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<p><i>In this survey, we overview the celebrated Ramanujan’s congruences modulo 5, 7 and 11 for the partition function and how the classical combinatorial statistics rank and crank introduced by Dyson, Andrews and Garvan explain them. We also present some recent results on the explicit calculation of the dissections of the deviations of rank and crank and how these computations help to deduce unexpected arithmetic properties of these combinatorial statistics. In addition, we posed some open problems in this area. Bibliography: 36 titles.</i></p>

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SURVEY ON DEVIATIONS AND ARITHMETIC PROPERTIES OF COMBINATORIAL STATISTICS AFTER DYSON, ANDREWS, AND GARVAN

  • 期刊论文

  • 发表日期：

    2025年8月22日

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  • Nikolay E. Borozenets

摘要

In this survey, we overview the celebrated Ramanujan’s congruences modulo 5, 7 and 11 for the partition function and how the classical combinatorial statistics rank and crank introduced by Dyson, Andrews and Garvan explain them. We also present some recent results on the explicit calculation of the dissections of the deviations of rank and crank and how these computations help to deduce unexpected arithmetic properties of these combinatorial statistics. In addition, we posed some open problems in this area. Bibliography: 36 titles.