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A180188
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Triangle read by rows: T(n,k) is the number of permutations of [n] with k circular successions (0<=k<=n-1). A circular succession in a permutation p of [n] is either a pair p(i), p(i+1), where p(i+1)=p(i)+1 or the pair p(n), p(1) if p(1)=p(n)+1.
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4
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1, 0, 2, 3, 0, 3, 8, 12, 0, 4, 45, 40, 30, 0, 5, 264, 270, 120, 60, 0, 6, 1855, 1848, 945, 280, 105, 0, 7, 14832, 14840, 7392, 2520, 560, 168, 0, 8, 133497, 133488, 66780, 22176, 5670, 1008, 252, 0, 9, 1334960, 1334970, 667440, 222600, 55440, 11340, 1680, 360, 0
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OFFSET
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1,3
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COMMENTS
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For example, p=(4,1,2,5,3) has 2 circular successions: (1,2) and (3,4).
Sum of entries in row n = n! = A000142(n).
Sum(k*T(n,k), k>=0)=n! = A000142(n) if n>=2.
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LINKS
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FORMULA
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T(n,k) = n*C(n-1,k)*d(n-1-k), where d(j) = A000166(j) are the derangement numbers (see Prop. 1 of the Tanny reference).
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EXAMPLE
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T(3,2) = 3 because we have 123, 312, and 231.
The triangle starts:
1;
0, 2;
3, 0, 3;
8, 12, 0, 4;
45, 40, 30, 0, 5;
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MAPLE
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for n to 10 do seq(A180188(n, k), k = 0 .. n-1) end do; # yields sequence in triangular form
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MATHEMATICA
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T[n_, k_] := n*Binomial[n-1, k]*Subfactorial[n-1-k]; Table[T[n, k], {n, 0, 10}, {k, 0, n-1}] // Flatten (* Jean-François Alcover, Feb 19 2017 *)
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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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