Pop Stacks with a Bypass

TL;DR

This work advances the theory of sorting permutations using a pop stack with a bypass (PSB). It characterizes sortable permutations via pattern avoidance ($\mathrm{Av}_n(231,4213)$), and proves a Motzkin-path bijection underpinning their enumeration by the odd-indexed Fibonacci sequence. It then develops a recursive preimage algorithm to enumerate and classify PSB preimages, analyzes preimages of permutation classes, and studies compositions with classic sorting devices, yielding detailed forbidden-pattern characterizations. Finally, it extends the framework to two pop stacks in parallel with a bypass, obtaining the basis of sortable permutations and a rational generating function for the inverse class, plus intriguing connections to Fibonacci numbers in the simple permutation case.

Abstract

We consider sorting procedures for permutations making use of pop stacks with a bypass operation, and explore the combinatorial properties of the associated algorithms.

Theorems & Definitions (9)

• Proposition 2.1
• Proposition 2.2
• Proposition 3.1
• Proposition 3.2
• Proposition 3.3
• Proposition 4.1
• Proposition 5.1
• Proposition 6.1
• Conjecture