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A279557 Number of length n inversion sequences avoiding the patterns 110, 120, and 021. 25
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 (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 and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 110, 120, and 021.
LINKS
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
Sequence in context: A295873 A006012 A127152 * A363182 A150120 A360219
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

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Last modified April 15 15:27 EDT 2024. Contains 371692 sequences. (Running on oeis4.)