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A208528
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Number of permutations of n>1 having exactly 3 points P on the boundary of their bounding square.
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4
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0, 4, 16, 72, 384, 2400, 17280, 141120, 1290240, 13063680, 145152000, 1756339200, 22992076800, 323805081600, 4881984307200, 78460462080000, 1339058552832000, 24186745110528000, 460970906812416000, 9245027631071232000, 194632160654131200000
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OFFSET
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2,2
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COMMENTS
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A bounding square for a permutation of n is the square with sides parallel to the coordinate axis containing (1,1) and (n,n), and the set of points P of a permutation p is the set {(k,p(k)) for 0<k<n+1}.
a(n) is the number of permutations of n symbols that 3-commute with a transposition (see A233440 for definition): a permutation p of {1,...,n} has exactly three points on the boundary of their bounding square if and only if p 3-commutes with transposition (1, n). - Luis Manuel Rivera Martínez, Feb 27 2014
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LINKS
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FORMULA
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a(n) = (4*n-8) * (n-2)!
Sum_{n>=3} 1/a(n) = (Ei(1) - gamma)/4, where Ei(1) = A091725 and gamma = A001620.
Sum_{n>=3} (-1)^(n+1)/a(n) = (gamma - Ei(-1))/4, where Ei(-1) = -A099285. (End)
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EXAMPLE
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a(3) = 4 because {(1,1),(2,3),(3,2)}, {(1,3),(2,1),(3,2)}, {(1,2),(2,3),(3,1)} and {(1,2),(2,1),(3,3)} each have three points on the bounding square.
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MATHEMATICA
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Table[(4n-8)(n-2)!, {n, 2, 10}]
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PROG
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(Python)
import math
def a(n):
return (4*n-8)*math.factorial(n-2)
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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