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| Main Authors: | , , , |
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| Format: | Preprint |
| Published: |
2026
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2605.04889 |
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| _version_ | 1866910194507710464 |
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| author | Hao, Robert X. J. Niu, Xiaorui Sang, Doris D. M. Shi, Diane Y. H. |
| author_facet | Hao, Robert X. J. Niu, Xiaorui Sang, Doris D. M. Shi, Diane Y. H. |
| contents | Andrews investigated parity conditions in the Rogers-Ramanujan-Gordon theorem.
Under the conditions that even parts or odd parts appear an even number of times,
Andrews discovered two Rogers-Ramanujan-Gordon type partition theorems
and derived corresponding generating functions. In the Rogers-Ramanujan-Gordon
theorem, there are two parameters $k$ and $a$, where $k-1$ is the maximum
number of consecutive parts $l$ and $l+1$, and $a-1$ is the maximum number of
parts equal to $1$. Andrews' first theorem deals with the case
$k\equiv a \;(\rm{mod}\;2)$, while the second theorem concerns the case
where $k$ is even and $a$ is odd. These two partition identities have different
infinite product forms on the right-hand side. In this paper, we consider the case
$k\not\equiv a \;(\rm{mod}\;2)$ and use Bailey's lemma to obtain an Andrews-Gordon
type identity whose right-hand side coincides with that of Andrews' identity for the
case $k\equiv a \;(\rm{mod}\;2)$. We were unable to find a suitable combinatorial
interpretation of the infinite sum form of this expression in terms of partitions,
but with the help of lattice paths, we provide an appropriate combinatorial interpretation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2605_04889 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | An Andrews-Gordon Type Identity Related to Andrews' Parity Consideration Hao, Robert X. J. Niu, Xiaorui Sang, Doris D. M. Shi, Diane Y. H. Combinatorics Andrews investigated parity conditions in the Rogers-Ramanujan-Gordon theorem. Under the conditions that even parts or odd parts appear an even number of times, Andrews discovered two Rogers-Ramanujan-Gordon type partition theorems and derived corresponding generating functions. In the Rogers-Ramanujan-Gordon theorem, there are two parameters $k$ and $a$, where $k-1$ is the maximum number of consecutive parts $l$ and $l+1$, and $a-1$ is the maximum number of parts equal to $1$. Andrews' first theorem deals with the case $k\equiv a \;(\rm{mod}\;2)$, while the second theorem concerns the case where $k$ is even and $a$ is odd. These two partition identities have different infinite product forms on the right-hand side. In this paper, we consider the case $k\not\equiv a \;(\rm{mod}\;2)$ and use Bailey's lemma to obtain an Andrews-Gordon type identity whose right-hand side coincides with that of Andrews' identity for the case $k\equiv a \;(\rm{mod}\;2)$. We were unable to find a suitable combinatorial interpretation of the infinite sum form of this expression in terms of partitions, but with the help of lattice paths, we provide an appropriate combinatorial interpretation. |
| title | An Andrews-Gordon Type Identity Related to Andrews' Parity Consideration |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2605.04889 |