A279557 Number of length n inversion sequences avoiding the patterns 110, 120, and 021.

1, 1, 2, 6, 20, 68, 233, 805, 2807, 9879, 35073, 125513, 452389, 1641029, 5986994, 21954974, 80884424, 299233544, 1111219334, 4140813374, 15478839554, 58028869154, 218123355524, 821908275548, 3104046382352, 11747506651600, 44546351423300, 169227201341652

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 and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 110, 120, and 021.

Links

Alois P. Heinz, Table of n, a(n) for n = 0..1668

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

Murray Tannock, Equivalence classes of mesh patterns with a dominating pattern, MSc Thesis, Reykjavik Univ., May 2016.

Formula

a(n) = 1 + Sum_{t=1..n-1} Sum_{k=t+2..n+1} (k-t-1)*(k-t)/(n-t+1) * binomial(2n-k-t+1,n-k+1).

Conjecture: a(n) = C_{n+1}-Sum_{i=1..n} C_i where C_i is the i-th Catalan number, binomial(2i,i)/(i+1).

Assuming the conjecture a(n) ~ (64/3)*4^n/((4*n+7)^(3/2)*sqrt(Pi)). - Peter Luschny, Feb 24 2017

From Alois P. Heinz, Mar 11 2017: (Start)

a(n) = 1 + A114277(n-2) for n>1.

G.f.: (sqrt(1-4*x)+2*x-1)*(2*x-1)/(2*(1-x)*x^2). (End)

D-finite with recurrence: (n+2)*a(n) +(-7*n-4)*a(n-1) +2*(7*n-5)*a(n-2) +4*(-2*n+3)*a(n-3)=0. - R. J. Mathar, Feb 21 2020

Example

The length 4 inversion sequences avoiding (110, 120, 021) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0022, 0023, 0100, 0101, 0102, 0103, 0111, 0112, 0113, 0122, 0123.

Maple

a:= proc(n) option remember; `if`(n<3, n!,
      ((5*n^2-6*n-2)*a(n-1)-(4*n-2)*(n-1)*a(n-2))/(n^2-4))
    end:
seq(a(n), n=0..30);  # Alois P. Heinz, Mar 11 2017

Mathematica

a[n_] := 1 + Sum[(k - t - 1) (k - t)/(n - t + 1)* Binomial[2 n - k - t + 1, n - k + 1], {t, n - 1}, {k, t + 2, n + 1}]; Array[a, 28, 0] (* Robert G. Wilson v, Feb 25 2017 *)

Crossrefs

Cf. A000108, A114277, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Sequence in context: A295873 A006012 A127152 * A363182 A150120 A360219

Adjacent sequences: A279554 A279555 A279556 * A279558 A279559 A279560

Keyword

nonn

Author

Megan A. Martinez, Jan 16 2017

Extensions

a(10)-a(12) from Alois P. Heinz, Feb 24 2017

a(13) onward Robert G. Wilson v, Feb 25 2017

Status

approved