# Compositions with restricted parts

## TL;DR

This paper extends classical partition theorems to compositions with restricted parts, providing new bijective and analytical formulas, and confirming conjectures related to compositions with specific restrictions.

## Contribution

It introduces a new generalization of composition theorems analogous to Franklin's, extending previous bijections and formulas, and confirms related conjectures.

## Key findings

- Derived two closed formulas for compositions with restricted parts.
- Extended Sills' bijection to a broader class of compositions.
- Confirmed conjectures of Beck in the context of compositions.

## Abstract

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.