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A279565
Number of length n inversion sequences avoiding the patterns 100, 110, 120, 201, and 210.
25
1, 1, 2, 6, 21, 81, 332, 1420, 6266, 28318, 130412, 609808, 2887582, 13818590, 66726628, 324713196, 1590853485, 7840315329, 38843186366, 193342353214, 966409013021, 4848846341569, 24412146213116, 123290812268404, 624448756434476, 3171046361310556
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_k. This is the same as the set of length n inversion sequences avoiding 100, 110, 120, 201, and 210.
LINKS
Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
FORMULA
G.f.: 3/(4-4*sin(asin((27*x+11)/16)/3)). - Vladimir Kruchinin, Mar 25 2019
a(n) = (1/n)*Sum_{m=1..n} m*Sum_{k=0..n-m} C(k,n-m-k)*C(n+k-1,k), n>0, a(0)=1. - Vladimir Kruchinin, Mar 26 2019
a(n) ~ 3^(3*n + 1/2) / (2^(7/2) * sqrt(Pi) * n^(3/2) * 5^(n - 1/2)). - Vaclav Kotesovec, Oct 07 2021
Conjecture: a(n) = (v_n + v_{n+1})/2 for n > 0 with a(0) = 1 where we start with vector v of fixed length m with elements v_i = 1 and for i=1..m-2, for j=i+2..m apply v_j := Sum_{k=0..2} v_{j-k}. - Mikhail Kurkov, Sep 03 2024
EXAMPLE
The length 4 inversion sequences avoiding (100, 110, 120, 201, 210) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0102, 0103, 0111, 0112, 0113, 0121, 0122, 0123.
MAPLE
a:= proc(n) option remember; `if`(n<3, n!,
((n-1)*(17*n-28)*a(n-1) +(49*n^2-185*n+196)*a(n-2)
+(3*(3*n-7))*(3*n-8)*a(n-3)) / (5*n*(n-1)))
end:
seq(a(n), n=0..30); # Alois P. Heinz, Feb 22 2017
MATHEMATICA
a[n_] := a[n] = If[n < 3, n!, (((n - 1)*(17*n - 28)*a[n-1] + (49*n^2 - 185*n + 196)*a[n-2] + (3*(3*n - 7))*(3*n - 8)*a[n-3]) / (5*n*(n - 1)))]; Array[a, 30, 0] (* Jean-François Alcover, Nov 06 2017, after Alois P. Heinz *)
Join[{1}, Table[(1/n)*Sum[m*Sum[Binomial[k, n-m-k]*Binomial[n+k-1, k], {k, 0, n-m}], {m, 1, n}], {n, 1, 30}]] (* G. C. Greubel, Mar 29 2019 *)
PROG
(Maxima)
a(n):=if n=0 then 1 else sum(m*sum(binomial(k, n-m-k)*binomial(n+k-1, k), k, 0, n-m), m, 1, n)/n /* Vladimir Kruchinin, Mar 26 2019 */
(PARI) my(x='x+O('x^30)); Vec(round(3/(4-4*sin(asin((27*x+11)/16)/3)))) \\ G. C. Greubel, Mar 29 2019
(Magma) I:=[6, 21, 81]; [1, 1, 2] cat [n le 3 select I[n] else ( (n+1)*(17*n+6)*Self(n-1) +(49*n^2+11*n+22)*Self(n-2) +3*(3*n-1)*(3*n-2)*Self(n-3) )/(5*(n+2)*(n+1)) : n in [1..30]]; // G. C. Greubel, Mar 29 2019
(Sage) [1] +[(1/n)*(sum(sum(k*binomial(j, n-k-j)*binomial(n+j-1, j) for j in (0..n-k)) for k in (1..n))) for n in (1..30)] # G. C. Greubel, Mar 29 2019
KEYWORD
nonn
AUTHOR
Megan A. Martinez, Feb 09 2017
EXTENSIONS
a(10)-a(25) from Alois P. Heinz, Feb 22 2017
STATUS
approved