Geodesic
Rational Dyck Paths in the Non Relatively Prime Case
Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020).

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We study the relationship between rational slope Dyck paths and invariant subsets in 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 first steps towards understanding the relationship between the rational slope Catalan combinatorics in non relatively prime case and the geometry of affine Springer fibers and representation theory. 
@article{DMTCS_2020_special_379_a69,
     author = {Gorsky, Eugene and Mazin, Mikhail and Vazirani, Monica},
     title = {Rational {Dyck} {Paths} in the {Non} {Relatively} {Prime} {Case}},
     journal = {Discrete mathematics & theoretical computer science},
     publisher = {mathdoc},
     volume = {DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)},
     year = {2020},
     doi = {10.46298/dmtcs.6387},
     language = {en},
     url = {http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6387/}
}
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%D 2020
%V DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016)
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Gorsky, Eugene; Mazin, Mikhail; Vazirani, Monica. Rational Dyck Paths in the Non Relatively Prime Case. Discrete mathematics & theoretical computer science, DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016), DMTCS Proceedings, 28th International Conference on Formal Power Series and Algebraic Combinatorics (FPSAC 2016) (2020). doi : 10.46298/dmtcs.6387. http://geodesic.mathdoc.fr/articles/10.46298/dmtcs.6387/

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