A328437 Number of inversion sequences of length n avoiding the consecutive pattern 001.

1, 1, 2, 4, 11, 42, 210, 1292, 9352, 77505, 722294, 7470003, 84854788, 1049924370, 14052654158, 202271440732, 3115338658280, 51118336314648, 890201500701303, 16397264064993185, 318505677099378561, 6506565509515408206, 139449260758011488550, 3128599281190613701180

Offset

0,3

Comments

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}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 001.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..448

Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.

Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.

Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.

Formula

a(n) ~ n! * c / sqrt(n), where c = 0.549342310436989831962783548104445992522... - Vaclav Kotesovec, Oct 18 2019

Example

The a(4)=11 length 4 inversion sequences avoiding the consecutive pattern 001 are 0000, 0100, 0110, 0120, 0101, 0111, 0121, 0102, 0122, 0103, and 0123.

Maple

# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
       `if`(t and i = x, 0, b(n - 1, i, i < x)), i = 0 .. n - 1))
     end proc:
a := n -> b(n, -1, false):
seq(a(n), n = 0 .. 24);

Mathematica

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}]];
a[n_] := b[n, -1, False];
a /@ Range[0, 24] (* Jean-François Alcover, Mar 02 2020, after Alois P. Heinz in A328357 *)

Crossrefs

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328438, A328439, A328440, A328441, A328442.

Sequence in context: A067353 A105996 A213937 * A107703 A140838 A114954

Adjacent sequences: A328434 A328435 A328436 * A328438 A328439 A328440

Keyword

nonn

Author

Juan S. Auli, Oct 17 2019

Status

approved