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A008305
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Triangle read by rows: a(n,k) = number of permutations of [n] allowing i->i+j (mod n), j=0..k-1.
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17
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1, 1, 2, 1, 2, 6, 1, 2, 9, 24, 1, 2, 13, 44, 120, 1, 2, 20, 80, 265, 720, 1, 2, 31, 144, 579, 1854, 5040, 1, 2, 49, 264, 1265, 4738, 14833, 40320, 1, 2, 78, 484, 2783, 12072, 43387, 133496, 362880, 1, 2, 125, 888, 6208, 30818, 126565, 439792, 1334961, 3628800
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OFFSET
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1,3
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COMMENTS
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The point is, we are counting permutations of [n] = {1,2,...,n} with the restriction that i cannot move by more than k places. Hence the phrase "permutations with restricted displacements". - N. J. A. Sloane, Mar 07 2014
The triangle could have been defined as an infinite square array by setting a(n,k) = n! for k >= n.
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REFERENCES
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H. Minc, Permanents, Encyc. Math. #6, 1978, p. 48
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LINKS
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FORMULA
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a(n,k) = per(sum(P^j, j=0..k-1)), where P is n X n, P[ i, i+1 (mod n) ]=1, 0's otherwise.
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EXAMPLE
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a(4,3) = 9 because 9 permutations of {1,2,3,4} are allowed if each i can be placed on 3 positions i+0, i+1, i+2 (mod 4): 1234, 1423, 1432, 3124, 3214, 3412, 4123, 4132, 4213.
Triangle begins:
1,
1, 2,
1, 2, 6,
1, 2, 9, 24,
1, 2, 13, 44, 120,
1, 2, 20, 80, 265, 720,
1, 2, 31, 144, 579, 1854, 5040,
1, 2, 49, 264, 1265, 4738, 14833, 40320,
1, 2, 78, 484, 2783, 12072, 43387, 133496, 362880,
1, 2, 125, 888, 6208, 30818, 126565, 439792, 1334961, 3628800,
...
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MAPLE
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with(LinearAlgebra):
a:= (n, k)-> Permanent(Matrix(n,
(i, j)-> `if`(0<=j-i and j-i<k or j-i<k-n, 1, 0))):
seq(seq(a(n, k), k=1..n), n=1..10);
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MATHEMATICA
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a[n_, k_] := Permanent[Table[If[0 <= j-i && j-i < k || j-i < k-n, 1, 0], {i, 1, n}, {j, 1, n}]]; Table[Table[a[n, k], {k, 1, n}], {n, 1, 10}] // Flatten (* Jean-François Alcover, Mar 10 2014, after Alois P. Heinz *)
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CROSSREFS
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KEYWORD
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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