Stochastic Couplings and Bijections from the Symmetric Group to Itself

William Chang

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arXiv · 2207.08231

Stochastic Couplings and Bijections from the Symmetric Group to Itself

William Chang

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Abstract

Inspired by the Stochastic processes described by the Feller Coupling and Chinese Restaurant Processes, we create four different bijections from words in the set $[1]\times [2] \times\cdot \times[n]$ to $S_n$. We then compose these maps with their inverse to obtain a toal of six bijections $S_n \to S_n$. Following that, we investigate the fixed points ($1$-cycle) and higher $k$-cycles of these maps. We characterized some of their properties completely as well as empirically showing the complexity of the higher $k$-cycle structures for these maps.

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BibTeX RIS

William Chang. 2022-07-17. Stochastic Couplings and Bijections from the Symmetric Group to Itself. https://arxiv.org/abs/2207.08231

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