTHOR
| A. Umar (aumarh(AT)squ.edu.om), Sep 30 2008
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