[Paper Review] Stackable and queueable permutations

Peter G. Doyle|arXiv (Cornell University)|Jan 31, 2012

Algorithms and Data Compression11 references3 citations

TL;DR

This paper establishes a natural bijection between stackable permutations (312-avoiding) and queueable permutations (321-avoiding), both enumerated by the Catalan number Cₙ. The bijection arises from a shared correspondence to peakless weak Dyck paths, where the stack height or queue length record uniquely determines each permutation; the authors argue this bijection—equivalent to Simion and Schmidt’s 1985 result—could have been discovered earlier by readers of Knuth’s 1968 volume 1 had exercises not been difficulty-rated, which may have discouraged exploration of alternative interpretations.

ABSTRACT

There is a natural bijection between permutations obtainable using a stack (those avoiding the pattern 312) and permutations obtainable using a queue (those avoiding 321). This bijection is equivalent to one described by Simion and Schmidt in 1985. We argue that this bijection might well have been found back in 1968 by readers of volume 1 of Knuth's *The Art of Computer Programming*, if Knuth had not assigned difficulty ratings to his exercises.

Motivation & Objective

• To identify and formalize a natural bijection between permutations obtainable via a stack (312-avoiding) and those obtainable via a queue (321-avoiding).
• To demonstrate that this bijection, equivalent to Simion and Schmidt’s 1985 construction, arises naturally from shared correspondence to peakless weak Dyck paths.
• To argue that this bijection might have been discovered earlier in 1968 by readers of Knuth’s The Art of Computer Programming, Volume 1, had the difficulty rating of exercises not discouraged exploratory interpretation.

Proposed method

• Represent stackable and queueable permutations via their stack height and queue length records, respectively, both of which form peakless weak Dyck paths.
• Use the unique correspondence between peakless weak Dyck paths and stackable (or queueable) permutations to define a bijection between the two sets.
• Construct the bijection by simulating the stack-based generation of a permutation using a queue instead, or vice versa, showing that the same path structure yields the corresponding permutation in the other model.
• Define idempotent functions 'stackit' and 'queueit' that project arbitrary permutations onto the sets of stackable and queueable permutations, respectively, based on their set-size records.
• Show that the restriction of 'queueit' to stackable permutations yields a bijection to queueable permutations, with the inverse being the restriction of 'stackit' to queueable permutations.
• Use the Robinson-Schensted correspondence and Young tableaux as a historical alternative route to the Catalan count, though the paper favors the more direct path via Dyck paths.

Experimental results

Research questions

• RQ1 What is the natural bijection between 312-avoiding (stackable) and 321-avoiding (queueable) permutations, and how does it relate to Dyck paths?
• RQ2 Why is the number of stackable and queueable permutations both equal to the Catalan number Cₙ, and what structural feature underlies this equality?
• RQ3 Could this bijection have been discovered earlier by readers of Knuth’s 1968 volume 1, and if so, why was it not?

Key findings

• There exists a natural bijection between stackable permutations (312-avoiding) and queueable permutations (321-avoiding), both of which are counted by the Catalan number Cₙ.
• This bijection is equivalent to the one found by Simion and Schmidt in 1985 and arises from a shared correspondence to peakless weak Dyck paths.
• The stack height record of a stackable permutation and the queue length record of a queueable permutation both form peakless weak Dyck paths, and each such path corresponds to exactly one permutation in each class.
• The bijection can be realized by simulating stack operations using a queue: if one attempts to generate a stackable permutation using a queue instead, the result is the corresponding queueable permutation.
• The idempotent functions 'stackit' and 'queueit' project arbitrary permutations onto the sets of stackable and queueable permutations, respectively, based on their set-size records, and their restrictions form the inverse bijection.
• The paper argues that the difficulty rating of Knuth’s exercises in 1968 may have discouraged readers from exploring alternative interpretations, potentially delaying the discovery of this bijection.