A simple optimal binary representation of mosaic floor plans and Baxter permutations

Mosaic floorplans are rectangular structures subdivided into smaller rectangular sections and are widely used in VLSI circuit design. Baxter permutations are a set of permutations that have been shown to have a one-to-one correspondence to objects in the Baxter combinatorial family, which includes mosaic floorplans. An important problem in this area is to find short binary string representations of the set of n-block mosaic floorplans and Baxter permutations of length n. The best known representation is the Quarter-State Sequence which uses 4n bits. This paper introduces a simple binary representation of n-block mosaic floorplan using 3n−3 bits. It has been shown that any binary representation of n-block mosaic floorplans must use at least (3n−o(n)) bits. Therefore, the representation presented in this paper is optimal (up to an additive lower order term)

Enumeration and Asymptotic Formulas for Rectangular Partitions of the Hypercube

We study a two-parameter generalization of the Catalan numbers: $C_{d,p}(n)$ is the number of ways to subdivide the $d$-dimensional hypercube into $n$ rectangular blocks using orthogonal partitions of fixed arity $p$. Bremner \& Dotsenko introduced $C_{d,p}(n)$ in their work on Boardman--Vogt tensor products of operads; they used homological algebra to prove a recursive formula and a functional equation. We express $C_{d,p}(n)$ as simple finite sums, and determine their growth rate and asymptotic behaviour. We give an elementary proof of the functional equation, using a bijection between hypercube decompositions and a family of full $p$-ary trees. Our results generalize the well-known correspondence between Catalan numbers and full binary trees