7 research outputs found
Egge triples and unbalanced Wilf-equivalence
Egge conjectured that permutations avoiding the set of patterns
, where ,
are enumerated by the large Schr\"oder numbers. Consequently,
with as above is Wilf-equivalent to the set of
patterns . Burstein and Pantone proved the case of
. We prove the remaining four cases. As a byproduct of our proof,
we also enumerate the case .Comment: 20 pages, 6 figures (published version
Minimal classes of graphs of unbounded clique-width defined by finitely many forbidden induced subgraphs
We discover new hereditary classes of graphs that are minimal (with respect
to set inclusion) of unbounded clique-width. The new examples include split
permutation graphs and bichain graphs. Each of these classes is characterised
by a finite list of minimal forbidden induced subgraphs. These, therefore,
disprove a conjecture due to Daligault, Rao and Thomasse from 2010 claiming
that all such minimal classes must be defined by infinitely many forbidden
induced subgraphs.
In the same paper, Daligault, Rao and Thomasse make another conjecture that
every hereditary class of unbounded clique-width must contain a labelled
infinite antichain. We show that the two example classes we consider here
satisfy this conjecture. Indeed, they each contain a canonical labelled
infinite antichain, which leads us to propose a stronger conjecture: that every
hereditary class of graphs that is minimal of unbounded clique-width contains a
canonical labelled infinite antichain.Comment: 17 pages, 7 figure
Pattern-Avoiding Involutions: Exact and Asymptotic Enumeration
We consider the enumeration of pattern-avoiding involutions, focusing in
particular on sets defined by avoiding a single pattern of length 4. As we
demonstrate, the numerical data for these problems demonstrates some surprising
behavior. This strange behavior even provides some very unexpected data related
to the number of 1324-avoiding permutations
Permutation classes
This is a survey on permutation classes for the upcoming book Handbook of
Enumerative Combinatorics