Type B Set partitions, an analogue of restricted growth functions
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866909178008698880 |
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| author | Acharyya, Amrita |
| author_facet | Acharyya, Amrita |
| contents | In this work, we study type B set partitions for a given specific positive integer $k$ defined over $\langle n\rangle=\{-n, -(n-1),\cdots -1,0,1,\cdots n-1,n\}$. We found a few generating functions of type B analogue for some of the set partition statistics defined by Wachs, White and Steingrimsson for partitions over positive integers $[n] =\{1,2,\cdots n\}$, both for standard and ordered set partitions respectively. We extended the idea of restricted growth functions utilized by Wachs and White for set partitions over $[n]$, in the scenario of $\langle n\rangle$ and called the analogue as Signed Restricted Growth Function (SRGF).
We discussed analogues of major index for type B partitions in terms of SRGF. We found an analogue of Foata bijection and reduced matrix for type B set partitions as done by Sagan for set partitions of $[n]$ with sepcific number of blocks $k$. We conclude with some open questions regarding the type B analogue of some well known results already done in case of set partitions of $[n]$. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2404_14439 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Type B Set partitions, an analogue of restricted growth functions Acharyya, Amrita Combinatorics In this work, we study type B set partitions for a given specific positive integer $k$ defined over $\langle n\rangle=\{-n, -(n-1),\cdots -1,0,1,\cdots n-1,n\}$. We found a few generating functions of type B analogue for some of the set partition statistics defined by Wachs, White and Steingrimsson for partitions over positive integers $[n] =\{1,2,\cdots n\}$, both for standard and ordered set partitions respectively. We extended the idea of restricted growth functions utilized by Wachs and White for set partitions over $[n]$, in the scenario of $\langle n\rangle$ and called the analogue as Signed Restricted Growth Function (SRGF). We discussed analogues of major index for type B partitions in terms of SRGF. We found an analogue of Foata bijection and reduced matrix for type B set partitions as done by Sagan for set partitions of $[n]$ with sepcific number of blocks $k$. We conclude with some open questions regarding the type B analogue of some well known results already done in case of set partitions of $[n]$. |
| title | Type B Set partitions, an analogue of restricted growth functions |
| topic | Combinatorics |
| url | https://arxiv.org/abs/2404.14439 |