%I #21 Jan 16 2024 17:31:49
%S 1,1,2,6,22,88,362,1488,6034,24024,93830,359824,1357088,5043260,
%T 18501562,67120024,241169322,859450004,3041415520,10699090888,
%U 37448249502,130518538696,453276141238,1569476495000,5420784841936,18683861676756,64286814548706
%N 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.
%C Equivalently, for n>0, the number of separable permutations of [n] that avoid 2-1-3-5-4 and 4-5-3-1-2.
%C 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".
%H Andrei Asinowski and Cyril Banderier, <a href="https://arxiv.org/abs/2401.05558">From geometry to generating functions: rectangulations and permutations</a>, arXiv:2401.05558 [cs.DM], 2024. See page 2.
%H Arturo Merino and Torsten Mütze. <a href="https://doi.org/10.1007/s00454-022-00393-w">Combinatorial generation via permutation languages. III. Rectangulations</a>. Discrete & Computational Geometry, 70 (2023), 51-122. Preprint: arXiv:2103.09333 [math.CO], 2021.
%H <a href="/index/Rec#order_10">Index entries for linear recurrences with constant coefficients</a>, signature (18,-141,630,-1767,3224,-3834,2896,-1312,320,-32).
%F 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).
%t 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 *)
%Y Other entries including the patterns 1, 2, 3, 4 in the Merino and Mütze reference: A006318, A106228, A363809, A078482, A033321, A363810, A363812, A363813, A006012.
%K nonn,easy
%O 0,3
%A _Andrei Asinowski_, Jun 23 2023
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