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%I #22 Oct 07 2021 04:07:23
%S 1,1,2,6,21,76,277,1016,3756,13998,52554,198568,754316,2878552,
%T 11027384,42384412,163372325,631290168,2444700421,9485463044,
%U 36866810877,143508889270,559399074443,2183269032876,8530724152279,33366805383326,130633854520329,511889287682280
%N Number of length n inversion sequences avoiding the patterns 100, 210, 201, and 102.
%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_i > e_j and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 100, 210, 201, and 102.
%H Alois P. Heinz, <a href="/A279560/b279560.txt">Table of n, a(n) for n = 0..1665</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 a(n) = binomial(2n-2,n-1) + Sum_{k=2..n-2} Sum_{i=1..k-1} Sum_{u=1..i} Sum_{d=0..u-1} ((i-d+1)/(i+1)*binomial(i+d,d)) for n>0, a(0)=1.
%F a(n) ~ 4^(n-1) / sqrt(Pi*n). - _Vaclav Kotesovec_, Oct 07 2021
%e The length 4 inversion sequences avoiding (100, 210, 201, 102) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
%p a:= proc(n) option remember; `if`(n<4, n!,
%p ((6*(9*n^4-61*n^3+100*n^2+52*n-140))*a(n-1)
%p -(3*(3*n-8))*(9*n^3-38*n^2+3*n+70)*a(n-2)
%p +(2*(2*n-7))*(9*n^3-31*n^2-2*n+60)*a(n-3))
%p / ((9*n^3-58*n^2+87*n+22)*n))
%p end:
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Feb 24 2017
%t a[0] = 1; a[n_] := Binomial[2n-2, n-1] + Sum[(4i Binomial[2i+1, i+1]) / ((i+2)(i+3)), {k, 2, n-2}, {i, 1, k-1}]; Array[a, 30, 0] (* _Jean-François Alcover_, Nov 06 2017 *)
%o (PARI) a(n) = if (n==0, 1, binomial(2*n-2,n-1) + sum(k=2, n-2, sum(i=1,k-1, sum(u=1, i, sum(d=0, u-1, ((i-d+1)/(i+1)*binomial(i+d,d))))))); \\ _Michel Marcus_, Jan 18 2017
%Y Cf. A000108, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.
%K nonn
%O 0,3
%A _Megan A. Martinez_, Jan 17 2017
%E More terms from _Michel Marcus_, Jan 18 2017
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