OFFSET
0,3
COMMENTS
A length n inversion sequence e_1, e_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 100.
The term a(n) also 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) also counts the inversion sequences of length n avoiding the consecutive pattern 110, see the Auli and Elizalde links.
LINKS
Vaclav Kotesovec, Table of n, a(n) for n = 0..448
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.
Juan S. Auli and Sergi Elizalde, Wilf equivalences between vincular patterns in inversion sequences, arXiv:2003.11533 [math.CO], 2020.
FORMULA
a(n) ~ n! * c / sqrt(n), where c = 2.428754692682297906864850201408427747198... - Vaclav Kotesovec, Oct 18 2019
EXAMPLE
Note that a(4)=23. Indeed, of the 24 inversion sequences of length 4, the only one that does not avoid the consecutive pattern 100 is 0100.
Similarly, 0110 is the only inversion sequence of length 4 that does not avoid the consecutive pattern 110.
MAPLE
# after Alois P. Heinz in A328357
b := proc(n, x, t) local i; option remember; `if`(n = 0, 1, add(`if`(t and x < i, 0, b(n - 1, i, i = x)), i = 0 .. n - 1)); end proc;
a := n -> b(n, -1, false);
seq(a(n), n = 0 .. 24);
MATHEMATICA
i100[1] = 1; i100[2] = 2; i100[n_] := i100[n] = Sum[s100[n, k], {k, 0, n - 1}]; s100[n_, k_] := s100[n, k] = i100[n - 1] - Sum[s100[n - 2, j], {j, k + 1, n - 3}]; Flatten[{1, Table[i100[m], {m, 1, 25}]}] (* Vaclav Kotesovec, Oct 18 2019 *)
CROSSREFS
KEYWORD
nonn
AUTHOR
Juan S. Auli, Oct 17 2019
STATUS
approved