  Tesler matrices are certain integral matrices counted by the Kostant
partition function and have appeared recently in Haglund's study of diagonal
harmonics. In 2014, Drew Armstrong defined a poset on such matrices and
conjectured that the characteristic polynomial of this poset is a power of
$(q-1)$. We use a method of Hallam and Sagan to prove a stronger version of
this conjecture for posets of a certain class of generalized Tesler matrices.
We also study bounds for the number of Tesler matrices and how they compare to
the number of parking functions, the dimension of the space of diagonal
harmonics.
