6 research outputs found
Random partitions with non negative rth differences
Let be the set of partitions of n with non negative rth differences.
Let be a partition chosen uniformly at random among the set .
Let be a positive rth difference chosen uniformly at random in
. The aim of this work is to show that for every , the
probability that approaches as . To
prove this result we use bijective, asymptotic/analytic, and probabilistic
combinatorics
Five Guidelines for Partition Analysis with Applications to Lecture Hall-type Theorems
Five simple guidelines are proposed to compute the generating function for
the nonnegative integer solutions of a system of linear inequalities. In
contrast to other approaches, the emphasis is on deriving recurrences. We show
how to use the guidelines strategically to solve some nontrivial enumeration
problems in the theory of partitions and compositions. This includes a
strikingly different approach to lecture hall-type theorems, with new
-series identities arising in the process. For completeness, we prove that
the guidelines suffice to find the generating function for any system of
homogeneous linear inequalities with integer coefficients. The guidelines can
be viewed as a simplification of MacMahon's partition analysis with ideas from
matrix techiniques, Elliott reduction, and ``adding a slice''