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| Format: | Preprint |
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2025
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| Online Access: | https://arxiv.org/abs/2502.02262 |
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| _version_ | 1866913908791443456 |
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| author | Ogievetsky, Oleg Shlosman, Senya |
| author_facet | Ogievetsky, Oleg Shlosman, Senya |
| contents | We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape $λ/μ.$ The partition function of this particle system gives the generating function of the SsYT of skew shape $λ/μ.$ Thus we obtain a bijective proof of the Stanley formula for the SsYT generating function.
To do this we define for every SsYT $T$ its plinth, $\mathsf{p}\left( T\right) ,$ which is a SsYT of the same shape $λ/μ.$ The set of plinths is finite. Our bijection associates to every SsYT $T$ a pair $\left( \mathsf{p}\left( T\right) ,Y\left( T-\mathsf{p}\left( T\right) \right) \right) ,$ where $Y\left( T-\mathsf{p}\left( T\right) \right) $ is the reading Young diagram of the SsYT $\left( T-\mathsf{p}\left( T\right) \right) $. \newline In particular, every Standard Young Tableau (SYT) $P$ has its plinth, $\mathsf{p}\left( P\right) $. The two statistics of SYT-s -- the volume $\left\vert \mathsf{p}\left( P\right) \right\vert $ and $\mathsf{maj}\left( P\right) $ -- are related via the Schützenberger involution $Sch:$% \[ \left\vert \mathsf{p}\left( P\right) \right\vert =\mathsf{maj}\left( Sch\left( P\right) \right) . \] |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2502_02262 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The major index (maj) and its Schützenberger dual Ogievetsky, Oleg Shlosman, Senya Combinatorics 05A19 (Primary) 05A15 (Secondary) We construct the independent particle representation for the Semistandard Young Tableaux (SsYT) of skew shape $λ/μ.$ The partition function of this particle system gives the generating function of the SsYT of skew shape $λ/μ.$ Thus we obtain a bijective proof of the Stanley formula for the SsYT generating function. To do this we define for every SsYT $T$ its plinth, $\mathsf{p}\left( T\right) ,$ which is a SsYT of the same shape $λ/μ.$ The set of plinths is finite. Our bijection associates to every SsYT $T$ a pair $\left( \mathsf{p}\left( T\right) ,Y\left( T-\mathsf{p}\left( T\right) \right) \right) ,$ where $Y\left( T-\mathsf{p}\left( T\right) \right) $ is the reading Young diagram of the SsYT $\left( T-\mathsf{p}\left( T\right) \right) $. \newline In particular, every Standard Young Tableau (SYT) $P$ has its plinth, $\mathsf{p}\left( P\right) $. The two statistics of SYT-s -- the volume $\left\vert \mathsf{p}\left( P\right) \right\vert $ and $\mathsf{maj}\left( P\right) $ -- are related via the Schützenberger involution $Sch:$% \[ \left\vert \mathsf{p}\left( P\right) \right\vert =\mathsf{maj}\left( Sch\left( P\right) \right) . \] |
| title | The major index (maj) and its Schützenberger dual |
| topic | Combinatorics 05A19 (Primary) 05A15 (Secondary) |
| url | https://arxiv.org/abs/2502.02262 |