A363811 Number of permutations of [n] that avoid the patterns 2-41-3, 3-14-2, 2-1-3-5-4, and 4-5-3-1-2.

1, 1, 2, 6, 22, 88, 362, 1488, 6034, 24024, 93830, 359824, 1357088, 5043260, 18501562, 67120024, 241169322, 859450004, 3041415520, 10699090888, 37448249502, 130518538696, 453276141238, 1569476495000, 5420784841936, 18683861676756, 64286814548706

Offset: 0

Author

Andrei Asinowski, Jun 23 2023

Keywords

• nonn
• easy

Comments

Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-3-5-4 and 4-5-3-1-2.

The number of guillotine rectangulations (with respect to the weak equivalence) that avoid the geometric patterns "7" and "8". See the Merino and Mütze reference, Table 3, entry "123478".

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.
• Index entries for linear recurrences with constant coefficients, signature (18,-141,630,-1767,3224,-3834,2896,-1312,320,-32).

Crossrefs

Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363812, A363813, A006012.

Programs

Mathematica

CoefficientList[Series[(1 - x)*(1 - 16*x + 109*x^2 - 410*x^3 + 923*x^4 - 1256*x^5 + 988*x^6 - 400*x^7 + 66*x^8 - 2*x^9)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2*(1 - 2*x)^4),{x,0,26}],x] (* Stefano Spezia, Jun 24 2023 *)

Formula

G.f.: (1 - x)*(1 - 16*x + 109*x^2 - 410*x^3 + 923*x^4 - 1256*x^5 + 988*x^6 - 400*x^7 + 66*x^8 - 2*x^9)/((1 - 4*x + 2*x^2)*(1 - 3*x + x^2)^2*(1 - 2*x)^4).