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| Format: | Preprint |
| Published: |
2024
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2408.09166 |
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Table of Contents:
- A composition $π=π_1π_2\cdotsπ_k$ of a positive integer $n$ is an ordered collection of one or more positive integers whose sum is $n$ . The number of summands, namely $k$, is called the number of parts of $π$. In this paper, we introduce two statistics over compositions of an integer $n$ with exactly $k$ parts: heights of symmetric peaks and depths of symmetric valleys over all compositions of $n$. We derive an explicit formula for the generating functions of compositions of $n$ with exactly $k$ parts according to the number of symmetric peaks (valleys) and the total heights (depths) of peaks (valleys).