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A108723
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Triangle read by rows: T(n,k) is number of permutations of [n] with ascending runs consisting of consecutive integers and having k fixed points.
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0
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1, 0, 1, 1, 0, 1, 2, 1, 0, 1, 6, 0, 1, 0, 1, 10, 4, 0, 1, 0, 1, 26, 0, 4, 0, 1, 0, 1, 42, 16, 0, 4, 0, 1, 0, 1, 106, 0, 16, 0, 4, 0, 1, 0, 1, 170, 64, 0, 16, 0, 4, 0, 1, 0, 1, 426, 0, 64, 0, 16, 0, 4, 0, 1, 0, 1, 682, 256, 0, 64, 0, 16, 0, 4, 0, 1, 0, 1, 1706, 0, 256, 0, 64, 0, 16, 0, 4, 0, 1, 0, 1
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OFFSET
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0,7
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COMMENTS
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T(n,0)=A061547(n). Sum of row n is 2^(n-1) (n>=1).
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LINKS
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FORMULA
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T(n, 0)=(3/8)2^n + (1/24)(-2)^n - 2/3; T(n, n)=1; T(n, k)=2^(n-k-2) if k>0 and n-k is even; T(n, k)=0 if k>0 and n-k is odd or if k>n.
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EXAMPLE
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T(3,0)=2 because we have (23)(1) and (3)(12); T(3,1)=1 because we have (3)(2)(1); T(3,3)=1 because we have (123) (the ascending runs are shown between parentheses).
Triangle begins:
1;
0,1;
1,0,1;
2,1,0,1;
6,0,1,0,1;
10,4,0,1,0,1;
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MAPLE
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T:=proc(n, k) if k=n then 1 elif k>n then 0 elif k=0 then 3*2^n/8+(-2)^n/24-2/3 elif k>0 and n-k mod 2 = 0 then 2^(n-k-2) else 0 fi end: for n from 0 to 13 do seq(T(n, k), k=0..n) od; # yields sequence in triangular form
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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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