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%I A279564 #37 Jul 11 2024 14:27:40
%S A279564 1,1,2,5,16,60,260,1267,6850,40572,260812,1805646,13377274,105487540,
%T A279564 881338060,7770957903,72060991394,700653026744,7123871583656,
%U A279564 75561097962918,834285471737784,9570207406738352,113855103776348136,1402523725268921870,17863056512845724036,234910502414771617316,3185732802058088068444,44501675392317774477088
%N A279564 Number of length n inversion sequences avoiding the patterns 000 and 100.
%C A279564 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_i >= e_j = e_k. This is the same as the set of length n inversion sequences avoiding 000 and 100.
%H A279564 Benjamin Testart, <a href="/A279564/b279564.txt">Table of n, a(n) for n = 0..540</a>
%H A279564 Megan A. Martinez and 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 A279564 Benjamin Testart, <a href="https://arxiv.org/abs/2407.07701">Completing the enumeration of inversion sequences avoiding one or two patterns of length 3</a>, arXiv:2407.07701 [math.CO], 2024.
%H A279564 Chunyan Yan and Zhicong Lin, <a href="https://arxiv.org/abs/1912.03674">Inversion sequences avoiding pairs of patterns</a>, arXiv:1912.03674 [math.CO], 2019.
%F A279564 The length 4 inversion sequences avoiding (000,100) are 0011, 0012, 0013, 0021, 0022, 0023, 0101, 0102, 0103, 0110, 0112, 0113, 0120, 0121, 0122, 0123.
%p A279564 b:= proc(n, i, m, s) option remember; `if`(n=0, 1, add(
%p A279564       `if`(j in s, 0, b(n-1, i+1, max(m, j),
%p A279564       `if`(j<=m, s union {j}, s))), j=1..i))
%p A279564     end:
%p A279564 a:= n-> b(n, 1, 0, {}):
%p A279564 seq(a(n), n=0..15);  # _Alois P. Heinz_, Feb 22 2017
%t A279564 b[n_, i_, m_, s_List] := b[n, i, m, s] = If[n == 0, 1, Sum[If[MemberQ[s, j], 0, b[n-1, i+1, Max[m, j], If[j <= m, s ~Union~ {j}, s]]], {j, 1, i}] ]; a[n_] := b[n, 1, 0, {}]; Table[a[n], {n, 0, 15}] (* _Jean-François Alcover_, Jul 10 2017, after _Alois P. Heinz_ *)
%Y A279564 Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.
%K A279564 nonn
%O A279564 0,3
%A A279564 _Megan A. Martinez_, Feb 09 2017
%E A279564 a(10)-a(23) from _Alois P. Heinz_, Feb 22 2017
%E A279564 a(24)-a(27) from _Vaclav Kotesovec_, Oct 08 2021

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