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

0,3

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<j<k) such that e_j > 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<j<k) such that e_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.

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.

Formula

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

Example

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.

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 *)

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.

Sequence in context: A124293 A341382 A107591 * A155866 A150273 A342292

Adjacent sequences: A279566 A279567 A279568 * A279570 A279571 A279572

Keyword

nonn

Author

Megan A. Martinez, Feb 21 2017

Extensions

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

Status

approved