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%I #22 Jan 25 2023 11:10:43
%S 1,1,1,2,5,14,43,140,477,1683,6106,22664,85735,329572,1284440,5065828,
%T 20188877,81201801,329281059,1345059602,5530600618,22876354484,
%U 95137126194,397610249052,1669285639455,7037395810149,29782584966376
%N Number of valid hook configurations of 132-avoiding permutations of [n].
%C Also the number of valid hook configurations of 231-avoiding permutations of [n].
%C For n > 0, a(n) is the number of intervals in the Motzkin-Tamari poset introduced by Fang.
%H Colin Defant, <a href="http://arxiv.org/abs/1904.10451">Motzkin intervals and valid hook configurations</a>, arXiv preprint arXiv:1904.10451 [math.CO], 2019.
%H Wenjie Fang, <a href="http://arxiv.org/abs/1801.04809">A partial order on Motzkin paths</a>, arXiv preprint arXiv:1801.04809 [math.CO], 2018.
%F O.g.f. A(x) satisfies (-1 + 6 x + 15 x^2 + 8 x^3) + (1 - 11 x + 28 x^3 + 16 x^4)*A(x) + (4 x - 19 x^2 - 14 x^3)*A(x)^2 + (6 x^2 - 9 x^3 + 8 x^4)*A(x)^3 + 4 x^3*A(x)^4 + x^4*A(x)^5 = 0.
%F a(n) ~ (b*r^n)/((Pi*n^5)^(1/2)), where b = 0.805810... is the unique positive real root of 41472*x^6 - 34749*x^4 + 5472*x^2 - 256 and r = 4.658905... is the unique real root of 256*x^3 - 645*x^2 - 2112*x - 2048.
%F D-finite with recurrence -40*(148331*n-97009)*(4*n+3)*(2*n+3)*(4*n+5)*(n+2) *a(n) +(416275187*n^5 +1198175440*n^4 +714804925*n^3 -286654300*n^2 -249009732*n -17183760)*a(n-1) +60*(28713647*n^5 -13727276*n^4 -42761251*n^3 +18500340*n^2 +16274828*n -6917680)*a(n-2) +128*(n-2)*(2*n-3) *(7934261*n^3 -428899*n^2 -4370812*n -1585650)*a(n-3) +2048*(2*n-5)*(60637*n +16808)*(n-2)*(n-3)*(2*n-3)*a(n-4)=0. - _R. J. Mathar_, Jan 25 2023
%t m = 30; A[_] = 0;
%t Do[A[x_] = (-x^4 A[x]^5 - 4x^3 A[x]^4 + x^2 (-8x^2 + 9x - 6) A[x]^3 + x (14x^2 + 19x - 4) A[x]^2 - (x + 1)^2 (8x - 1))/(16x^4 + 28x^3 - 11x + 1) + O[x]^m, {m}];
%t CoefficientList[A[x], x] (* _Jean-François Alcover_, Sep 28 2019 *)
%Y Cf. A001006, A344498.
%K nonn
%O 0,4
%A _Colin Defant_, Apr 28 2019
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