A279561 Number of length n inversion sequences avoiding the patterns 101, 102, 201, and 210.

1, 1, 2, 6, 21, 77, 287, 1079, 4082, 15522, 59280, 227240, 873886, 3370030, 13027730, 50469890, 195892565, 761615285, 2965576715, 11563073315, 45141073925, 176423482325, 690215089745, 2702831489825, 10593202603775, 41550902139551, 163099562175851

Offset: 0

Author

Megan A. Martinez, Jan 17 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_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 e_k and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 021 and 120.

Examples

			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.
		

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.

Crossrefs

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Programs

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 *)

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