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A328442 Number of inversion sequences of length n avoiding the consecutive pattern 210.

%I A328442 #16 Aug 21 2020 17:32:55
%S A328442 1,1,2,6,24,118,684,4554,34192,285558,2624496,26315990,285828324,
%T A328442 3342566724,41869664320,559265742918,7934746600620,119162454310392,
%U A328442 1888417811354292,31492626988890798,551302582228438512,10107905106374914860,193700015975819881008,3872391687779493752340,80623321999146782133372
%N A328442 Number of inversion sequences of length n avoiding the consecutive pattern 210.
%C A328442 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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 210.
%H A328442 Vaclav Kotesovec, <a href="/A328442/b328442.txt">Table of n, a(n) for n = 0..460</a>
%H A328442 Juan S. Auli, <a href="https://search.proquest.com/openview/3f0cef1fbdb016d61e16412e4b855969/1">Pattern Avoidance in Inversion Sequences</a>, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
%H A328442 Juan S. Auli and Sergi Elizalde, <a href="https://arxiv.org/abs/1904.02694">Consecutive Patterns in Inversion Sequences</a>, arXiv:1904.02694 [math.CO], 2019.
%H A328442 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 A328442 a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n * n^(2*Pi/3^(3/2)), where c = 0.24427562500895080639039917229089... - _Vaclav Kotesovec_, Oct 19 2019
%e A328442 Note that a(5)=118. Indeed, of the 120 inversion sequences of length 5, the only ones that do not avoid the consecutive patterns 210 are 00210 and 01210.
%p A328442 # after _Alois P. Heinz_ in A328357
%p A328442 b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
%p A328442        `if`(t and x < i, 0, b(n - 1, i, x < i)), i = 0 .. n - 1))
%p A328442      end proc:
%p A328442 a := n -> b(n, n, false):
%p A328442 seq(a(n), n = 0 .. 24);
%t A328442 b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && x < i, 0, b[n - 1, i, x < i]], {i, 0, n - 1}]];
%t A328442 a[n_] := b[n, n, False];
%t A328442 a /@ Range[0, 24] (* _Jean-François Alcover_, Mar 02 2020, after _Alois P. Heinz_ in A328357 *)
%Y A328442 Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441.
%K A328442 nonn
%O A328442 0,3
%A A328442 _Juan S. Auli_, Oct 17 2019