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Poincaré polynomial at \(-1\) associated with a Young diagram of three rows

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Abstract

In this paper, we give an explicit formula for the Poincaré polynomial \(P_\lambda (x)\) for the Betti numbers of the Springer fibers over nilpotent elements in \(gl_n(\mathbb {C})\) of Jordan form \(\lambda =abc\) with \(a\ge b\ge c\ge 0\) at \(x=-1\). In particular, we introduce \(\lambda \)-vacillating diagrams and show that \(P_{ab}(-1)\) is equal to the number of restricted Dyck paths.

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Acknowledgements

The author thanks Anna Melnikov for reading the previous version of the present paper and for her helpful comments.

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  1. AL-Qasemi Academy, (R.A)-Academic College of Education, 3010000, Baqa-El-Gharbia, Israel

    Ronit Mansour

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  1. Ronit Mansour
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Correspondence to Ronit Mansour.

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Mansour, R. Poincaré polynomial at \(-1\) associated with a Young diagram of three rows. J Algebr Comb 56, 1011–1021 (2022). https://doi.org/10.1007/s10801-022-01142-1

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  • DOI: https://doi.org/10.1007/s10801-022-01142-1

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