New combinatorial proof of Gaussian polynomial and the monotonicity of the Garvan's $k$-rank
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866911200211632128 |
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| author | Qu, Wenxia Zang, Wenston J. T. |
| author_facet | Qu, Wenxia Zang, Wenston J. T. |
| contents | Gaussian polynomial, which is also known as $q$-binomial coefficient, is one of the fundamental concepts in the theory of partitions. Zeilberger provided a combinatorial proof of Gaussian polynomial, which is called Algorithm Z by Andrews and Bressoud. In this paper, we provide a new bijection on Gaussian polynomial, which leads to a refinement of Algorithm Z. Moreover, using this bijection, we provide an alternative proof of generalized Rogers-Ramanujan identity, which was first proved by Bressoud and Zeilberger. Furthermore, we give a combinatorial proof of the monotonicity property of Garvan's $k$-rank, which is a generalization of Dyson's rank and Andrews-Garvan's crank. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2510_08262 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | New combinatorial proof of Gaussian polynomial and the monotonicity of the Garvan's $k$-rank Qu, Wenxia Zang, Wenston J. T. Combinatorics Gaussian polynomial, which is also known as $q$-binomial coefficient, is one of the fundamental concepts in the theory of partitions. Zeilberger provided a combinatorial proof of Gaussian polynomial, which is called Algorithm Z by Andrews and Bressoud. In this paper, we provide a new bijection on Gaussian polynomial, which leads to a refinement of Algorithm Z. Moreover, using this bijection, we provide an alternative proof of generalized Rogers-Ramanujan identity, which was first proved by Bressoud and Zeilberger. Furthermore, we give a combinatorial proof of the monotonicity property of Garvan's $k$-rank, which is a generalization of Dyson's rank and Andrews-Garvan's crank. |
| title | New combinatorial proof of Gaussian polynomial and the monotonicity of the Garvan's $k$-rank |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2510.08262 |