A279564 Number of length n inversion sequences avoiding the patterns 000 and 100.

1, 1, 2, 5, 16, 60, 260, 1267, 6850, 40572, 260812, 1805646, 13377274, 105487540, 881338060, 7770957903, 72060991394, 700653026744, 7123871583656, 75561097962918, 834285471737784, 9570207406738352, 113855103776348136, 1402523725268921870, 17863056512845724036, 234910502414771617316, 3185732802058088068444, 44501675392317774477088

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. This is the same as the set of length n inversion sequences avoiding 000 and 100.

Links

Table of n, a(n) for n=0..27.

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

Chunyan Yan, Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.

Formula

The length 4 inversion sequences avoiding (000,100) are 0011, 0012, 0013, 0021, 0022, 0023, 0101, 0102, 0103, 0110, 0112, 0113, 0120, 0121, 0122, 0123.

Maple

b:= proc(n, i, m, s) option remember; `if`(n=0, 1, add(
      `if`(j in s, 0, b(n-1, i+1, max(m, j),
      `if`(j<=m, s union {j}, s))), j=1..i))
    end:
a:= n-> b(n, 1, 0, {}):
seq(a(n), n=0..15);  # Alois P. Heinz, Feb 22 2017

Mathematica

b[n_, i_, m_, s_List] := b[n, i, m, s] = If[n == 0, 1, Sum[If[MemberQ[s, j], 0, b[n-1, i+1, Max[m, j], If[j <= m, s ~Union~ {j}, s]]], {j, 1, i}] ]; a[n_] := b[n, 1, 0, {}]; Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Jul 10 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000108, A057552, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279565, A279566, A279567, A279568, A279569, A279570, A279571, A279572, A279573.

Sequence in context: A000764 A205486 A210668 * A005036 A012051 A012159

Adjacent sequences: A279561 A279562 A279563 * A279565 A279566 A279567

Keyword

nonn

Author

Megan A. Martinez, Feb 09 2017

Extensions

a(10)-a(23) from Alois P. Heinz, Feb 22 2017

a(24)-a(27) from Vaclav Kotesovec, Oct 08 2021

Status

approved