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%I #13 Jun 01 2022 01:56:09
%S 1,1,2,5,18,81,448,2920,21955,186981,1779170,18706222,215364181,
%T 2694650157,36408144034,528302958022,8193953571315,135277259197031,
%U 2368556730208679,43838335667451773,855200666797199814,17538187897491897945,377199969925672569364,8489656058119117230574
%N Number of inversion sequences of length n avoiding the consecutive patterns 000 and 100.
%C A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i >= e_{i+1} = e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 100.
%C The term a(n) also counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i = e_{i+1} >= e_{i+2}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 000 and 110, see the Auli and Elizalde reference.
%H Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/1906.07365">Consecutive patterns in inversion sequences II: avoiding patterns of relations</a>, arXiv:1906.07365 [math.CO], 2019.
%e The a(4)=18 length 4 inversion sequences avoiding the consecutive patterns 000 and 100 are 0010, 0110, 0020, 0120, 0101, 0011, 0021, 0121, 0102, 0012, 0112, 0022, 0122, 0103, 0013, 0113, 0023, and 0123.
%p # after _Alois P. Heinz_ in A328357
%p b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
%p `if`(t and x <= i, 0, b(n - 1, i, i = x)), i = 0 .. n - 1))
%p end proc:
%p a := n -> b(n, -1, false):
%p seq(a(n), n = 0 .. 24);
%t b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && x <= i, 0, b[n - 1, i, i == x]], {i, 0, n - 1}]];
%t a[n_] := b[n, -1, False];
%t a /@ Range[0, 24] (* _Jean-François Alcover_, Mar 02 2020 after _Alois P. Heinz_ in A328357 *)
%Y Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328441, A328442.
%K nonn
%O 0,3
%A _Juan S. Auli_, Oct 17 2019
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