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

Convex Rank Tests and Semigraphoids

Convex rank tests are partitions of the symmetric group which have desirable geometric properties. The statistical tests defined by such partitions involve counting all permutations in the equivalence classes. Each class consists of the linear extensions of a partially ordered set specified by data. Our methods refine existing rank tests of non-parametric statistics, such as the sign test and the runs test, and are useful for exploratory analysis of ordinal data. We establish a bijection between convex rank tests and probabilistic conditional independence structures known as semigraphoids. The subclass of submodular rank tests is derived from faces of the cone of submodular functions, or from Minkowski summands of the permutohedron. We enumerate all small instances of such rank tests. Of particular interest are graphical tests, which correspond to both graphical models and to graph associahedra

Accession Games: A Dynamic Per-Member Partition Function Aapproach

In this paper we define and solve the accession game, a dynamic game containing a union and a set of applicants with a per-member partition function satisfying the conditions of Yi [17] to include negative externalities. The solution gives an equilibrium partition of the players as well as, after Morelli and Penelle [12], the optimal path, a subgame-perfect sequence of partitions, where each player maximises the present value of its payo.s subject to others’ moves. While this game can be applied in general our motivation was to model the ongoing extensions of the European Union.partition function, externalities, path dependence