[Paper Review] Another bijection for 021-avoiding ascent sequences

David Callan|arXiv (Cornell University)|Feb 24, 2014

Coding theory and cryptography2 references3 citations

TL;DR

This paper presents a new algorithmic bijection between 021-avoiding ascent sequences and Dyck paths, constructing the Dyck path iteratively by inserting up-down pairs based on sequence entries. The bijection preserves key statistics such as number of ascents, descents, and terminal zeros, offering a direct alternative to Chen et al.'s recursive construction with distinct structural properties.

ABSTRACT

Chen and collaborators give a recursively defined bijection from 021-avoiding ascent sequences to 021-avoiding (aka 132-avoiding) permutations. Here we give an algorithmic bijection from 021-avoiding ascent sequences to Dyck paths. Our bijection does not appear to be closely related to the Chen bijection but, like the Chen bijection, it preserves several interesting statistics.

Motivation & Objective

• To provide a new, algorithmic bijection between 021-avoiding ascent sequences and Dyck paths, distinct from Chen et al.'s recursive approach.
• To preserve meaningful combinatorial statistics such as number of ascents, descents, and terminal zeros across the bijection.
• To establish a constructive, iterative method for path generation that builds the Dyck path step-by-step from the sequence.
• To clarify the structural relationship between ascent sequences and Dyck paths through explicit insertion rules based on sequence values and path features.

Proposed method

• The bijection constructs the Dyck path incrementally by processing each entry in the 021-avoiding ascent sequence from left to right, starting with the path UD.
• For each entry $ u_i $, the algorithm applies one of four cases based on the value of $ u_i $, its relation to previous entries, and the current path structure.
• Case 1 ( $ u_i = 0 $ ) inserts UD at the last peak, creating a long last ascent.
• Case 2 ( $ u_i = u_{i-1} \neq 0 $ ) elevates the path by prepending and appending UD, increasing its height.
• Case 3 ( $ u_i = a+1 $, where $ a $ is the number of ascents so far) appends UD to the end of the path.
• Case 4 ( $ u_i $ in a specific range) inserts UD at the $ j $-th key downstep, transferring upsteps from the start to maintain path balance and ensure correct insertion position.

Experimental results

Research questions

• RQ1 Can a direct, non-recursive bijection be constructed between 021-avoiding ascent sequences and Dyck paths?
• RQ2 Which combinatorial statistics are preserved under such a bijection, and how do they relate to path features like valleys, descents, and key downsteps?
• RQ3 How does the structure of the ascent sequence, particularly the values and positions of entries, determine the shape and features of the resulting Dyck path?
• RQ4 Is the new bijection structurally distinct from Chen et al.'s recursive construction, and if so, in what ways?

Key findings

• The bijection is algorithmic and iterative, building the Dyck path step-by-step from the ascent sequence, with each insertion rule based on the current sequence value and path state.
• The number of ascents in the sequence corresponds exactly to the number of valleys (DU pairs) in the Dyck path.
• The number of descents in the sequence corresponds to the number of DUU substrings in the Dyck path.
• The number of terminal zeros in the sequence corresponds to the length of the last ascent minus one in the Dyck path.
• The number of initial zeros corresponds to the length of the first descent in the Dyck path.
• The degree of elevation of the Dyck path corresponds to the number of entries immediately preceding the last non-zero entry and equal to it.