Noncrossing partitions and nonnesting partitions are both counted by Catalan numbers. Noncrossing partitions on [n] admit a natural cyclic action of order 2n, induced by the Kreweras complement. Nonnesting partitions admit a natural toggle-based action; in fact, they admit one such action for each choice of Coxeter element of the symmetric group. We prove that the latter actions all have order 2n by constructing a family of bijections between noncrossing and nonnesting partitions, equivariant with respect to the cyclic actions on either side. This talk is based on arXiv:2212.14831, joint with Benjamin Dequêne, Gabriel Frieden, Alessandro Iraci, Florian Schreier-Aigner, and Nathan Williams. Our results were presented at FPSAC in July; our extended abstract (and Nathan Williams’s slides) are available from the FPSAC website.

Speaker’s website: https://professeurs.uqam.ca/professeur/thomas.hugh_r/