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_i > e_j <> e_k. This is the same as the set of length n inversion sequences avoiding 101, 102, 201, and 210.
It is conjectured 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 021 and 120.
LINKS
Alois P. Heinz, Table of n, a(n) for n = 0..1664
Shane Chern, On 0012-avoiding inversion sequences and a Conjecture of Lin and Ma, arXiv:2006.04318 [math.CO], 2020.
A. V. Kitaev and A. Vartanian, Algebroid Solutions of the Degenerate Third Painlevé Equation for Vanishing Formal Monodromy Parameter, arXiv:2304.05671 [math.CA], 2023.
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.
FORMULA
a(n) = 1 + Sum_{i=1..n-1} binomial(2i, i-1).
a(n) = 1 + A057552(n-2).
G.f.: (1-4*x+sqrt(-16*x^3+20*x^2-8*x+1))/(2*(x-1)*(4*x-1)).
D-finite with recurrence: n*a(n) +(-7*n+6)*a(n-1) +2*(7*n-13)*a(n-2) +4*(-2*n+5)*a(n-3)=0. - R. J. Mathar, Feb 21 2020
EXAMPLE
The length 4 inversion sequences avoiding (101, 102, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0100, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.
The length 4 inversion sequences avoiding (021, 120) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0022, 0023, 0100, 0101, 0102, 0103, 0110, 0111, 0112, 0113, 0122, 0123.
MAPLE
a:= proc(n) option remember; `if`(n<3, 1+n*(n-1)/2,
((5*n^2-12*n+6)*a(n-1)-(4*n^2-10*n+6)*a(n-2))/((n-2)*n))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Jan 18 2017
MATHEMATICA
a[n_] := 1 + Sum[Binomial[2i, i-1], {i, 0, n-1}];
Table[a[n], {n, 0, 30}] (* Jean-François Alcover, Mar 28 2017 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Megan A. Martinez, Jan 17 2017
STATUS
approved