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A279561 Number of length n inversion sequences avoiding the patterns 101, 102, 201, and 210. 23
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=0..26.

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

Chunyan Yan, 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-Fran├žois Alcover, Mar 28 2017 *)

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.

Sequence in context: A101265 A101879 A242622 * A294048 A063023 A150188

Adjacent sequences:  A279558 A279559 A279560 * A279562 A279563 A279564

KEYWORD

nonn,changed

AUTHOR

Megan A. Martinez, Jan 17 2017

STATUS

approved

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Last modified February 28 22:24 EST 2020. Contains 332335 sequences. (Running on oeis4.)