A328430 - OEIS

%I #16 Jun 01 2022 01:55:15

%S 1,1,2,3,7,18,70,317,1825,11805,88212,727731,6660103,66377942,

%T 718681969,8376682083,104703957902,1395883946839,19777652272297,

%U 296686846198829,4697959440255354,78299282813403618,1370127872827224359,25114095425698971152,481202765468970358153

%N Number of inversion sequences of length n avoiding the consecutive patterns 001 and 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}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 001 and 012.

%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.

%e The a(4)=7 length 4 inversion sequences avoiding the consecutive patterns 001 and 012 are 0000, 0100, 0110, 0101, 0111, 0102, and 0103.

%p # after _Alois P. Heinz_ in A328357

%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 = 0 .. n - 1))

%p      end proc:

%p a := n -> b(n, -1, false):

%p seq(a(n), n = 0 .. 24);

%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, 0, n - 1}]];

%t a[n_] := b[n, -1, False];

%t a /@ Range[0, 24] (* _Jean-François Alcover_, Mar 02 2020 after _Alois P. Heinz_ in A328357 *)

%Y Cf. A328357, A328358, A328429, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328439, A328440, A328441, A328442.

%K nonn

%O 0,3

%A _Juan S. Auli_, Oct 15 2019