  We give an explicit description of toric sheaves on the weighted projective
plane $\mathbb{P}(a,b,c)$ viewed as a toric Deligne-Mumford stack. The integers
$(a,b,c)$ are not necessarily chosen coprime or mutually coprime allowing for
gerbe and root stack structures.
  As an application, we describe the fixed point locus of the moduli scheme of
stable rank 1 and 2 torsion free sheaves on $\mathbb{P}(a,b,c)$ with fixed
$K$-group class. Summing over all $K$-group classes, we obtain explicit
formulae for generating functions of the topological Euler characteristics.
  In the case of stable rank 2 locally free sheaves on $\mathbb{P}(a,b,c)$ with
$a,b,c \leq 2$ the generating functions can be expressed in terms of Hurwitz
class numbers and give rise to modular forms of weight $3/2$. This generalizes
Klyachko's computation on $\mathbb{P}^2$ and is consistent with $S$-duality
predictions from physics.
