A279571 Number of length n inversion sequences avoiding the patterns 100, 101, and 201.

1, 1, 2, 6, 22, 92, 424, 2106, 11102, 61436, 353980, 2110366, 12955020, 81569168, 525106698, 3447244188, 23028080268, 156246994264, 1075127143948, 7492458675666, 52820934349420, 376331681648402, 2707312468516446, 19650530699752470, 143807774782994412, 1060472244838174574, 7875713244761349666, 58876660310205135380, 442862775457168812898, 3350397169412102710198

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 and e_i >= e_k. This is the same as the set of length n inversion sequences avoiding 100, 101, and 201.

Links

Vaclav Kotesovec, Table of n, a(n) for n = 0..32

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

Example

The length 4 inversion sequences avoiding (100,101,201) are 0000, 0001, 0002, 0003, 0010, 0011, 0012, 0013, 0020, 0021, 0022, 0023, 0102, 0103, 0110, 0111, 0112, 0113, 0120, 0121, 0122, 0123.

Maple

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

Mathematica

b[n_, i_, s_, m_] := b[n, i, s, m] = If[n == 0, 1, Sum[b[n-1, i+1, s  ~Complement~ Range[j, m - If[j == m, 1, 0]] ~Union~ {i+1}, Max[m, j]], {j, s}]];
a[n_] := b[n, 1, {1}, 0];
Table[a[n], {n, 0, 15}] (* Jean-François Alcover, Oct 27 2017, after Alois P. Heinz *)

Crossrefs

Cf. A000108, A057552, A108307, A117106, A263777, A263778, A263779, A263780, A279551, A279552, A279553, A279554, A279555, A279556, A279557, A279558, A279559, A279560, A279561, A279562, A279563, A279564, A279565, A279566, A279567, A279568, A279569, A279570, A279572, A279573.

Sequence in context: A279570 A338748 A107945 * A014330 A225294 A124294

Adjacent sequences: A279568 A279569 A279570 * A279572 A279573 A279574

Keyword

nonn

Author

Megan A. Martinez, Feb 21 2017

Extensions

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

a(26)-a(29) from Vaclav Kotesovec, Oct 07 2021

Status

approved