Abstract
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.
Highlights
Catalan numbers, in one of their incarnations, count the number of Dyck paths, that is, the lattice paths in a square which never cross the diagonal
For coprime m and n there are a number of interesting maps involving Yn,m, see Figure 1: (a) J
(b) Armstrong, Loehr, and Warrington defined a “sweep” map ζ : Yn,m → Yn,m and conjectured that it is bijective. This conjecture was proved by Thomas and Williams in [26]
Summary
In one of their incarnations, count the number of Dyck paths, that is, the lattice paths in a square which never cross the diagonal. (b) Armstrong, Loehr, and Warrington defined a “sweep” map ζ : Yn,m → Yn,m and conjectured that it is bijective. This conjecture was proved by Thomas and Williams in [26]. We will want to shift or translate each a fixed amount This defines a map ǫ : MN,M → (Mn,m)d and different shifts correspond to different preimages under ǫ. To resolve this problem, we introduce a certain equivalence relation ∼ on MN,M. For general M and N, it fits into the framework of conjectures of [1, 16, 17], and we refer the reader to these references for more details
Published Version
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