A307787 -id:A307787 - cp's OEIS frontend

A307788 Number of valid hook configurations of permutations of [n] that avoid the patterns 231 and 321.

1, 1, 1, 2, 5, 12, 29, 72, 182, 466, 1207, 3158, 8334, 22158, 59299, 159614, 431838, 1173710, 3203244, 8774780, 24118522, 66497316, 183858411, 509670494, 1416231616, 3944027402, 11006186760, 30772507128, 86191006746, 241815195292

Offset: 0

Colin Defant, Apr 28 2019

nonn

Essentially the same as A217333 and A025273. - R. J. Mathar, May 17 2019

Colin Defant, Motzkin intervals and valid hook configurations, arXiv preprint arXiv:1904.10451 [math.CO], 2019.

Cf. A307786, A307787.

my(x='x+O('x^35)); Vec((1 - 2*x + 2*x^2 - sqrt(1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4))/(2*x^2)) \\ Michel Marcus, May 08 2019

G.f.: (1 - 2*x + 2*x^2 - sqrt(1 - 4*x + 4*x^2 - 4*x^3 + 4*x^4)) / (2*x^2).

D-finite with recurrence: (n+2)*a(n) +2*(-2*n-1)*a(n-1) +4*(n-1)*a(n-2) +2*(-2*n+5)*a(n-3) +4*(n-4)*a(n-4)=0. - R. J. Mathar, May 17 2019

A334498 Number of intervals in Fang's Schroeder-Tamari poset.

Original entry on oeis.org

2, 8, 46, 320, 2500, 21120, 188758, 1760256, 16969756, 168022016, 1700483916, 17527963648, 183499999368, 1946861076480, 20896083575142, 226570927865856, 2478789884919084, 27336509563600896, 303635676268456996, 3394385993908879360, 38168423356190965688, 431472747874361540608

Offset: 1

N. J. A. Sloane, May 07 2020

nonn

Fang (2020), Theorem 4.2, gives a generating function.

Wenjie Fang, A partial order on Motzkin paths, Discrete Math., 343 (2020), #111802. See Section 4.

Wenjie Fang, A partial order on Motzkin paths, arXiv preprint arXiv:1801.04809 [math.CO], 2018.

Cf. A307787.

Rest[CoefficientList[Series[(-1 + w^4*x^2 + w*(1 + 2*x) - w^3*(3*x + 2*x^2)) / (w*x*(-1 + w^2*x)) /. w -> Root[-1 + #1 + 2*x*#1^2 - 2*x*#1^3 - x*(1 + x)*#1^4 + x^2*#1^5 &, 1], {x, 0, 30}], x]] (* Vaclav Kotesovec, May 07 2020 *)

a(n) ~ sqrt(5/9 + 1/sqrt(3)) * (4*sqrt(45 + 26*sqrt(3))/3)^n / (sqrt(Pi)*n^(5/2)). - Vaclav Kotesovec, May 07 2020

More terms from Vaclav Kotesovec, May 07 2020

cp's OEIS Frontend

A307788 Number of valid hook configurations of permutations of [n] that avoid the patterns 231 and 321.

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