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A183153
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T(n,k) is the number of order-preserving partial isometries of an n-chain of height k (height of alpha = |Im(alpha)|).
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1
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1, 1, 1, 1, 4, 1, 1, 9, 5, 1, 1, 16, 14, 6, 1, 1, 25, 30, 20, 7, 1, 1, 36, 55, 50, 27, 8, 1, 1, 49, 91, 105, 77, 35, 9, 1, 1, 64, 140, 196, 182, 112, 44, 10, 1, 1, 81, 204, 336, 378, 294, 156, 54, 11, 1, 1, 100, 285, 540, 714, 672, 450, 210, 65, 12, 1, 1, 121, 385, 825, 1254, 1386, 1122, 660
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OFFSET
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0,5
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COMMENTS
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The matrix inverse starts
1;
-1,1;
3,-4,1;
-7,11,-5,1;
15,-26,16,-6,1;
-31,57,-42,22,-7,1;
63,-120,99,-64,29,-8,1;
-127,247,-219,163,-93,37,-9,1;
255,-502,466,-382,256,-130,46,-10,1;
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LINKS
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FORMULA
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T(n,0)=1. T(n,k)=(2*n-k+1)*C(n,k)/(k+1) if k>0.
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EXAMPLE
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T(3,2)=5 because there are exactly 5 order-preserving partial isometries (on a 3-chain) of height 2, namely: (1,2)-->(1,2); (1,2)-->(2,3); (2,3)-->(1,2); (2,3)-->(2,3); (1,3)-->(1,3), the mappings are coordinate-wise.
Triangle begins as:
1;
1, 1;
1, 4, 1;
1, 9, 5, 1;
1, 16, 14, 6, 1;
1, 25, 30, 20, 7, 1;
1, 36, 55, 50, 27, 8, 1;
1, 49, 91, 105, 77, 35, 9, 1;
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PROG
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(PARI) T(n, k)=if(k==0, 1, (2*n-k+1)*binomial(n, k)/(k+1));
for(n=0, 17, for(k=0, n, print1(T(n, k), ", ")))
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CROSSREFS
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KEYWORD
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AUTHOR
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STATUS
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approved
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