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A Fast Algorithm for Computing the Number of Magic Series

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Abstract

We present a fast algorithm for computing the number of magic series, an enumeration problem of a certain integer partition. Kinnaes showed that the number appears as a coefficient in a Gaussian polynomial and that the exact value can be efficiently extracted with a finite variant of Cauchy’s integral formula. The algorithm requires a bit time complexity of \(O(m^4 M(m \log m))\) and \(O(m \log m)\)-bit space, where m is the order of the magic series and \(M(n)\) is the time complexity of multiplying two n-bit numbers. Through our analysis, we confirm that this is the most efficient among previously reported algorithms. In addition, we show that the number can be computed with a bit time complexity of \(O(m^3 \log m M(m \log m))\) by directly carrying out polynomial multiplication and division on the Gaussian polynomial. Though the space consumption increases to \(O(m^3 \log m)\) bits, we demonstrate that our method actually computes the number faster for large orders.

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Acknowledgements

The authors are grateful to the anonymous reviewers for their insightful comments and suggestions. This research in part used the computational resources of the Cygnus provided by the Multidisciplinary Cooperative Research Program in the Center for Computational Sciences, University of Tsukuba.

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Authors and Affiliations

  1. Graduate School of Science and Technology, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki,  305-8573, Japan

    Yukimasa Sugizaki

  2. Center for Computational Sciences, University of Tsukuba, 1-1-1 Tennodai, Tsukuba, Ibaraki,  305-8577, Japan

    Daisuke Takahashi

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  1. Yukimasa Sugizaki
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Sugizaki, Y., Takahashi, D. A Fast Algorithm for Computing the Number of Magic Series. Ann. Comb. 26, 511–532 (2022). https://doi.org/10.1007/s00026-022-00584-5

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