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arXiv · 2304.13947

Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix

Amritanshu Prasad·Samrith Ram
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Abstract

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.

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Amritanshu Prasad, Samrith Ram. 2024-01-17. Enumeration of Anti-Invariant Subspaces and Touchard's Formula for the Entries of the $q$-Hermite Catalan Matrix. https://doi.org/10.1016/j.aam.2023.102654

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