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Number of length n inversion sequences avoiding the patterns 110, 210, 120, 201, and 010.
23

%I #39 Feb 21 2020 07:32:36

%S 1,1,2,5,15,50,178,663,2552,10071,40528,165682,686151,2872576,

%T 12137278,51690255,221657999,956265050,4147533262,18074429421,

%U 79102157060,347519074010,1532070899412,6775687911920,30052744139440,133649573395725,595816470717728

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

%C 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.

%C 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.

%H Alois P. Heinz, <a href="/A279553/b279553.txt">Table of n, a(n) for n = 0..1489</a>

%H Megan A. Martinez, Carla D. Savage, <a href="https://arxiv.org/abs/1609.08106">Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations</a>, arXiv:1609.08106 [math.CO], 2016.

%F G.f.: 1 + Series_Reversion(x*A094373(-x)). - _Gheorghe Coserea_, Jul 11 2018

%F 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

%F 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

%e The length 3 inversion sequences avoiding (110, 210, 120, 201, 010) are 000, 001, 002, 011, 012.

%e 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.

%p a:= proc(n) option remember; `if`(n<4, [1, 1, 2, 5][n+1],

%p ((12*(n-1))*(182*n^3-1659*n^2+4628*n-3756)*a(n-1)

%p -(4*(91*n^4-1057*n^3+3812*n^2-4046*n-906))*a(n-2)

%p +(6*(n-4))*(182*n^3-1659*n^2+4901*n-4630)*a(n-3)

%p -(4*(n-4))*(n-5)*(91*n^2-511*n+690)*a(n-4))

%p /(5*n*(n-1)*(91*n^2-693*n+1292)))

%p end:

%p seq(a(n), n=0..30); # _Alois P. Heinz_, Feb 22 2017

%t 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_ *)

%o (PARI)

%o seq(N) = my(x='x+O('x^N)); Vec(1+serreverse((-x^3+x^2+x)/(2*x^2+3*x+1)));

%o seq(27) \\ _Gheorghe Coserea_, Jul 11 2018

%Y 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.

%K nonn

%O 0,3

%A _Megan A. Martinez_, Dec 15 2016

%E a(10)-a(16) from _Lars Blomberg_, Feb 02 2017

%E a(17)-a(26) from _Alois P. Heinz_, Feb 22 2017