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A328431 Number of inversion sequences of length n avoiding the consecutive patterns 010, 021, 120, and 210.

%I A328431 #17 Jun 01 2022 01:55:23
%S A328431 1,1,2,5,15,53,214,960,4701,24873,141147,853641,5472642,37024569,
%T A328431 263342224,1962835806,15288074104,124120865849,1048092680689,
%U A328431 9186689045482,83435365244510,783923558286071,7608398620990535,76177574145052258,785853360840424425
%N A328431 Number of inversion sequences of length n avoiding the consecutive patterns 010, 021, 120, and 210.
%C A328431 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 010, 021, 120, and 210.
%H A328431 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 A328431 The a(4)=15 length 4 inversion sequences avoiding the consecutive patterns 010, 021, 120, and 210 are 0000, 0110, 0001, 0011, 0111, 0002, 0012, 0112, 0022, 0122, 0003, 0013, 0113, 0023, and 0123.
%p A328431 # after _Alois P. Heinz_ in A328357
%p A328431 b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
%p A328431        `if`(t and i <> x, 0, b(n - 1, i, x < i)), i = 0 .. n - 1))
%p A328431      end proc:
%p A328431 a := n -> b(n, n, false):
%p A328431 seq(a(n), n = 0 .. 24);
%t A328431 b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i != x, 0, b[n - 1, i, x < i]], {i, 0, n - 1}]];
%t A328431 a[n_] := b[n, n, False];
%t A328431 a /@ Range[0, 24] (* _Jean-François Alcover_, Mar 02 2020 after _Alois P. Heinz_ in A328357 *)
%Y A328431 Cf. A328357, A328358, A328429, A328430, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.
%K A328431 nonn
%O A328431 0,3
%A A328431 _Juan S. Auli_, Oct 16 2019