A328439 Number of inversion sequences of length n avoiding the consecutive pattern 011.

1, 1, 2, 5, 17, 75, 407, 2621, 19524, 165090, 1561900, 16345264, 187452475, 2337729329, 31497068553, 455930417721, 7056447326642, 116279714536838, 2032547040624336, 37563420959431569, 731810131489893185, 14989602024463575408, 322032777284323744894, 7240745954488939549295

Offset: 0

Author

Juan S. Auli, Oct 17 2019

Keywords

• nonn

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 011.

Examples

			The a(4)=17 length 4 inversion sequences avoiding the consecutive pattern 011 are 0000, 0100, 0010, 0020, 0120, 0001, 0101, 0021, 0121, 0002, 0102, 0012, 0003, 0103, 0013, 0023, and 0123.
		

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.

Crossrefs

Cf. A328357, A328358, A328429, A328430, A328431, A328432, A328433, A328434, A328435, A328436, A328437, A328438, A328440, A328441, A328442.

Programs

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 *)

Formula

a(n) ~ n! * c / sqrt(n), where c = 1.3306953765239857433314976921138998977998... - Vaclav Kotesovec, Oct 19 2019