• Corpus ID: 56183772

Whirling injections, surjections, and other functions between finite sets

  title={Whirling injections, surjections, and other functions between finite sets},
  author={Michael Joseph and James Gary Propp and Tom Roby},
  journal={arXiv: Combinatorics},
This paper analyzes a certain action called "whirling" that can be defined on any family of functions between two finite sets equipped with a linear (or cyclic) ordering. As a map on injections and surjections, we prove that within any whirling-orbit, any two elements of the codomain appear as outputs of functions the same number of times. This result, can be stated in terms of the homomesy phenomenon, which occurs when a statistic has the same average across every orbit. We further explore… 

Antichain Toggling and Rowmotion

This paper examines the relationship between the toggle groups of antichains and order ideals, constructing an explicit isomorphism between the two groups (for a finite poset) and describes a piecewise-linear analogue of toggling to the Stanley’s chain polytope.

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We study a simple generalization of the rotation (or circular shift) of the binary sequences. In particular, we show each orbit of this generalized rotation has a certain statistical symmetry. This

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A generalization of a homomesy conjecture of Propp that for the action of a "Coxeter element" of vertex toggles, the difference of indicator functions of symmetrically-located vertices is 0-mesic is proved.

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We survey recent work within the area of algebraic combinatorics that has the flavor of discrete dynamical systems, with a particular focus on the homomesy phenomenon codified in 2013 by James Propp

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The notion of the toggle group is generalized from the set of order ideals of a poset to any family of subsets of a finite set, and structure theorems for certain finite generalized toggle groups are proved, similar to the theorem of Cameron and Fon-der-Flaass in the case ofOrder ideals.

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This work introduces natural involutions ("toggles") on the set of noncrossing partitions of size n, and shows that for many operations $T$ of this kind, a surprisingly large family of functions exhibits the homomesy phenomenon.

Homomesy in Products of Two Chains

A theoretical framework for results of this kind is described and old and new results for the actions of promotion and rowmotion on the poset that is the product of two chains are discussed.

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q-Stirling numbers: A new view

Parking Functions of Types A and B

    P. Biane
    Electron. J. Comb.
  • 2002
The lattice of noncrossing partitions can be embedded into the Cayley graph of the symmetric group. This allows us to rederive connections between noncrossing partitions and parking functions. We use

Orbits of antichains revisited

Enumerative Combinatorics: Volume 1

Richard Stanley's two-volume basic introduction to enumerative combinatorics has become the standard guide to the topic for students and experts alike. This thoroughly revised second edition of