A307787 Number of valid hook configurations of 132-avoiding permutations of [n].

1, 1, 1, 2, 5, 14, 43, 140, 477, 1683, 6106, 22664, 85735, 329572, 1284440, 5065828, 20188877, 81201801, 329281059, 1345059602, 5530600618, 22876354484, 95137126194, 397610249052, 1669285639455, 7037395810149, 29782584966376

Offset: 0

Author

Colin Defant, Apr 28 2019

Keywords

• nonn

Comments

Also the number of valid hook configurations of 231-avoiding permutations of [n].

For n > 0, a(n) is the number of intervals in the Motzkin-Tamari poset introduced by Fang.

Links

• Colin Defant, Motzkin intervals and valid hook configurations, arXiv preprint arXiv:1904.10451 [math.CO], 2019.
• Wenjie Fang, A partial order on Motzkin paths, arXiv preprint arXiv:1801.04809 [math.CO], 2018.

Crossrefs

Cf. A001006, A344498.

Programs

Mathematica

m = 30; A[_] = 0;
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}];
CoefficientList[A[x], x] (* Jean-François Alcover, Sep 28 2019 *)

Formula

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.

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.

D-finite with recurrence -40*(148331*n-97009)*(4*n+3)*(2*n+3)*(4*n+5)*(n+2) *a(n) +(416275187*n^5 [phone]*n^4 [phone]*n^3 -286654300*n^2 -249009732*n -17183760)*a(n-1) +60*(28713647*n^5 -13727276*n^4 -42761251*n^3 [phone]*n^2 [phone]*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