A328433 Number of inversion sequences of length n avoiding the consecutive patterns 011 and 012.

1, 1, 2, 4, 11, 37, 157, 791, 4676, 31490, 238814, 2009074, 18585645, 187366675, 2045016693, 24018394333, 302051731428, 4049206907012, 57642586053512, 868375941780450, 13801973373609889, 230808858283551859, 4051069379668626948, 74459335679007458268

Offset: 0

Author

Juan S. Auli, Oct 16 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}. This is the same as the set of inversion sequences of length n avoiding the consecutive patterns 011 and 012.

Examples

			The a(4)=11 length 4 inversion sequences avoiding the consecutive patterns 011 and 012 are 0000, 0100, 0010, 0020, 0001, 0101, 0021, 0002, 0102, 0003, and 0103.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..464
• 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, A328434, A328435, A328436, A328437, A328438, A328439, 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 * (3^(3/2)/(2*Pi))^n / n^alfa, where alfa = A073016 = Sum_{k>=1} 1/binomial(2*k, k) = 1/3 + 2*Pi/3^(5/2) = 0.73639985871871507790... and c = 2.21611825460684222558745179... - Vaclav Kotesovec, Oct 19 2019