Compositions with restricted parts
Published 2018-12-28Version 1
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. Extending a bijection due to Sills, we generalize this result similarly as Glaisher and Franklin generalized Euler's partition theorem. We prove, both analytically and combinatorially, two closed formulas for the number of compositions with restricted parts appearing in our generalization. We also obtain some composition analogues for the conjectures of Beck.