Abstract
We generalize the generating series of the Dyson ranks and \(M_2\)-ranks of overpartitions to obtain k-fold variants and give a combinatorial interpretation of each. The k-fold generating series correspond to the full ranks of two families of buffered Frobenius representations, which generalize Lovejoy’s first and second Frobenius representations of overpartitions, respectively.
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Notes
This convention ensures that mirroring the diagram across its main diagonal will produce the Young diagram of another overpartition, more commonly known as conjugating the overpartition.
When unspecified, the terms partition and overpartition should be taken to mean partitions and overpartitions into positive parts.
Note that Lovejoy uses Andrews’ convention \(|\nu | = k + \sum (a_i + b_i)\) in his earlier work [12]. Statements of these results have been adjusted for consistency.
This is why only \(\alpha _1\) and \(\beta _1\) may be overpartitions.
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Acknowledgements
The author is very grateful to the referee for uncovering multiple errors and suggesting improvements in the presentation of the results, to Jeremy Lovejoy for careful reading of earlier drafts and many helpful comments, and to Thomas Schmidt for a useful observation for future work.
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Morrill, T. Two Families of Buffered Frobenius Representations of Overpartitions. Ann. Comb. 23, 103–141 (2019). https://doi.org/10.1007/s00026-019-00419-w
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DOI: https://doi.org/10.1007/s00026-019-00419-w