OFFSET
0,3
COMMENTS
A length n inversion sequence e_1e_2...e_n is a sequence of integers such that 0 <= e_i < i. The term a(n) counts the inversion sequences of length n with no entries e_i, e_{i+1}, e_{i+2} such that e_i > e_{i+1} > e_{i+2}. That is, a(n) counts the inversion sequences of length n avoiding the consecutive pattern 210.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..460
Juan S. Auli, Pattern Avoidance in Inversion Sequences, Ph. D. thesis, Dartmouth College, ProQuest Dissertations Publishing (2020), 27964164.
Juan S. Auli and Sergi Elizalde, Consecutive Patterns in Inversion Sequences, arXiv:1904.02694 [math.CO], 2019.
Juan S. Auli and Sergi Elizalde, Consecutive patterns in inversion sequences II: avoiding patterns of relations, arXiv:1906.07365 [math.CO], 2019.
FORMULA
a(n) ~ n! * c * (3^(3/2)/(2*Pi))^n * n^(2*Pi/3^(3/2)), where c = 0.24427562500895080639039917229089... - Vaclav Kotesovec, Oct 19 2019
EXAMPLE
Note that a(5)=118. Indeed, of the 120 inversion sequences of length 5, the only ones that do not avoid the consecutive patterns 210 are 00210 and 01210.
MAPLE
# after Alois P. Heinz in A328357
b := proc(n, x, t) option remember; `if`(n = 0, 1, add(
`if`(t and x < i, 0, b(n - 1, i, x < i)), i = 0 .. n - 1))
end proc:
a := n -> b(n, n, false):
seq(a(n), n = 0 .. 24);
MATHEMATICA
b[n_, x_, t_] := b[n, x, t] = If[n == 0, 1, Sum[If[t && x < i, 0, b[n - 1, i, x < i]], {i, 0, n - 1}]];
a[n_] := b[n, n, False];
CROSSREFS
KEYWORD
nonn
AUTHOR
Juan S. Auli, Oct 17 2019
STATUS
approved