Abstract

Two permutation classes, the X-class and subpermutations of the increasing oscillation are shown to exhibit an exponential Wilf-collapse. This means that the number of distinct enumerations of principal subclasses of each of these classes grows much more slowly than the class itself whereas a priori, based only on symmetries of the class, there is no reason to expect this. The underlying cause of the collapse in both cases is the ability to apply some form of local symmetry which, combined with a greedy algorithm for detecting patterns in these classes, yields a Wilf-collapse.

Highlights

  • The coincidences that arise when two unrelated, or peripherally related collections of combinatorial structures have the same generating function have always attracted interest

  • A permutation π of size k is contained in another permutation τ if there is a subsequence of k elements in τ whose relative ordering is the same as that of π

  • In X it will turn out to be easiest to compute generating functions for Inv(α) and Inv(β), while in SIO we demonstrate the existence of implicitly defined bijections

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Summary

Introduction

The coincidences that arise when two unrelated, or peripherally related collections of combinatorial structures have the same generating function have always attracted interest. In this paper we are concerned with certain families of such coincidences which are called Wilf-equivalence These arise when the two classes in question both occur as subsets of some universal class carrying a partial order (always interpreted as containment of structures, and presumed to satisfy the obvious conditions that such an order should have). The independent interest of these two particular examples is that neither X nor SIO satisfies those conditions Both admit a natural representation of structures as words over an infinite alphabet which combines with a greedy algorithm to decide the containment relation to allow “local” applications of symmetries in creating Wilf-equivalent structures. It is in exploring this central thesis with new examples that we believe the value of this paper lies

Permutation classes
The X-class
Sum closure of increasing oscillations
Conclusion

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