  Euler showed that the number of partitions of $n$ into distinct parts equals
the number of partitions of $n$ into odd parts. This theorem was generalized by
Glaisher and further by Franklin. Recently, Beck made three conjectures on
partitions with restricted parts, which were confirmed analytically by Andrews
and Chern and combinatorially by Yang.
  Analogous to Euler's partition theorem, it is known that the number of
compositions of $n$ with odd parts equals the number of compositions of $n+1$
with parts greater than one, as both numbers equal the Fibonacci number $F_n$.
Recently, Sills provided a bijective proof for this result using binary
sequences, and Munagi proved a generalization similar to Glaisher's result
using the zigzag graphs of compositions. Extending Sills' bijection, we obtain
a further generalizaiton which is analogous to Franklin's result. We establish,
both analytically and combinatorially, two closed formulas for the number of
compositions with restricted parts appearing in our generalization. We also
prove some composition analogues for the conjectures of Beck.
