A279553 Number of length n inversion sequences avoiding the patterns 110, 210, 120, 201, and 010.

1, 1, 2, 5, 15, 50, 178, 663, 2552, 10071, 40528, 165682, 686151, 2872576, 12137278, 51690255, 221657999, 956265050, 4147533262, 18074429421, 79102157060, 347519074010, 1532070899412, 6775687911920, 30052744139440, 133649573395725, 595816470717728

Offset: 0

Author

Megan A. Martinez, Dec 15 2016

Keywords

• nonn

Comments

A length n inversion sequence e_1e_2...e_n is a sequence of integers where 0 <= e_i <= i-1. The term a(n) counts those length n inversion sequences with no entries e_i, e_j, e_k (where i e_k and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 010, 110, 120, 201, and 210.

It can be shown that this sequence also counts the length n inversion sequences with no entries e_i, e_j, e_k (where i e_j and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 010, 100, 120, 201, and 210.

Examples

			The length 3 inversion sequences avoiding (110, 210, 120, 201, 010) are 000, 001, 002, 011, 012.
The length 4 inversion sequences avoiding (110, 210, 120, 201, 010) are 0000, 0001, 0002, 0003, 0011, 0012, 0013, 0021, 0022, 0023, 0111, 0112, 0113, 0122, 0123.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..1489
• Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Crossrefs

Cf. A263777, A263778, A263779, A263780, A279551, A279552, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Programs

Maple

a:= proc(n) option remember; `if`(n<4, [1, 1, 2, 5][n+1],
      ((12*(n-1))*(182*n^3-1659*n^2+4628*n-3756)*a(n-1)
      -(4*(91*n^4-1057*n^3+3812*n^2-4046*n-906))*a(n-2)
      +(6*(n-4))*(182*n^3-1659*n^2+4901*n-4630)*a(n-3)
      -(4*(n-4))*(n-5)*(91*n^2-511*n+690)*a(n-4))
       /(5*n*(n-1)*(91*n^2-693*n+1292)))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 22 2017

Mathematica

a[n_] := a[n] = If[n < 4, {1, 1, 2, 5}[[n + 1]], ((12*(n - 1))*(182*n^3 - 1659*n^2 + 4628*n - 3756)*a[n - 1] - (4*(91*n^4 - 1057*n^3 + 3812*n^2 - 4046*n - 906))*a[n - 2] + (6*(n - 4))*(182*n^3 - 1659*n^2 + 4901*n - 4630)*a[n - 3] - (4*(n - 4))*(n - 5)*(91*n^2 - 511*n + 690)*a[n - 4]) / (5*n*(n - 1)*(91*n^2 - 693*n + 1292))]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)

PARI

seq(N) = my(x='x+O('x^N)); Vec(1+serreverse((-x^3+x^2+x)/(2*x^2+3*x+1)));
seq(27) \\ Gheorghe Coserea, Jul 11 2018

Formula

G.f.: 1 + Series_Reversion(x*A094373(-x)). - Gheorghe Coserea, Jul 11 2018

a(n) ~ c * d^n / (sqrt(Pi) * n^(3/2)), where d = 4.730576939379622099763633264641101585205420756515858657461873... is the greatest real root of the equation 4 - 12*d + 4*d^2 - 24*d^3 + 5*d^4 = 0 and c = 0.3916760466183576202289779130261876915536170330427700961416097... is the positive real root of the equation -5 - 64*c^2 - 33728*c^4 + 209664*c^6 + 93184*c^8 = 0. - Vaclav Kotesovec, Jul 12 2018

D-finite with recurrence: 45*n*(n-1)*a(n) -4*(n-1)*(49*n-66)*a(n-1) +2*(-25*n^2+157*n-264)*a(n-2) +2*(-70*n^2+445*n-714)*a(n-3) -4*(n-6)*(n-13)*a(n-4) -4*(n-6)*(2*n-17)*a(n-5) +8*(n-6)*(n-7)*a(n-6)=0. - R. J. Mathar, Feb 21 2020

Extensions

a(10)-a(16) from Lars Blomberg, Feb 02 2017

a(17)-a(26) from Alois P. Heinz, Feb 22 2017