A279568 Number of length n inversion sequences avoiding the patterns 110, 120, 201, and 210.

1, 1, 2, 6, 22, 90, 396, 1833, 8801, 43441, 219092, 1124201, 5850414, 30805498, 163824559, 878655117, 4747341879, 25815026491, 141173582016, 775920816789, 4283833709457, 23746640019657, 132116647765569, 737485227605338, 4129174120158569, 23183379592361839

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, 201, 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 and e_i > e_k. This is the same as the set of length n inversion sequences avoiding 100, 120, 201, and 210.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1294

Megan A. Martinez, Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.

Formula

a(n) ~ c * d^n / n^(3/2), where d = 5.98041772076926677236919875200507... is the positive root of the equation -32 - 195*d - 12*d^2 - 112*d^3 + 20*d^4 = 0 and c = 0.1056946795054351807407212356928404107733262398133039312067247126343... - Vaclav Kotesovec, Oct 07 2021

Example

The length 4 inversion sequences avoiding (110, 120, 201, 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, 201, 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, l) option remember; `if`(n=0, 1, add((h->
      b(n-1, i-h+2, j-h+1))(max(1, `if`(j=l, 0, l))), j=1..i))
    end:
a:= n-> b(n, 1$2):
seq(a(n), n=0..30);  # Alois P. Heinz, Feb 23 2017

Mathematica

b[n_, i_, l_] := b[n, i, l] = If[n == 0, 1, Sum[b[n-1, i-#+2, j-#+1]& @ Max[1, If[j == l, 0, l]], {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, A279569, A279570, A279571, A279572, A279573.

Sequence in context: A103137 A340892 A165546 * A053617 A089449 A264601

Adjacent sequences: A279565 A279566 A279567 * A279569 A279570 A279571

Keyword

nonn

Author

Megan A. Martinez, Feb 21 2017

Extensions

a(10)-a(25) from Alois P. Heinz, Feb 23 2017

Status

approved