Rational Dyck Paths in the Non Relatively Prime Case

Eugene Gorsky, Mikhail Mazin, Monica Vazirani


We study the relationship between rational slope Dyck paths and invariant subsets of $\mathbb{Z},$ extending the work of the first two authors in the relatively prime case. We also find a bijection between $(dn,dm)$–Dyck paths and $d$-tuples of $(n,m)$-Dyck paths endowed with certain gluing data. These are the first steps towards understanding the relationship between rational slope Catalan combinatorics and the geometry of affine Springer fibers and knot invariants in the non relatively prime case.


Rational Dyck paths; Rational Catalan combinatorics; Simultaneous core partitions; Invariant integer subsets; Semigroups

Full Text: