Extreme diagonally and antidiagonally symmetric alternating sign matrices of odd order

For each $\alpha \in \{0,1,-1 \}$, we count diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order with a maximal number of $\alpha$'s along the diagonal and the antidiagonal, as well as DASASMs of fixed odd order with a minimal number of $0$'s along the diagonal and the antidiagonal. In these enumerations, we encounter product formulas that have previously appeared in plane partition or alternating sign matrix counting, namely for the number of all alternating sign matrices, the number of cyclically symmetric plane partitions in a given box, and the number of vertically and horizontally symmetric ASMs. We also prove several refinements. For instance, in the case of DASASMs with a maximal number of $-1$'s along the diagonal and the antidiagonal, these considerations lead naturally to the definition of alternating sign triangles. These are new objects that are equinumerous with ASMs, and we are able to prove a two parameter refinement of this fact, involving the number of $-1$'s and the inversion number on the ASM side. To prove our results, we extend techniques to deal with triangular six-vertex configurations that have recently successfully been applied to settle Robbins' conjecture on the number of all DASASMs of odd order. Importantly, we use a general solution of the reflection equation to prove the symmetry of the partition function in the spectral parameters. In all of our cases, we derive determinant or Pfaffian formulas for the partition functions, which we then specialize in order to obtain the product formulas for the various classes of extreme odd DASASMs under consideration.Comment: 41 pages, several minor improvements in response to referee's comments. Final version. Matches published version except for very minor change

Forest Certification: Toward Common Standards?

The forestry industry provides a good illustration of the active roles that industry associations, environmental nongovernmental organizations (NGOs), national governments, and international organizations can play in developing and promoting codes of conduct that are formally sanctioned and certified. It also reflects some of the challenges of disseminating codes of conduct in developing countries and ensuring market benefits from certification. We describe the emergence of forest certification standards, outline current certification schemes, and discuss the role of major corporations in creating demand for certified products. We also discuss the limited success of certification and some of the obstacles to its adoption in developing countries. The current diversity of forest certification programs and ecolabeling schemes has created a costly, less-than-transparent system that has been largely ineffective in terms of the initial goals of reducing tropical deforestation and illegal logging. Some steps have been taken toward harmonization of different certification criteria as well as endorsement and mutual recognition among existing forest certification programs. However, it is unlikely that standardization alone can overcome other, more serious barriers to certification in developing countries.forest certification, codes of conduct, Forest Stewardship Council, PEFC, Sustainable Forestry Initiative, sustainable forest management

New Zealand Handbook, 5th Edition (Review)

New Zealand Handbook, 5th edition. by Jane King, 1999; 512 pages, illustrations, photos and maps. US$18.95; UK £ 12.99; CAN$ 29.50 ISBN: 1-56691-165-6. Moon Travel Handbooks, Emeryville, CA Review by Steven Roger Fischer</p

Diagonally and antidiagonally symmetric alternating sign matrices of odd order

We study the enumeration of diagonally and antidiagonally symmetric alternating sign matrices (DASASMs) of fixed odd order by introducing a case of the six-vertex model whose configurations are in bijection with such matrices. The model involves a grid graph on a triangle, with bulk and boundary weights which satisfy the Yang-Baxter and reflection equations. We obtain a general expression for the partition function of this model as a sum of two determinantal terms, and show that at a certain point each of these terms reduces to a Schur function. We are then able to prove a conjecture of Robbins from the mid 1980's that the total number of (2n+1)x(2n+1) DASASMs is \prod_{i=0}^n (3i)!/(n+i)!, and a conjecture of Stroganov from 2008 that the ratio between the numbers of (2n+1)x(2n+1) DASASMs with central entry -1 and 1 is n/(n+1). Among the several product formulae for the enumeration of symmetric alternating sign matrices which were conjectured in the 1980's, that for odd-order DASASMs is the last to have been proved