|
%I #15 Oct 07 2021 04:29:21
%S 1,1,2,6,21,82,343,1509,6893,32419,156058,765578,3815062,19263736,
%T 98368919,507197436,2637242188,13814247530,72834238423,386244387688,
%U 2058933104170,11026807340592,59304897232442,320181600386661,1734685419170666,9428340999504441
%N Number of length n inversion sequences avoiding the patterns 100, 110, 120, 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 100, 110, 120, and 210.
%H Alois P. Heinz, <a href="/A279567/b279567.txt">Table of n, a(n) for n = 0..400</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 * (1 + sqrt(2))^(2*n) / n^(3/2), where c = 0.066085708825649431003670013119332303648755519420440375... - _Vaclav Kotesovec_, Oct 07 2021
%e The length 4 inversion sequences avoiding (100, 110, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.
%p b:= proc(n, i, m) option remember; `if`(n=0, 1, add((h->
%p b(n-1, i-h+1, max(m, j)-h))(max(0, min(m-1, j))), j=1..i))
%p end:
%p a:= n-> b(n, 1, 0):
%p seq(a(n), n=0..30); # _Alois P. Heinz_, Feb 23 2017
%t b[n_, i_, m_] := b[n, i, m] = If[n == 0, 1, Sum[b[n-1, i-#+1, Max[m, j]-#]& @ Max[0, Min[m-1, j]], {j, 1, i}]]; a[n_] := b[n, 1, 0]; 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, A279568, A279569, A279570, A279571, A279572, A279573.
%K nonn
%O 0,3
%A _Megan A. Martinez_, Feb 09 2017
%E a(10)-a(25) from _Alois P. Heinz_, Feb 23 2017
|
