4 research outputs found
An algorithmic approach based on generating trees for enumerating pattern-avoiding inversion sequences
We introduce an algorithmic approach based on generating tree method for
enumerating the inversion sequences with various pattern-avoidance
restrictions. For a given set of patterns, we propose an algorithm that outputs
either an accurate description of the succession rules of the corresponding
generating tree or an ansatz. By using this approach, we determine the
generating trees for the pattern-classes ,
, , ,
and . Then we use the kernel method, obtain generating functions
of each class, and find enumerating formulas. Lin and Yan studied the
classification of the Wilf-equivalences for inversion sequences avoiding pairs
of length-three patterns and showed that there are 48 Wilf classes among 78
pairs. In this paper, we solve six open cases for such pattern classes.Comment: 20 pages, 2 figure
Inversion sequences avoiding pairs of patterns
The enumeration of inversion sequences avoiding a single pattern was
initiated by Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck
independently. Their work has sparked various investigations of generalized
patterns in inversion sequences, including patterns of relation triples by
Martinez and Savage, consecutive patterns by Auli and Elizalde, and vincular
patterns by Lin and Yan. In this paper, we carried out the systematic study of
inversion sequences avoiding two patterns of length . Our enumerative
results establish further connections to the OEIS sequences and some classical
combinatorial objects, such as restricted permutations, weighted ordered trees
and set partitions. Since patterns of relation triples are some special
multiple patterns of length , our results complement the work by Martinez
and Savage. In particular, one of their conjectures regarding the enumeration
of -avoiding inversion sequences is solved