We define the operation of composing two hereditary classes of permutations using the standard composition of permutations as functions and we explore properties and structure of permutation classes considering this operation. We mostly concern ourselves with the problem of whether permutation classes can be composed from their proper subclasses. We provide examples of classes which can be composed from two proper subclasses, classes which can be composed from three but not from two proper subclasses and classes which cannot be composed from any finite number of proper subclasses.
A skew shape is the difference of two top-left justified Ferrers shapes
sharing the same top-left corner. We study integer fillings of skew shapes. As
our first main result, we show that for a specific hereditary class of skew
shapes, which we call D-free shapes, the fillings that avoid a north-east chain
of size $k$ are in bijection with fillings that avoid a south-east chain of the
same size. Since Ferrers shapes are a subclass of D-free shapes, this result
can be seen as a generalization of previous analogous results for Ferrers
shapes.
As our second main result, we construct a bijection between 01-fillings of an
arbitrary skew shape that avoid a south-east chain of size 2, and the
01-fillings of the same shape that simultaneously avoid a north-east chain of
size 2 and a particular non-square subfilling. This generalizes a previous
result for transversal fillings.
Comment: 23 pages, 14 figures; formatting changes for publication in DMTCS, no
changes in content
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