Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix
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arXiv
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866914642810372096 |
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| author | Prasad, Amritanshu Ram, Samrith |
| author_facet | Prasad, Amritanshu Ram, Samrith |
| contents | We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the $q$-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2304_13947 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix Prasad, Amritanshu Ram, Samrith Combinatorics 05A15, 05A19, 33C45 We express the number of anti-invariant subspaces for a linear operator on a finite vector space in terms of the number of its invariant subspaces. When the operator is diagonalizable with distinct eigenvalues, our formula gives a finite-field interpretation for the entries of the $q$-Hermite Catalan matrix. We also obtain an interesting new proof of Touchard's formula for these entries. |
| title | Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix |
| topic | Combinatorics 05A15, 05A19, 33C45 |
| url | https://arxiv.org/abs/2304.13947 |