A Subfamily of Skew Dyck Paths Related to k-ary Trees
Yuxuan Zhang
Department of Statistical Science
Duke University
Durham, NC 27708
USA
Yan Zhuang
Department of Mathematics and Computer Science
Davidson College
405 N. Main St.
Davidson, NC 28035
USA
Abstract:
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
kt-Dyck paths, as well as a bijection with a subfamily of
(k,l)-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.
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(Concerned with sequences
A128728
A143603.)
Received August 1 2023; revised version received November 21 2023.
Published in Journal of Integer Sequences,
January 19 2024.
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