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A279569 Number of length n inversion sequences avoiding the patterns 110, 120, and 210.

%I A279569 #20 Oct 07 2021 03:40:05
%S A279569 1,1,2,6,22,91,409,1953,9763,50583,269697,1472080,8193306,46359256,
%T A279569 266023710,1545165168,9070274236,53739936609,321025143482,
%U A279569 1931764542709,11700651842997,71288958790413,436662467207291,2687623420862395,16615163817647042,103131646740020637
%N A279569 Number of length n inversion sequences avoiding the patterns 110, 120, and 210.
%C A279569 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 110, 120, and 210.
%C A279569 It was shown that a_n also 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 >= e_k and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 100, 120, and 210.
%H A279569 Alois P. Heinz, <a href="/A279569/b279569.txt">Table of n, a(n) for n = 0..400</a>
%H A279569 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.
%H A279569 Hanna Mularczyk, <a href="https://arxiv.org/abs/1908.04025">Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations</a>, arXiv:1908.04025 [math.CO], 2019.
%F A279569 a(n) ~ c * (27/4)^n / n^(3/2), where c = 0.0111684107126703379786799829348... - _Vaclav Kotesovec_, Oct 07 2021
%e A279569 The length 4 inversion sequences avoiding (110, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.
%e A279569 The length 4 inversion sequences avoiding (100, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0121, 0122, 0123.
%p A279569 b:= proc(n, i, t) option remember; `if`(n=0, 1,
%p A279569       add(b(n-1, i-min(t, j)+2, abs(t-j)+1), j=1..i))
%p A279569     end:
%p A279569 a:= n-> b(n, 1$2):
%p A279569 seq(a(n), n=0..30);  # _Alois P. Heinz_, Feb 21 2017
%t A279569 b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Sum[b[n - 1, i - Min[t, j] + 2, Abs[t-j]+1], {j, 1, i}]]; a[n_] :=  b[n, 1, 1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 10 2017, after _Alois P. Heinz_ *)
%Y A279569 Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279570, A279571, A279572, A279573.
%K A279569 nonn
%O A279569 0,3
%A A279569 _Megan A. Martinez_, Feb 21 2017
%E A279569 a(10)-a(25) from _Alois P. Heinz_, Feb 21 2017