Two-stack-sorting with pop stacks

We consider the set of permutations that are sorted after two passes through a pop stack. We characterize these permutations in terms of forbidden patterns (classical and barred) and enumerate them according to the ascent statistic. Then we show these permutations to be in bijection with a special family of polyominoes. As a consequence, the permutations sortable by this machine are shown to have the same enumeration as three classical permutation classes.Comment: 18 pages, 7 figure

Flip-sort and combinatorial aspects of pop-stack sorting

Flip-sort is a natural sorting procedure which raises fascinating combinatorial questions. It finds its roots in the seminal work of Knuth on stack-based sorting algorithms and leads to many links with permutation patterns. We present several structural, enumerative, and algorithmic results on permutations that need few (resp. many) iterations of this procedure to be sorted. In particular, we give the shape of the permutations after one iteration, and characterize several families of permutations related to the best and worst cases of flip-sort. En passant, we also give some links between pop-stack sorting, automata, and lattice paths, and introduce several tactics of bijective proofs which have their own interest.Comment: This v3 just updates the journal reference, according to the publisher wis

Commutative combinatorial Hopf algebras

We propose several constructions of commutative or cocommutative Hopf algebras based on various combinatorial structures, and investigate the relations between them. A commutative Hopf algebra of permutations is obtained by a general construction based on graphs, and its non-commutative dual is realized in three different ways, in particular as the Grossman-Larson algebra of heap ordered trees. Extensions to endofunctions, parking functions, set compositions, set partitions, planar binary trees and rooted forests are discussed. Finally, we introduce one-parameter families interpolating between different structures constructed on the same combinatorial objects.Comment: 29 pages, LaTEX; expanded and updated version of math.CO/050245

Dynamic Programming Methodologies in Very Large Scale Neighborhood Search Applied to the Traveling Salesman Problem

We provide two different neighborhood construction techniques for creating exponentially large neighborhoods that are searchable in polynomial time using dynamic programming. We illustrate both of these approaches on very large scale neighborhood search techniques for the traveling salesman problem. Our approaches are intended both to unify previously known results as well as to offer schemas for generating additional exponential neighborhoods that are searchable in polynomial time. The first approach is to define the neighborhood recursively. In this approach, the dynamic programming recursion is a natural consequence of the recursion that defines the neighborhood. In particular, we show how to create the pyramidal tour neighborhood, the twisted sequences neighborhood, and dynasearch neighborhoods using this approach. In the second approach, we consider the standard dynamic program to solve the TSP. We then obtain exponentially large neighborhoods by selecting a polynomially bounded number of states, and restricting the dynamic program to those states only. We show how the Balas and Simonetti neighborhood and the insertion dynasearch neighborhood can be viewed in this manner. We also show that one of the dynasearch neighborhoods can be derived directly from the 2-exchange neighborhood using this approach

Design and construction of a Cherenkov imager for charge measurement of nuclear cosmic rays

A proximity focusing Cherenkov imager called CHERCAM, has been built for the charge measurement of nuclear cosmic rays with the CREAM instrument. It consists of a silica aerogel radiator plane across from a detector plane equipped with 1,600 1" diameter photomultipliers. The two planes are separated by a ring expansion gap. The Cherenkov light yield is proportional to the charge squared of the incident particle. The expected relative light collection accuracy is in the few percents range. It leads to an expected single element separation over the range of nuclear charge Z of main interest 1 < Z < 26. CHERCAM is designed to fly with the CREAM balloon experiment. The design of the instrument and the implemented technical solutions allowing its safe operation in high altitude conditions (radiations, low pressure, cold) are presented.Comment: 24 pages, 19 figure