Subclasses of Baxter Permutations Based on Pattern Avoidance

  • Shankar Balachandran
  • Sajin KorothEmail author
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9691)


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 Hierarchical Floorplans of Order 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.


Floorplanning Pattern avoidance Baxter permutation 


  1. 1.
    Ackerman, E., Barequet, G., Pinter, R.Y.: A bijection between permutations and floorplans, and its applications. Discrete Appl. Math. 154(12), 1674–1684 (2006)MathSciNetCrossRefzbMATHGoogle Scholar
  2. 2.
    Albert, M.H., Atkinson, M.D.: Simple permutations and pattern restricted permutations. Discrete Math. 300(1–3), 1–15 (2005)MathSciNetCrossRefzbMATHGoogle Scholar
  3. 3.
    Balachandran, S., Koroth, S.: Sub-families of baxter permutations based on pattern avoidance. CoRR, abs/1112.1374 (2011)Google Scholar
  4. 4.
    De Almeida, A.M., Rodrigues, R.: Trees, slices, and wheels: on the floorplan area minimization problem. Networks 41(4), 235–244 (2003)MathSciNetCrossRefzbMATHGoogle Scholar
  5. 5.
    Guillemot, S., Marx, D.: Finding small patterns in permutations in linear time. In: Proceedings of the Twenty-Fifth Annual ACM-SIAM Symposium on Discrete Algorithms, SODA, Portland, Oregon, USA, pp. 82–101. SIAM, 5–7 January 2014Google Scholar
  6. 6.
    Hart, J.M.: Fast recognition of baxter permutations using syntactical and complete bipartite composite dag’s. Int. J. Comput. Inf. Sci. 9(4), 307–321 (1980)CrossRefzbMATHGoogle Scholar
  7. 7.
    Law, S., Reading, N.: The hopf algebra of diagonal rectangulations. J. Comb. Theory Ser. A 119(3), 788–824 (2011)MathSciNetCrossRefzbMATHGoogle Scholar
  8. 8.
    Sait, S.M., Youssef, H.: VLSI Physical Design Automation: Theory and Practice. Lecture Notes Series on Computing. World Scientific, Singapore (1999)CrossRefGoogle Scholar
  9. 9.
    Shen, Z.C., Chu, C.C.N.: Bounds on the number of slicing, mosaic, and general floorplans. IEEE Trans. CAD Integr. Circ. Syst. 22(10), 1354–1361 (2003)CrossRefGoogle Scholar
  10. 10.
    Stockmeyer, L.: Optimal orientations of cells in slicing floorplan designs. Inf. Control 57(2), 91–101 (1983)MathSciNetCrossRefzbMATHGoogle Scholar
  11. 11.
    Wong, D.F., Liu, C.L.: A new algorithm for floorplan design. In: 23rd Conference on Design Automation, pp. 101–107 (1986)Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of Computer Science and EngineeringIndian Institute of Technology MadrasChennaiIndia

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