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

Author

Megan A. Martinez, Jan 16 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_k and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 110, 120, and 021.

Examples

			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.
		

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.

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.

Programs

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 *)

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

Extensions

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

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