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%I #15 Mar 02 2020 09:40:11
%S 1,1,2,4,11,37,157,791,4676,31490,238814,2009074,18585645,187366675,
%T 2045016693,24018394333,302051731428,4049206907012,57642586053512,
%U 868375941780450,13801973373609889,230808858283551859,4051069379668626948,74459335679007458268
%N Number of inversion sequences of length n avoiding the consecutive patterns 011 and 012.
%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 011 and 012.
%H Alois P. Heinz, <a href="/A328433/b328433.txt">Table of n, a(n) for n = 0..464</a>
%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.
%F a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n / n^alfa, where alfa = A073016 = Sum_{k>=1} 1/binomial(2*k, k) = 1/3 + 2*Pi/3^(5/2) = 0.73639985871871507790... and c = 2.21611825460684222558745179... - _Vaclav Kotesovec_, Oct 19 2019
%e The a(4)=11 length 4 inversion sequences avoiding the consecutive patterns 011 and 012 are 0000, 0100, 0010, 0020, 0001, 0101, 0021, 0002, 0102, 0003, and 0103.
%p # after _Alois P. Heinz_ in A328357
%p b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
%p `if`(t and i < x, 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 && i < x, 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, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.
%K nonn
%O 0,3
%A _Juan S. Auli_, Oct 16 2019
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