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

0,3

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<j<k) such that e_j <> 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<j<k) such that e_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.

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.

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

Example

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.

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 *)

Prog

(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

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.

Sequence in context: A196836 A365247 A369212 * A007853 A337522 A149952

Adjacent sequences: A279550 A279551 A279552 * A279554 A279555 A279556

Keyword

nonn

Author

Megan A. Martinez, Dec 15 2016

Extensions

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

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

Status

approved