A279565 Number of length n inversion sequences avoiding the patterns 100, 110, 120, 201, and 210.

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

Alois P. Heinz, Table of n, a(n) for n = 0..1372

Megan A. Martinez, 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

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

Crossrefs

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Sequence in context: A148494 A150212 A150213 * A150214 A150215 A328434

Adjacent sequences: A279562 A279563 A279564 * A279566 A279567 A279568

Keyword

nonn

Author

Megan A. Martinez, Feb 09 2017

Extensions

a(10)-a(25) from Alois P. Heinz, Feb 22 2017

Status

approved