A128652 Number of square permutations of length n.

1, 2, 6, 24, 104, 464, 2088, 9392, 42064, 187296, 828776, 3644912, 15937776, 69317984, 300009744, 1292654304, 5547021728, 23715100480, 101046014952, 429209373296, 1817975905456, 7680278380512, 32368750662320

Offset

1,2

Links

Table of n, a(n) for n=1..23.

Michael Albert, Steve Linton, Nik Ruskuc, Vincent Vatter, Steve Waton, On convex permutations, preprint.

Michael Albert, Steve Linton, Nik Ruskuc, Vincent Vatter, Steve Waton, On convex permutations, Discrete Mathematics, vol.311, pp.715-722, (2011).

A. Bernini, F. Disanto, R. Pinzani and S. Rinaldi, Permutations defining convex permutominoes, J. Int. Seq. 10 (2007) # 07.9.7.

Enrica Duchi, A code for square permutations and convex permutominoes, arXiv:1904.02691 [math.CO], 2019.

Sergey Kitaev and Jeffrey Remmel, Simple marked mesh patterns, arXiv preprint arXiv:1201.1323 [math.CO], 2012.

S. Kitaev, J. Remmel, Quadrant Marked Mesh Patterns, J. Int. Seq. 15 (2012) # 12.4.7

T. Mansour and S. Severini, Grid polygons from permutations and their enumeration by the kernel method, arXiv:math/0603225 [math.CO], 2006.

Formula

a(n) = 2*(n+2) * 4^(n-3) - 4*(2*n-5) * C(2*n-6,n-3) for n>=2, a(1)=1.

G.f.: x*(1-6*x+10*x^2-4*x^2*sqrt(1-4*x))/(1-4*x)^2 (See theorem 3.1 in Albert et al. reference). [Joerg Arndt, Jun 21 2011]

Conjecture: +(n-3)*(n-8)*a(n) +2*(-4*n^2+43*n-96)*a(n-1) +8*(2*n-7)*(n-7)*a(n-2)=0. - R. J. Mathar, Oct 16 2017

Mathematica

a[1] = 1; a[n_] := 2(n+2) * 4^(n-3) - 4(2n-5) * Binomial[2n-6, n-3];
Array[a, 30] (* Jean-François Alcover, Jul 22 2018 *)

Prog

(PARI) a(n) = if(n<=1, n, 2*(n+2) * 4^(n-3) - 4*(2*n-5) * binomial(2*n-6, n-3)); /* Joerg Arndt, Jun 21 2011 */

Crossrefs

Sequence in context: A129817 A230797 A376585 * A152316 A177520 A152326

Adjacent sequences: A128649 A128650 A128651 * A128653 A128654 A128655

Keyword

nonn

Author

Ralf Stephan, May 08 2007

Status

approved