  We consider finite sums of counting functions on the free group $F_n$ and the
free monoid $M_n$ for $n \geq 2$. Two such sums are considered equivalent if
they differ by a bounded function. We find the complete set of linear relations
between equivalence classes of sums of counting functions and apply this result
to construct an explicit basis for the vector space of such equivalence
classes. Moreover, we provide a graphical algorithm to determine whether two
given sums of counting functions are equivalent. In particular, this yields an
algorithm to decide whether two sums of Brooks quasimorphisms on $F_n$
represent the same class in bounded cohomology.
