A328442 Number of inversion sequences of length n avoiding the consecutive pattern 210.

1, 1, 2, 6, 24, 118, 684, 4554, 34192, 285558, 2624496, 26315990, 285828324, 3342566724, 41869664320, 559265742918, 7934746600620, 119162454310392, 1888417811354292, 31492626988890798, 551302582228438512, 10107905106374914860, 193700015975819881008, 3872391687779493752340, 80623321999146782133372

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 210.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..460

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 * (3^(3/2)/(2*Pi))^n * n^(2*Pi/3^(3/2)), where c = 0.24427562500895080639039917229089... - Vaclav Kotesovec, Oct 19 2019

Example

Note that a(5)=118. Indeed, of the 120 inversion sequences of length 5, the only ones that do not avoid the consecutive patterns 210 are 00210 and 01210.

Maple

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

Mathematica

b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && x < i, 0, b[n - 1, i, x < i]], {i, 0, n - 1}]];
a[n_] := b[n, n, 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, A328437, A328438, A328439, A328440, A328441.

Sequence in context: A298432 A336072 A328501 * A135106 A248837 A005394

Adjacent sequences: A328439 A328440 A328441 * A328443 A328444 A328445

Keyword

nonn

Author

Juan S. Auli, Oct 17 2019

Status

approved