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

Author

Megan A. Martinez, Feb 09 2017

Keywords

• nonn

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

Links

• Benjamin Testart, Table of n, a(n) for n = 0..540
• Megan A. Martinez and Carla D. Savage, Patterns in Inversion Sequences II: Inversion Sequences Avoiding Triples of Relations, arXiv:1609.08106 [math.CO], 2016.
• Benjamin Testart, Completing the enumeration of inversion sequences avoiding one or two patterns of length 3, arXiv:2407.07701 [math.CO], 2024.
• Chunyan Yan and Zhicong Lin, Inversion sequences avoiding pairs of patterns, arXiv:1912.03674 [math.CO], 2019.

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.

Programs

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 *)

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.

Extensions

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

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