A279569 Number of length n inversion sequences avoiding the patterns 110, 120, and 210.

1, 1, 2, 6, 22, 91, 409, 1953, 9763, 50583, 269697, 1472080, 8193306, 46359256, 266023710, 1545165168, 9070274236, 53739936609, 321025143482, 1931764542709, 11700651842997, 71288958790413, 436662467207291, 2687623420862395, 16615163817647042, 103131646740020637

Offset: 0

Author

Megan A. Martinez, Feb 21 2017

Keywords

• nonn

Comments

A length n inversion sequence e_1e_2...e_n is a sequence of integers where 0 <= e_i <= i-1. The term a(n) counts those length n inversion sequences with no entries e_i, e_j, e_k (where i e_k and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 110, 120, and 210.

It was shown that a_n also counts those length n inversion sequences with no entries e_i, e_j, e_k (where i e_j >= e_k and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 100, 120, and 210.

Examples

			The length 4 inversion sequences avoiding (110, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.
The length 4 inversion sequences avoiding (100, 120, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0121, 0122, 0123.
		

Links

• Alois P. Heinz, Table of n, a(n) for n = 0..400
• Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
• Hanna Mularczyk, Lattice Paths and Pattern-Avoiding Uniquely Sorted Permutations, arXiv:1908.04025 [math.CO], 2019.

Crossrefs

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279570, A279571, A279572, A279573.

Programs

Maple

b:= proc(n, i, t) option remember; `if`(n=0, 1,
      add(b(n-1, i-min(t, j)+2, abs(t-j)+1), j=1..i))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 21 2017

Mathematica

b[n_, i_, t_] := b[n, i, t] = If[n == 0, 1, Sum[b[n - 1, i - Min[t, j] + 2, Abs[t-j]+1], {j, 1, i}]]; a[n_] :=  b[n, 1, 1]; Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)

Formula

a(n) ~ c * (27/4)^n / n^(3/2), where c = 0.0111684107126703379786799829348... - Vaclav Kotesovec, Oct 07 2021

Extensions

a(10)-a(25) from Alois P. Heinz, Feb 21 2017