Abstract

 The concept of prolificity was previously introduced by the authors in the context of compositions of integers. We give a general interpretation of prolificity that applies across a range of relational structures defined in terms of counting embeddings. We then proceed to classify prolificity in permutation classes with bases consisting of permutations of length 2, or 3; completely classifying all such classes except ${\rm Av}(321)$. We then show a number of interesting properties that arise when studying prolificity in ${\rm Av}(321)$, concluding by showing that the class of permutations that are not prolific for any increasing permutation in ${\rm Av}(321)$ form a polynomial subclass.
Highlights
The notion of prolificity was introduced by the authors [3] in the context of compositions of integers
Equivalent to that original context is the class of layered permutations and so considering prolific structures in permutation classes is a logical step
Somewhat following the lead of Simion and Schmidt [7] we provide a classification of prolificity in permutation classes whose bases consist of permutations of length up to 3
Summary
The notion of prolificity was introduced by the authors [3] in the context of compositions of integers. A combinatorial structure, T , is prolific for another structure, P , if every proper extension of T contains more occurrences of P than T did (formal definitions follow below). This natural property does not seem to have been considered previously, so the step in its exploration is to consider how it behaves in other types of structures. Equivalent to that original context is the class of layered permutations (which, in one line notation are represented by an increasing sequence of decreasing intervals, e.g., 215436987 corresponding to the composition 2, 3, 1, 3) and so considering prolific structures in permutation classes is a logical step. Note that the notion of prolificity discussed here is distinct from that presented by Bevan, Homberger and Tenner [4]
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