cp's OEIS Frontend

A208528 Number of permutations of n>1 having exactly 3 points P on the boundary of their bounding square.

%I A208528 #30 May 17 2022 07:26:09
%S A208528 0,4,16,72,384,2400,17280,141120,1290240,13063680,145152000,
%T A208528 1756339200,22992076800,323805081600,4881984307200,78460462080000,
%U A208528 1339058552832000,24186745110528000,460970906812416000,9245027631071232000,194632160654131200000
%N A208528 Number of permutations of n>1 having exactly 3 points P on the boundary of their bounding square.
%C A208528 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}.
%C A208528 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
%H A208528 Vincenzo Librandi, <a href="/A208528/b208528.txt">Table of n, a(n) for n = 2..200</a>
%H A208528 Emeric Deutsch, <a href="http://webbox.lafayette.edu/~gordong/MathMagProblems.pdf">Permutations and their bounding squares</a>, Math Magazine, 85(1) (2012), p. 63.
%H A208528 Luis Manuel Rivera, <a href="http://arxiv.org/abs/1406.3081">Integer sequences and k-commuting permutations</a>, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.
%F A208528 a(n) = (4*n-8) * (n-2)!
%F A208528 From _Amiram Eldar_, May 17 2022: (Start)
%F A208528 Sum_{n>=3} 1/a(n) = (Ei(1) - gamma)/4, where Ei(1) = A091725 and gamma = A001620.
%F A208528 Sum_{n>=3} (-1)^(n+1)/a(n) = (gamma - Ei(-1))/4, where Ei(-1) = -A099285. (End)
%e A208528 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.
%t A208528 Table[(4n-8)(n-2)!, {n, 2, 10}]
%o A208528 (Python)
%o A208528 import math
%o A208528 def a(n):
%o A208528     return (4*n-8)*math.factorial(n-2)
%Y A208528 Cf. A098916, A208529.
%Y A208528 Cf. A001620, A091725, A099285.
%K A208528 nonn
%O A208528 2,2
%A A208528 _David Nacin_, Feb 27 2012