OFFSET
2,1
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 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
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.
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
EXAMPLE
a(2) = 2 because {(1,1),(2,2)} and {(1,2),(2,1)} each have two points on the bounding square.
MAPLE
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 *)
PROG
(Python)
import math
def a(n):
return 2*math.factorial(n-2)
(Magma) [2*Factorial(n)/n: n in [1..40]]; // Vincenzo Librandi, Apr 15 2014
(PARI) vector(33, n, 2*n!/n) /* Anders Hellström, Jul 07 2015 */
CROSSREFS
KEYWORD
nonn,base,easy
AUTHOR
David Nacin, Feb 27 2012
STATUS
approved