The number of Borel orbits in polarizations (the symmetric variety SL n /S(GL p × GL q )) is analyzed, various (bivariate) generating functions are found. Relations to lattice path combinatorics are explored.

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Antep cheese is a local Turkish cheese characterized by scalding during production and ripened in brine. In this study, Antep cheeses were produced using mixtures of different milk types (cow, sheep, and goat milk) and ripened for 5 months. The composition, proteolytic ripening extension index (REI), free fatty acid (FFA) content, and volatile compounds of the cheeses and the variation of the brines were analyzed for the 5-month ripening period. Low proteolytic activity in cheese during ripening caused the cheeses to have low REI values (3.92%-7.57%), although it was observed that some parts of the watersoluble nitrogen fractions diffused into the brine, causing a lower REI. As a result of lipolysis during ripening, total FFA (TFFA) concentrations in all cheeses were increased, whereas the highest increases were detected in the concentrations of short-chain FFAs. The highest FFA concentrations were determined in cheese produced using goat milk, and the volatile FFA ratio in TFFA exceeded 10% in the third month of ripening. Although it was observed that the milk types used in the production had significant effects on the change of volatile compounds of the produced cheeses and their brines, the impact of the ripening time was more important.

This is a continuation of our combinatorial program on the enumeration of Borel orbits in symmetric varieties of classical types. Here, we determine the generating series the numbers of Borel orbits in SO 2n+1 /S(O 2p ×O 2q+1 ) (type BI) and in Sp n /Sp p ×Sp q (type CII). In addition, we explore relations to lattice path enumeration.

We examine Borel subgroup orbits in the classical symmetric space of type CI, which are parametrized by skew symmetric (n, n)-clans. We describe bijections between such clans, certain weighted lattice paths, and pattern-avoiding signed involutions, and we give a cell decomposition of the symmetric space in terms of collections of clans called sects. The largest sect with a conjectural closure order is isomorphic (as a poset) to the Bruhat order on partial involutions.