A328357 Number of inversion sequences of length n avoiding the consecutive patterns 000, 001, 011, 012.

1, 1, 2, 1, 4, 6, 36, 117, 804, 4266, 33768, 249144, 2289348, 21353472, 227212824, 2533824900, 30914509212, 398623158096, 5508014798052, 80377645583430, 1242697826967816, 20218588415853480, 346035438765576720, 6206862951272939550, 116518581654518098332

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

Links

Alois P. Heinz, Table of n, a(n) for n = 0..465

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 * (3^(3/2)/(2*Pi))^n / n^(2*Pi/3^(3/2)), where c = 0.75844492121718325018323312623016463... - Vaclav Kotesovec, Oct 17 2019

Example

The a(4)=4 length 4 inversion sequences avoiding the consecutive patterns 000, 001, 011, 012 are 0100, 0101, 0102, 0103.
The a(5)=6 length 5 inversion sequences are 01010, 01020, 01021, 01030, 01031, 01032.

Maple

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=1..n))
    end:
a:= n-> b(n, 0, false):
seq(a(n), n=0..24);  # Alois P. Heinz, Oct 14 2019

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, 1, n}]];
a[n_] :=  b[n, 0, False];
a /@ Range[0, 24] (* Jean-François Alcover, Feb 25 2020, after Alois P. Heinz *)

Crossrefs

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

Sequence in context: A268573 A145858 A328491 * A341696 A052566 A234946

Adjacent sequences: A328354 A328355 A328356 * A328358 A328359 A328360

Keyword

nonn

Author

Juan S. Auli, Oct 13 2019

Extensions

Terms a(11)..a(16) from Joerg Arndt, Oct 14 2019

a(17)-a(24) from Alois P. Heinz, Oct 14 2019

Status

approved