Rational Dyck paths and decompositions
Abstract
We study combinatorial properties of a rational Dyck path by decomposing it into a tuple of Dyck paths. The combinatorial models such as $b$Stirling permutations, $(b+1)$ary trees, parenthesis presentations, and binary trees play central roles to establish a correspondence between the rational Dyck path and the tuple of Dyck paths. We reinterpret two orders, the Young and the rotation orders, on rational Dyck paths in terms of the tuple of Dyck paths by use of the decomposition. As an application, we show a duality between $(a,b)$Dyck paths and $(b,a)$Dyck paths in terms of binary trees.
 Publication:

arXiv eprints
 Pub Date:
 April 2021
 arXiv:
 arXiv:2104.01877
 Bibcode:
 2021arXiv210401877S
 Keywords:

 Mathematics  Combinatorics;
 Mathematical Physics
 EPrint:
 32 pages