[Paper Review] Two Bijections for Dyck Path Parameters

David Callan|ArXiv.org|Jun 19, 2004

Advanced Combinatorial Mathematics1 references19 citations

TL;DR

This paper presents two novel bijections between Dyck paths and Motzkin paths that preserve key parameters: the number of UDU substrings and the number of DDU substrings. The first bijection maps UUU-free Dyck paths to Motzkin paths, proving their counts are equal to the Motzkin numbers. The second bijection, using bicolored Motzkin paths, establishes exact distribution formulas for UDU and DDU occurrences in Dyck paths, linking them to Catalan and Motzkin numbers via combinatorial constructions with explicit step transformations and matching rules.

ABSTRACT

Here we give two bijections, one to show that the number of UUU-free Dyck n-paths is the Motzkin number M_n, the other to obtain the (known) distributions of the parameters "number of UDUs" and "number of DDUs" on Dyck n-paths. The first bijection is straightforward, the second not quite so obvious.

Motivation & Objective

• To establish a bijective correspondence between UUU-free Dyck paths and Motzkin paths, showing their counts equal the Motzkin number $ M_n $.
• To derive exact distribution formulas for the number of UDU substrings in Dyck paths using a bicolored Motzkin path construction.
• To derive exact distribution formulas for the number of DDU substrings in Dyck paths via a refined bijection involving step replacements and descent tracking.
• To characterize Dyck paths with no short nonterminal descents by relating them to Motzkin paths with no flatsteps at ground level.
• To provide invertible, explicit transformations between Dyck and Motzkin paths that preserve and map specific path parameters.

Proposed method

• Transform each UUD in a UUU-free Dyck path to a single U, and each remaining UD to a flatstep F, yielding a Motzkin path; the inverse process recovers the original Dyck path.
• Construct bicolored Motzkin paths by assigning two colors (green and black) to flatsteps, enabling parameter tracking via distinct step types.
• Apply a step-by-step transformation: leave U steps unchanged, replace D with UDD, green F with UD, and black F with U followed by inserting a D before its associated downstep.
• Append a D at the end of the Motzkin path to ensure every flatstep has a defined associated downstep, then remove it after transformation to maintain Dyck path structure.
• Use matching rules to invert the transformation: green F correspond to UD pairs followed by a U, black F to U steps whose matching D lies within a descent of length ≥3, and original U steps to those whose matching D is followed by a U.
• Leverage known counts of bicolored Motzkin paths by number of green F steps or D steps to derive distribution formulas for UDU and DDU in Dyck paths.

Experimental results

Research questions

• RQ1 What is the exact number of UUU-free Dyck $ n $-paths, and how can this be bijectively linked to Motzkin paths?
• RQ2 What is the distribution of the number of UDU substrings across all Dyck $ n $-paths, and how can this be derived combinatorially?
• RQ3 What is the distribution of the number of DDU substrings across all Dyck $ n $-paths, and how does it relate to known combinatorial sequences?
• RQ4 How can a bijection be constructed that maps the number of green flatsteps in a bicolored Motzkin path to the number of UDU substrings in a Dyck path?
• RQ5 What structural properties of Dyck paths (e.g., terminal descent, short descents) correspond to properties of Motzkin paths (e.g., flatsteps at ground level)?

Key findings

• The number of UUU-free Dyck $ n $-paths is exactly $ M_n $, the $ n $th Motzkin number, via a reversible transformation replacing UUD with U and UD with F.
• The number of Dyck $ n $-paths with exactly $ k $ UDU substrings is $ inom{n-1}{k} M_{n-1-k} $, known as the Donaghey distribution.
• The number of Dyck $ n $-paths with exactly $ k $ DDU substrings is $ inom{n-1}{2k} 2^{n-1-2k} C_k $, known as the Touchard distribution.
• The bijection maps green flatsteps in bicolored Motzkin paths to UDU substrings in Dyck paths, and D steps to DDU substrings, preserving path structure.
• Motzkin paths with no flatsteps at ground level correspond bijectively to Dyck paths with no short nonterminal descents, via deletion of the final UD pair.
• The inverse of the main bijection recovers original path types: green F steps are identified as UD pairs followed by a U, black F steps as U steps whose matching D lies in a descent of length ≥3.