A subfamily of skew Dyck paths related to $k$-ary trees

We introduce a subfamily of skew Dyck paths called box paths and show that they are in bijection with pairs of ternary trees, confirming an observation stated previously on the On-Line Encyclopedia of Integer Sequences. More generally, we define $k$-box paths, which are in bijection with $(k+1)$-tuples of $(k+2)$-ary trees. A bijection is given between $k$-box paths and a subfamily of $k_{t}$-Dyck paths, as well as a bijection with a subfamily of $(k,\ell)$-threshold sequences. We also study the refined enumeration of $k$-box paths by the number of returns and the number of long ascents. Notably, the distribution of long ascents over $k$-box paths generalizes the Narayana distribution on Dyck paths, and we find that $(k-3)$-box paths with exactly two long ascents provide a combinatorial model for the second $k$-gonal numbers.
Comment: 20 pages