1,194 research outputs found

    Generating Permutations with Restricted Containers

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    We investigate a generalization of stacks that we call C\mathcal{C}-machines. We show how this viewpoint rapidly leads to functional equations for the classes of permutations that C\mathcal{C}-machines generate, and how these systems of functional equations can frequently be solved by either the kernel method or, much more easily, by guessing and checking. General results about the rationality, algebraicity, and the existence of Wilfian formulas for some classes generated by C\mathcal{C}-machines are given. We also draw attention to some relatively small permutation classes which, although we can generate thousands of terms of their enumerations, seem to not have D-finite generating functions

    Avoiding patterns in irreducible permutations

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    International audienceWe explore the classical pattern avoidance question in the case of irreducible permutations, i.e., those in which there is no index ii such that σ(i+1)−σ(i)=1\sigma (i+1) - \sigma (i)=1. The problem is addressed completely in the case of avoiding one or two patterns of length three, and several well known sequences are encountered in the process, such as Catalan, Motzkin, Fibonacci, Tribonacci, Padovan and Binary numbers. Also, we present constructive bijections between the set of Motzkin paths of length n−1n-1 and the sets of irreducible permutations of length nn (respectively fixed point free irreducible involutions of length 2n2n) avoiding a pattern α\alpha for α∈{132,213,321}\alpha \in \{132,213,321\}. This induces two new bijections between the set of Dyck paths and some restricted sets of permutations

    Combinatorial generation via permutation languages. I. Fundamentals

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    In this work we present a general and versatile algorithmic framework for exhaustively generating a large variety of different combinatorial objects, based on encoding them as permutations. This approach provides a unified view on many known results and allows us to prove many new ones. In particular, we obtain the following four classical Gray codes as special cases: the Steinhaus-Johnson-Trotter algorithm to generate all permutations of an nn-element set by adjacent transpositions; the binary reflected Gray code to generate all nn-bit strings by flipping a single bit in each step; the Gray code for generating all nn-vertex binary trees by rotations due to Lucas, van Baronaigien, and Ruskey; the Gray code for generating all partitions of an nn-element ground set by element exchanges due to Kaye. We present two distinct applications for our new framework: The first main application is the generation of pattern-avoiding permutations, yielding new Gray codes for different families of permutations that are characterized by the avoidance of certain classical patterns, (bi)vincular patterns, barred patterns, boxed patterns, Bruhat-restricted patterns, mesh patterns, monotone and geometric grid classes, and many others. We also obtain new Gray codes for all the combinatorial objects that are in bijection to these permutations, in particular for five different types of geometric rectangulations, also known as floorplans, which are divisions of a square into nn rectangles subject to certain restrictions. The second main application of our framework are lattice congruences of the weak order on the symmetric group~SnS_n. Recently, Pilaud and Santos realized all those lattice congruences as (n−1)(n-1)-dimensional polytopes, called quotientopes, which generalize hypercubes, associahedra, permutahedra etc. Our algorithm generates the equivalence classes of each of those lattice congruences, by producing a Hamilton path on the skeleton of the corresponding quotientope, yielding a constructive proof that each of these highly symmetric graphs is Hamiltonian. We thus also obtain a provable notion of optimality for the Gray codes obtained from our framework: They translate into walks along the edges of a polytope
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