A208529 - OEIS

%I #95 May 25 2024 14:42:15

%S 2,2,4,12,48,240,1440,10080,80640,725760,7257600,79833600,958003200,

%T 12454041600,174356582400,2615348736000,41845579776000,

%U 711374856192000,12804747411456000,243290200817664000,4865804016353280000,102181884343418880000

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

%C 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 Sequences A098558 and A052849 have the same terms except for the first. - _Joerg Arndt_, Mar 03 2012

%C 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 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 Vincenzo Librandi, <a href="/A208529/b208529.txt">Table of n, a(n) for n = 2..200</a>

%H Emeric Deutsch, <a href="http://webbox.lafayette.edu/~gordong/MathMagProblems.pdf">Permutations and their bounding squares</a>, Math Magazine, 85(1) (2012), 63.

%H 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 a(n) = 2*(n-2)!.

%F 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 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 a(n) = 2*n!/(n*(n-1)). - _Vincenzo Librandi_, Apr 15 2014

%F E.g.f.: 2 - (1 - x)*(2 + log(1/(1 - x)^2)). - _Ilya Gutkovskiy_, Mar 21 2018

%F Sum_{n>=2} 1/a(n) = e/2. - _Amiram Eldar_, Feb 02 2023

%e a(2) = 2 because {(1,1),(2,2)} and {(1,2),(2,1)} each have two points on the bounding square.

%p A208529:=n->2*(n-2)!; seq(A208529(n), n=2..25); # _Wesley Ivan Hurt_, Feb 27 2014

%t Table[2(n-2)!, {n, 2, 10}]

%t FoldList[Times, 2, Range@21] (* _Arkadiusz Wesolowski_, May 08 2012 *)

%t Table[2 n!/n, {n, 1, 40}] (* _Vincenzo Librandi_, Apr 15 2014 *)

%o (Python)

%o import math

%o def a(n):

%o     return 2*math.factorial(n-2)

%o (Magma) [2*Factorial(n)/n: n in [1..40]]; // _Vincenzo Librandi_, Apr 15 2014

%o (PARI) vector(33,n,2*n!/n) /* _Anders Hellström_, Jul 07 2015 */

%Y Cf. A000142, A098916, A208528.

%K nonn,base,easy,changed

%O 2,1

%A _David Nacin_, Feb 27 2012