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arXiv · 2312.02508

The Diagonals of a Ferrers Diagram

Giuseppe Cotardo·Anina Gruica·Alberto Ravagnani
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Abstract

We propose and develop a theory of Ferrers diagrams and their $q$-rook polynomials solely based on their diagonals. We show that the cardinalities of the diagonals of a Ferrers diagram are equivalent information to their rook numbers, $q$-rook polynomials, and the rank distribution of matrices supported on the diagram. Our approach is based on the concept of \textit{canonical form} of a Ferrers diagrams, and on two simple diagram operations as the main proof tools. In the second part of the paper we develop the same theory for symmetric Ferrers diagrams, considering symmetric and alternating matrices supported on them. As an application of our results, we establish some combinatorial identities linking symmetric and alternating matrices, which do not appear to have an obvious bijective proof, and which generalize some curious results in enumerative combinatorics.

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Giuseppe Cotardo, Anina Gruica, Alberto Ravagnani. 2023-12-05. The Diagonals of a Ferrers Diagram. https://arxiv.org/abs/2312.02508

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