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

Author

Ralf Stephan, May 08 2007

Keywords

• nonn

Links

• 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.

Programs

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 *)

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 */

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