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A208528 Number of permutations of n>1 having exactly 3 points P on the boundary of their bounding square. 4
0, 4, 16, 72, 384, 2400, 17280, 141120, 1290240, 13063680, 145152000, 1756339200, 22992076800, 323805081600, 4881984307200, 78460462080000, 1339058552832000, 24186745110528000, 460970906812416000, 9245027631071232000, 194632160654131200000 (list; graph; refs; listen; history; text; internal format)
OFFSET

2,2

COMMENTS

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

LINKS

Vincenzo Librandi, Table of n, a(n) for n = 2..200

E. Deutsch, Permutations and their bounding squares, Math Magazine, 85(1) (2012), p. 63.

Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081, 2014

FORMULA

a(n) = (4*n-8) * (n-2)!

EXAMPLE

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.

MATHEMATICA

Table[(4n-8)(n-2)!, {n, 2, 10}]

PROG

(Python)

import math

def a(n):

.return (4*n-8)*math.factorial(n-2)

CROSSREFS

Cf. A098916, A208529.

Sequence in context: A217461 A129872 A059371 * A007234 A096244 A030131

Adjacent sequences:  A208525 A208526 A208527 * A208529 A208530 A208531

KEYWORD

nonn

AUTHOR

David Nacin, Feb 27 2012

STATUS

approved

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Last modified March 21 07:23 EDT 2019. Contains 321367 sequences. (Running on oeis4.)