A054394 Number of permutations with certain forbidden subsequences.

1, 1, 2, 5, 14, 42, 132, 429, 1429, 4847, 16660, 57820, 202086, 709928, 2503266, 8850681, 31355020, 111242127, 395091069, 1404332528, 4994581900, 17771328588, 63253477326, 225194224134, 801884971816, 2855809269782, 10171707099565

Offset: 0

Author

N. J. A. Sloane, Elisa Pergola (elisa(AT)dsi.unifi.it), May 21 2000

Keywords

• nonn

Links

• E. Barcucci et al., From Motzkin to Catalan Permutations, Discr. Math., 217 (2000), 33-49.
• Nickolas Hein, Jia Huang, Variations of the Catalan numbers from some nonassociative binary operations, arXiv:1807.04623 [math.CO], 2018.

Crossrefs

Interpolates between Motzkin numbers (A001006) and Catalan numbers (A000108). Cf. A005773, A054391-A054393.

Programs

Mathematica

a[0] = 1; a[n_] := Module[{M}, M = Table[If[j < i || i == j && i <= 6 || j == i+1, 1, 0], {i, 1, n}, {j, 1, n}]; MatrixPower[M, n][[1, 1]]];
Table[a[n], {n, 0, 26}] (* Jean-François Alcover, Aug 16 2018, after A054391 *)

Formula

Conjecture: g.f.(x)=1+z*(1-2z+z^2-z^3)/(1-3z+3z^2-3z^3+2z^4-z^5) where z=x*A001006(x) and A001006(x) is the g.f. of A001006. [R. J. Mathar, Jul 07 2009]