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A279560 Number of length n inversion sequences avoiding the patterns 100, 210, 201, and 102. 23
1, 1, 2, 6, 21, 76, 277, 1016, 3756, 13998, 52554, 198568, 754316, 2878552, 11027384, 42384412, 163372325, 631290168, 2444700421, 9485463044, 36866810877, 143508889270, 559399074443, 2183269032876, 8530724152279, 33366805383326, 130633854520329, 511889287682280 (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 and e_i <> e_k. This is the same as the set of length n inversion sequences avoiding 100, 210, 201, and 102.

LINKS

Alois P. Heinz, Table of n, a(n) for n = 0..1665

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

FORMULA

a(n) = binomial(2n-2,n-1) + Sum_{k=2..n-2} Sum_{i=1..k-1} Sum_{u=1..i} Sum_{d=0..u-1} ((i-d+1)/(i+1)*binomial(i+d,d)) for n>0, a(0)=1.

EXAMPLE

The length 4 inversion sequences avoiding (100, 210, 201, 102) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0101, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.

MAPLE

a:= proc(n) option remember; `if`(n<4, n!,

     ((6*(9*n^4-61*n^3+100*n^2+52*n-140))*a(n-1)

     -(3*(3*n-8))*(9*n^3-38*n^2+3*n+70)*a(n-2)

     +(2*(2*n-7))*(9*n^3-31*n^2-2*n+60)*a(n-3))

      / ((9*n^3-58*n^2+87*n+22)*n))

    end:

seq(a(n), n=0..30);  # Alois P. Heinz, Feb 24 2017

MATHEMATICA

a[0] = 1; a[n_] := Binomial[2n-2, n-1] + Sum[(4i Binomial[2i+1, i+1]) / ((i+2)(i+3)), {k, 2, n-2}, {i, 1, k-1}]; Array[a, 30, 0] (* Jean-Fran├žois Alcover, Nov 06 2017 *)

PROG

(PARI) a(n) = if (n==0, 1, binomial(2*n-2, n-1) + sum(k=2, n-2, sum(i=1, k-1, sum(u=1, i, sum(d=0, u-1, ((i-d+1)/(i+1)*binomial(i+d, d))))))); \\ Michel Marcus, Jan 18 2017

CROSSREFS

Cf. A000108, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Sequence in context: A112091 A108146 A116798 * A116821 A116772 A131792

Adjacent sequences:  A279557 A279558 A279559 * A279561 A279562 A279563

KEYWORD

nonn

AUTHOR

Megan A. Martinez, Jan 17 2017

EXTENSIONS

More terms from Michel Marcus, Jan 18 2017

STATUS

approved

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Last modified February 21 20:32 EST 2020. Contains 332111 sequences. (Running on oeis4.)