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%I #17 Oct 07 2021 04:37:42
%S 1,1,2,6,22,90,396,1833,8801,43441,219092,1124201,5850414,30805498,
%T 163824559,878655117,4747341879,25815026491,141173582016,775920816789,
%U 4283833709457,23746640019657,132116647765569,737485227605338,4129174120158569,23183379592361839
%N Number of length n inversion sequences avoiding the patterns 110, 120, 201, and 210.
%C 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_j <> e_k and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 110, 120, 201, and 210.
%C It was shown that a_n also 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 and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 100, 120, 201, and 210.
%H Alois P. Heinz, <a href="/A279568/b279568.txt">Table of n, a(n) for n = 0..1294</a>
%H Megan A. Martinez, 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.
%F a(n) ~ c * d^n / n^(3/2), where d = 5.98041772076926677236919875200507... is the positive root of the equation -32 - 195*d - 12*d^2 - 112*d^3 + 20*d^4 = 0 and c = 0.1056946795054351807407212356928404107733262398133039312067247126343... - _Vaclav Kotesovec_, Oct 07 2021
%e The length 4 inversion sequences avoiding (110, 120, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.
%e The length 4 inversion sequences avoiding (100, 120, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0121, 0122, 0123.
%p b:= proc(n, i, l) option remember; `if`(n=0, 1, add((h->
%p b(n-1, i-h+2, j-h+1))(max(1, `if`(j=l, 0, l))), j=1..i))
%p end:
%p a:= n-> b(n, 1$2):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Feb 23 2017
%t b[n_, i_, l_] := b[n, i, l] = If[n == 0, 1, Sum[b[n-1, i-#+2, j-#+1]& @ Max[1, If[j == l, 0, l]], {j, 1, i}]]; a[n_] := b[n, 1, 1]; Table[a[n], {n, 0, 30}] (* _Jean-François Alcover_, Jul 10 2017, after _Alois P. Heinz_ *)
%Y Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279569, A279570, A279571, A279572, A279573.
%K nonn
%O 0,3
%A _Megan A. Martinez_, Feb 21 2017
%E a(10)-a(25) from _Alois P. Heinz_, Feb 23 2017
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