Sub-families of Baxter Permutations Based on Pattern Avoidance

TL;DR

This paper explores a specific subclass of Baxter permutations linked to mosaic floorplans, providing combinatorial characterizations, algorithms for membership testing and enumeration, and insights into their algebraic and geometric properties.

Contribution

It introduces a new subclass of Baxter permutations characterized by pattern avoidance, establishes bijections with floorplans, and develops algorithms for enumeration and membership testing.

Findings

Derived an algebraic generating function for the class.

Provided a linear time algorithm for membership decision.

Proved closure of the class under permutation inverse.

Abstract

Baxter permutations are a class of permutations which are in bijection with a class of floorplans that arise in chip design called mosaic floorplans. We study a subclass of mosaic floorplans called $HFO_k$ defined from mosaic floorplans by placing certain geometric restrictions. This naturally leads to studying a subclass of Baxter permutations. This subclass of Baxter permutations are characterized by pattern avoidance. We establish a bijection, between the subclass of floorplans we study and a subclass of Baxter permutations, based on the analogy between decomposition of a floorplan into smaller blocks and block decomposition of permutations. Apart from the characterization, we also answer combinatorial questions on these classes. We give an algebraic generating function (but without a closed form solution) for the number of permutations, an exponential lower bound on growth rate, and a linear time algorithm for deciding membership in each subclass. Based on the recurrence relation describing the class, we also give a polynomial time algorithm for enumeration. We finally prove that Baxter permutations are closed under inverse based on an argument inspired from the geometry of the corresponding mosaic floorplans. This proof also establishes that the subclass of Baxter permutations we study are also closed under inverse. Characterizing permutations instead of the corresponding floorplans can be helpful in reasoning about the solution space and in designing efficient algorithms for floorplanning.