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)

Snow Leopard Permutations and Their Even and Odd Threads

Caffrey, Egge, Michel, Rubin and Ver Steegh recently introduced snow leopard permutations, which are the anti-Baxter permutations that are compatible with the doubly alternating Baxter permutations. Among other things, they showed that these permutations preserve parity, and that the number of snow leopard permutations of length $2n-1$ is the Catalan number $C_n$. In this paper we investigate the permutations that the snow leopard permutations induce on their even and odd entries; we call these the even threads and the odd threads, respectively. We give recursive bijections between these permutations and certain families of Catalan paths. We characterize the odd (resp. even) threads which form the other half of a snow leopard permutation whose even (resp. odd) thread is layered in terms of pattern avoidance, and we give a constructive bijection between the set of permutations of length $n$ which are both even threads and odd threads and the set of peakless Motzkin paths of length $n+1$.Comment: 25 pages, 6 figures. Version 3 is modified to use standard Discrete Mathematics and Theoretical Computer Science but is otherwise unchange

Geometric and algebraic properties of polyomino tilings

Thesis (Ph. D.)--Massachusetts Institute of Technology, Dept. of Mathematics, 2004.Includes bibliographical references (p. 165-167).This electronic version was submitted by the student author. The certified thesis is available in the Institute Archives and Special Collections.In this thesis we study tilings of regions on the square grid by polyominoes. A polyomino is any connected shape formed from a union of grid cells, and a tiling of a region is a collection of polyominoes lying in the region such that each square is covered exactly once. In particular, we focus on two main themes: local connectivity and tile invariants. Given a set of tiles T and a finite set L of local replacement moves, we say that a region [Delta] has local connectivity with respect to T and L if it is possible to convert any tiling of [Delta] into any other by means of these moves. If R is a set of regions (such as the set of all simply connected regions), then we say there is a local move property for T and R if there exists a finite set of moves L such that every r in R has local connectivity with respect to T and L. We use height function techniques to prove local move properties for several new tile sets. In addition, we provide explicit counterexamples to show the absence of a local move property for a number of tile sets where local move properties were conjectured to hold. We also provide several new results concerning tile invariants. If we let ai(t) denote the number of occurrences of the tile ti in a tiling t of a region [Delta], then a tile invariant is a linear combination of the ai's whose value depends only on t and not on r.(cont.) We modify the boundary-word technique of Conway and Lagarias to prove tile invariants for several new sets of tiles and provide specific examples to show that the invariants we obtain are the best possible. In addition, we prove some new enumerative results, relating certain tiling problems to Baxter permutations, the Tutte polynomial, and alternating-sign matrices.by Michael Robert Korn.Ph.D