Asymmetric function theory

The classical theory of symmetric functions has a central position in algebraic combinatorics, bridging aspects of representation theory, combinatorics, and enumerative geometry. More recently, this theory has been fruitfully extended to the larger ring of quasisymmetric functions, with corresponding applications. Here, we survey recent work extending this theory further to general asymmetric polynomials.Comment: 36 pages, 8 figures, 1 table. Written for the proceedings of the Schubert calculus conference in Guangzhou, Nov. 201

PESFOR-W: Improving the design and environmental effectiveness of woodlands for water Payments for Ecosystem Services

ABSTRACT: The EU Water Framework Directive aims to ensure restoration of Europe?s water bodies to ?good ecological status? by 2027. Many Member States will struggle to meet this target, with around half of EU river catchments currently reporting below standard water quality. Diffuse pollution from agriculture represents a major pressure, affecting over 90% of river basins. Accumulating evidence shows that recent improvements to agricultural practices are benefiting water quality but in many cases will be insufficient to achieve WFD objectives. There is growing support for land use change to help bridge the gap, with a particular focus on targeted tree planting to intercept and reduce the delivery of diffuse pollutants to water. This form of integrated catchment management offers multiple benefits to society but a significant cost to landowners and managers. New economic instruments, in combination with spatial targeting, need to be developed to ensure cost effective solutions - including tree planting for water benefits - are realised. Payments for Ecosystem Services (PES) are flexible, incentive-based mechanisms that could play an important role in promoting land use change to deliver water quality targets. The PESFOR-W COST Action will consolidate learning from existing woodlands for water PES schemes in Europe and help standardize approaches to evaluating the environmental effectiveness and cost-effectiveness of woodland measures. It will also create a European network through which PES schemes can be facilitated, extended and improved, for example by incorporating other ecosystem services linking with aims of the wider forestscarbon policy nexus

Recursive and Combinatorial Properties of Schubert Polynomials

We describe two recursive methods for the calculation of Schubert polynomials and use them to give new relatively simple proofs of their basic properties. Moreover

On The Expansion of Schur and Schubert Polynomials into Standard Elementary Monomials

Recently S. Fomin, S. Gelfand, and A. Postnikov discovered that quantum Schubert polynomials can be understood as a simple "quantisation" of the expansion of ordinary Schubert polynomials into standard elementary monomials (s.e.m.). It is therefore important to understand the structure of these s.e.m. expansions in more detail. We show that the s.e.m. expansions of Schur polynomials can be described by (1) a simple variant of the Jacobi-Trudi formula and (2) a combinatorial rule based on posets of staircase box diagrams. These posets are seen to be isomorphic to certain principal order ideals in the Bruhat order of symmetric groups. We prove and conjecture extensions of these results for general Schubert polynomials