A bijective proof of an identity of Berkovich and Uncu
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| Format: | Preprint |
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2023
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| _version_ | 1866916388969381888 |
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| author | Dhar, Aritram Mukhopadhyay, Avi |
| author_facet | Dhar, Aritram Mukhopadhyay, Avi |
| contents | The BG-rank BG($π$) of an integer partition $π$ is defined as $$\text{BG}(π) := i-j$$ where $i$ is the number of odd-indexed odd parts and $j$ is the number of even-indexed odd parts of $π$. In a recent work, Fu and Tang ask for a direct combinatorial proof of the following identity of Berkovich and Uncu $$B_{2N+ν}(k,q)=q^{2k^2-k}\left[\begin{matrix}2N+ν\\N+k\end{matrix}\right]_{q^2}$$ for any integer $k$ and non-negative integer $N$ where $ν\in \{0,1\}$, $B_N(k,q)$ is the generating function for partitions into distinct parts less than or equal to $N$ with BG-rank equal to $k$ and $\left[\begin{matrix}a+b\\b\end{matrix}\right]_q$ is a Gaussian binomial coefficient. In this paper, we provide a bijective proof of Berkovich and Uncu's identity along the lines of Vandervelde and Fu and Tang's idea. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2309_07785 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | A bijective proof of an identity of Berkovich and Uncu Dhar, Aritram Mukhopadhyay, Avi Combinatorics Number Theory 05A15, 05A17, 05A19, 11P81, 11P83, 11P84 The BG-rank BG($π$) of an integer partition $π$ is defined as $$\text{BG}(π) := i-j$$ where $i$ is the number of odd-indexed odd parts and $j$ is the number of even-indexed odd parts of $π$. In a recent work, Fu and Tang ask for a direct combinatorial proof of the following identity of Berkovich and Uncu $$B_{2N+ν}(k,q)=q^{2k^2-k}\left[\begin{matrix}2N+ν\\N+k\end{matrix}\right]_{q^2}$$ for any integer $k$ and non-negative integer $N$ where $ν\in \{0,1\}$, $B_N(k,q)$ is the generating function for partitions into distinct parts less than or equal to $N$ with BG-rank equal to $k$ and $\left[\begin{matrix}a+b\\b\end{matrix}\right]_q$ is a Gaussian binomial coefficient. In this paper, we provide a bijective proof of Berkovich and Uncu's identity along the lines of Vandervelde and Fu and Tang's idea. |
| title | A bijective proof of an identity of Berkovich and Uncu |
| topic | Combinatorics Number Theory 05A15, 05A17, 05A19, 11P81, 11P83, 11P84 |
| url | https://arxiv.org/abs/2309.07785 |