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 MartÃnez, Feb 27 2014
LINKS
Vincenzo Librandi, Table of n, a(n) for n = 2..200
Emeric 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 [math.CO], 2014-2015.
FORMULA
a(n) = (4*n-8) * (n-2)!
From Amiram Eldar, May 17 2022: (Start)
Sum_{n>=3} (-1)^(n+1)/a(n) = (gamma - Ei(-1))/4, where Ei(-1) = -A099285. (End)
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
KEYWORD
nonn
AUTHOR
David Nacin, Feb 27 2012
STATUS
approved