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

2, 2, 4, 12, 48, 240, 1440, 10080, 80640, 725760, 7257600, 79833600, 958003200, 12454041600, 174356582400, 2615348736000, 41845579776000, 711374856192000, 12804747411456000, 243290200817664000, 4865804016353280000, 102181884343418880000

Offset: 2

Author

David Nacin, Feb 27 2012

Keywords

• nonn
• base
• easy

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}.

Sequences A098558 and A052849 have the same terms except for the first. - Joerg Arndt, Mar 03 2012

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

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

Examples

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

Links

• Vincenzo Librandi, Table of n, a(n) for n = 2..200
• Emeric Deutsch, Permutations and their bounding squares, Math Magazine, 85(1) (2012), 63.
• Luis Manuel Rivera, Integer sequences and k-commuting permutations, arXiv preprint arXiv:1406.3081 [math.CO], 2014-2015.

Crossrefs

Cf. A000142, A098916, A208528.

Programs

Magma

[2*Factorial(n)/n: n in [1..40]]; // Vincenzo Librandi, Apr 15 2014

Maple

A208529:=n->2*(n-2)!; seq(A208529(n), n=2..25); # Wesley Ivan Hurt, Feb 27 2014

Mathematica

Table[2(n-2)!, {n, 2, 10}]
FoldList[Times, 2, Range@21] (* Arkadiusz Wesolowski, May 08 2012 *)
Table[2 n!/n, {n, 1, 40}] (* Vincenzo Librandi, Apr 15 2014 *)

PARI

vector(33,n,2*n!/n) /* Anders Hellström, Jul 07 2015 */

Python

import math
def a(n):
    return 2*math.factorial(n-2)

Formula

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

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

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

a(n) = 2*n!/(n*(n-1)). - Vincenzo Librandi, Apr 15 2014

E.g.f.: 2 - (1 - x)*(2 + log(1/(1 - x)^2)). - Ilya Gutkovskiy, Mar 21 2018

Sum_{n>=2} 1/a(n) = e/2. - Amiram Eldar, Feb 02 2023