Abstract: Let G be a transitive permutation group
on a set Q. The orbit
decompositions of the actions of G on the sets of ordered n-tuples
from Q with elements repeated at most m times are studied.
The
decompositions involve Stirling numbers and a new class of related
numbers, the so-called multi-restricted numbers. The talk presents
exponential generating functions for the numbers of orbits, and examines
relationships between various powers of the G-set involving
Stirling
numbers and the multi-restricted numbers. Moreover, a three-term
recurrence formula for the multi-restricted numbers is obtained.
Abstract: We begin by describing classical determinantal
inequalities involving
principal minors of positive definite matrices due to Hadamard, Fischer,
and
Koteljanskii. In the mid to later 1900's there was much work giving
particular
generalizations of these inequalities for positive definite matrices,
M-matrices, inverse M-matrices and totally positive matrices.
We finish by
describing a modern effort to characterize all bounded ratios of products
of
principal minors within a class of matrices, a major thrust of modern
matrix
theory.