  We investigate the Jordan-H\"older property (JHP) in exact categories. First,
we show that (JHP) holds in an exact category if and only if the Grothendieck
monoid introduced by Berenstein and Greenstein is free. Moreover, we give a
criterion for this which only uses the Grothendieck group and the number of
simple objects. Next, we apply these results to the representation theory of
artin algebras. For a large class of exact categories including functorially
finite torsion(-free) classes, (JHP) holds precisely when the number of
indecomposable projectives is equal to that of simples. We study torsion-free
classes in a quiver of type A in detail using the combinatorics of symmetric
groups. We introduce Bruhat inversions of permutations and show that simples in
a torsion-free class are in bijection with Bruhat inversions of the
corresponding $c$-sortable element. We use this to give a combinatorial
criterion for (JHP).
