OFFSET
0,3
COMMENTS
Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-14-3 and 2-1-3-5-4.
The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "5" and "8". See the Merino and Mütze reference, Table 3, entry "123458".
LINKS
Andrei Asinowski and Cyril Banderier, From geometry to generating functions: rectangulations and permutations, arXiv:2401.05558 [cs.DM], 2024. See page 2.
Arturo Merino and Torsten Mütze. Combinatorial generation via permutation languages. III. Rectangulations. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.
FORMULA
The generating function F=F(x) satisfies the equation x^8*(x - 2)^2*F^4 - x^3*(x - 1)*(x - 2)*(x^5 - 7*x^4 + 4*x^3 - 6*x^2 + 5*x - 1)*F^3 - x*(x - 1)*(4*x^7 - 22*x^6 + 37*x^5 - 42*x^4 + 53*x^3 - 35*x^2 + 10*x - 1)*F^2 - (5*x^6 - 16*x^5 + 15*x^4 - 28*x^3 + 23*x^2 - 8*x + 1)*(x - 1)^2*F - (2*x^5 - 5*x^4 + 4*x^3 - 10*x^2 + 6*x - 1)*(x - 1)^2 = 0.
MAPLE
with(gfun): seq(coeff(algeqtoseries(x^8*(-2+x)^2*F^4 - x^3*(x-1)*(-2+x)*(x^5-7*x^4+4*x^3-6*x^2+5*x-1)*F^3 - x*(x-1)*(4*x^7-22*x^6+37*x^5-42*x^4+53*x^3-35*x^2+10*x-1)*F^2 - (5*x^6-16*x^5+15*x^4-28*x^3+23*x^2-8*x+1)*(x-1)^2*F - (2*x^5-5*x^4+4*x^3-10*x^2+6*x-1)*(x-1)^2, x, F, 32, true)[1], x, n+1), n = 0..30); # Vaclav Kotesovec, Jun 24 2023
CROSSREFS
KEYWORD
nonn
AUTHOR
Andrei Asinowski, Jun 23 2023
STATUS
approved