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A307787 Number of valid hook configurations of 132-avoiding permutations of [n]. 2
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,4

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

Table of n, a(n) for n=0..26.

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.

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.

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-Fran├žois Alcover, Sep 28 2019 *)

CROSSREFS

Cf. A001006, A344498.

Sequence in context: A272461 A213264 A029889 * A221586 A258312 A123020

Adjacent sequences:  A307784 A307785 A307786 * A307788 A307789 A307790

KEYWORD

nonn

AUTHOR

Colin Defant, Apr 28 2019

STATUS

approved

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Last modified August 9 21:08 EDT 2022. Contains 356026 sequences. (Running on oeis4.)