  In the context of A. Eskin and A. Okounkov's approach to the calculation of
the volumes of the different strata of the moduli space of quadratic
differentials, two objects have a prominent role. Namely, the characters of
near-involutions and the pillowcase weights. For the former we give a fairly
explicit formula. On the other hand, the pillowcase weights induce a
distribution on the space of Young diagrams. We analyze this distribution and
prove several facts, including that its limit shape corresponds to the one
induced by the uniform distribution, that the probability concentrates on the
set of partitions with very similar 2-quotients, and that there is no hope for
a full Central Limit Theorem.
  This is a reformatted version of the author's Ph.D. thesis, advised by
Professor Andrei Okounkov. The results will be published in a forthcoming
paper.
