Abstract
The ordinary and symmetrized partition rank and crank moments of higher order have been extensively studied. If we assign a weight \(\sharp (\lambda )\) for the rank case and a weight \(\omega (\lambda )\) for the crank case, where \(\sharp (\lambda )\) and \(\omega (\lambda )\), respectively, denote the number of parts in the partition \(\lambda \) and the number of ones in \(\lambda \), it will be shown that such weighted ordinary and symmetrized rank and crank moments of higher order are closely related to the corresponding rank and crank moments when the order is odd.
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Notes
The definition of M(m, 1) is slightly notable. According to [5], apart from the only partition of 1 which has crank \(-1\) and hence gives \(M(-1,1)=1\), we also need to assume that \(M(0,1)=-1\), \(M(1,1)=1\) and \(M(m,1)=0\) otherwise.
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Acknowledgements
I would like to thank George Andrews for introducing this problem and having fruitful discussions.
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Chern, S. Weighted partition rank and crank moments II. Odd-order moments. Ramanujan J 57, 471–485 (2022). https://doi.org/10.1007/s11139-020-00365-9
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DOI: https://doi.org/10.1007/s11139-020-00365-9