A328357 - OEIS

%I #31 Feb 25 2020 08:14:47

%S 1,1,2,1,4,6,36,117,804,4266,33768,249144,2289348,21353472,227212824,

%T 2533824900,30914509212,398623158096,5508014798052,80377645583430,

%U 1242697826967816,20218588415853480,346035438765576720,6206862951272939550,116518581654518098332

%N Number of inversion sequences of length n avoiding the consecutive patterns 000, 001, 011, 012.

%C 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}. Alternatively, we can describe this as the set of inversion sequences of length n avoiding the consecutive patterns 000, 001, 011, 012.

%H Alois P. Heinz, <a href="/A328357/b328357.txt">Table of n, a(n) for n = 0..465</a>

%H 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 a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n / n^(2*Pi/3^(3/2)), where c = 0.75844492121718325018323312623016463... - _Vaclav Kotesovec_, Oct 17 2019

%e The a(4)=4 length 4 inversion sequences avoiding the consecutive patterns 000, 001, 011, 012 are 0100, 0101, 0102, 0103.

%e The a(5)=6 length 5 inversion sequences are 01010, 01020, 01021, 01030, 01031, 01032.

%p b:= proc(n, x, t) option remember; `if`(n=0, 1, add(

%p      `if`(t and i<=x, 0, b(n-1, i, i<=x)), i=1..n))

%p     end:

%p a:= n-> b(n, 0, false):

%p seq(a(n), n=0..24);  # _Alois P. Heinz_, Oct 14 2019

%t b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && i <= x, 0, b[n - 1, i, i <= x]], {i, 1, n}]];

%t a[n_] :=  b[n, 0, False];

%t a /@ Range[0, 24] (* _Jean-François Alcover_, Feb 25 2020, after _Alois P. Heinz_ *)

%Y Cf. A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

%K nonn

%O 0,3

%A _Juan S. Auli_, Oct 13 2019

%E Terms a(11)..a(16) from _Joerg Arndt_, Oct 14 2019

%E a(17)-a(24) from _Alois P. Heinz_, Oct 14 2019