cp's OEIS Frontend

A208529 Number of permutations of n > 1 having exactly 2 points on the boundary of their bounding square.

%I A208529 #95 May 25 2024 14:42:15
%S A208529 2,2,4,12,48,240,1440,10080,80640,725760,7257600,79833600,958003200,
%T A208529 12454041600,174356582400,2615348736000,41845579776000,
%U A208529 711374856192000,12804747411456000,243290200817664000,4865804016353280000,102181884343418880000
%N A208529 Number of permutations of n > 1 having exactly 2 points on the boundary of their bounding square.
%C A208529 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 A208529 Sequences A098558 and A052849 have the same terms except for the first. - _Joerg Arndt_, Mar 03 2012
%C A208529 a(n) is the number of permutations of n symbols that commute with a transposition: a permutation p of {1,...,n} has exactly two points on the boundary of their bounding square if and only if p commutes with transposition (1, n). - _Luis Manuel Rivera Martínez_, Feb 27 2014
%C A208529 a(n) is also the determinant of a matrix M each of whose elements M(i, j) is the result of a Reverse and Add operation (RADD) on i in base j: M(i,j) = i + (reverse(i) represented in base j), with 1 <= i < n and 1 < j <= n. - _Federico Provvedi_, May 10 2024
%H A208529 Vincenzo Librandi, <a href="/A208529/b208529.txt">Table of n, a(n) for n = 2..200</a>
%H A208529 Emeric Deutsch, <a href="http://webbox.lafayette.edu/~gordong/MathMagProblems.pdf">Permutations and their bounding squares</a>, Math Magazine, 85(1) (2012), 63.
%H A208529 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 A208529 a(n) = 2*(n-2)!.
%F A208529 G.f.: G(0), where G(k) = 1 + 1/(1 - x*(k+1)/(x*(k+1) + 1/G(k+1))); (continued fraction). - _Sergei N. Gladkovskii_, May 26 2013
%F A208529 G.f.: 2 + 2*x/Q(0), where Q(k) = 1 - 2*x*(2*k+1) - x^2*(2*k+1)*(2*k+2)/( 1 - 2*x*(2*k+2) - x^2*(2*k+2)*(2*k+3)/Q(k+1) ); (continued fraction). - _Sergei N. Gladkovskii_, Sep 23 2013
%F A208529 a(n) = 2*n!/(n*(n-1)). - _Vincenzo Librandi_, Apr 15 2014
%F A208529 E.g.f.: 2 - (1 - x)*(2 + log(1/(1 - x)^2)). - _Ilya Gutkovskiy_, Mar 21 2018
%F A208529 Sum_{n>=2} 1/a(n) = e/2. - _Amiram Eldar_, Feb 02 2023
%e A208529 a(2) = 2 because {(1,1),(2,2)} and {(1,2),(2,1)} each have two points on the bounding square.
%p A208529 A208529:=n->2*(n-2)!; seq(A208529(n), n=2..25); # _Wesley Ivan Hurt_, Feb 27 2014
%t A208529 Table[2(n-2)!, {n, 2, 10}]
%t A208529 FoldList[Times, 2, Range@21] (* _Arkadiusz Wesolowski_, May 08 2012 *)
%t A208529 Table[2 n!/n, {n, 1, 40}] (* _Vincenzo Librandi_, Apr 15 2014 *)
%o A208529 (Python)
%o A208529 import math
%o A208529 def a(n):
%o A208529     return 2*math.factorial(n-2)
%o A208529 (Magma) [2*Factorial(n)/n: n in [1..40]]; // _Vincenzo Librandi_, Apr 15 2014
%o A208529 (PARI) vector(33,n,2*n!/n) /* _Anders Hellström_, Jul 07 2015 */
%Y A208529 Cf. A000142, A098916, A208528.
%K A208529 nonn,base,easy
%O A208529 2,1
%A A208529 _David Nacin_, Feb 27 2012