不被 $2^l$、$2$ 和 $3^l$ 整除的指定被加数分拆关于模 $2$、$4$ 和 $3$ 的性质
Partitions With Designated Summands Not Divisible by $2^l$, $2$, and $3^l$ Modulo $2$, $4$, and $3$
AI总结：
本文明确刻画了部分不被 $2^l$、$2$ 和 $3^l$ 整除的指定被加数分拆数分别关于模 $2$、$4$ 和 $3$ 的性质，提供了同余式与快速递推应用，并揭示了其与奇重数分拆数的联系。
摘要：
自 Andrews、Lewis 和 Lovejoy 首次引入并研究指定被加数分拆以来，已证明了大量同余式。本文明确刻画了其部分不被 $2^\ell$、$2$ 和 $3^\ell$ 整除的指定被加数分拆数分别关于模 $2$、$4$ 和 $3$ 的性质，极大扩展了先前成果。我们以同余式和快速计算递推式的形式给出了这些刻画的应用。此外，我们展示了指定被加数分拆数与奇重数分拆数之间此前未被记录的联系。
英文摘要：
Numerous congruences for partitions with designated summands have been proven since first being introduced and studied by Andrews, Lewis, and Lovejoy. This paper explicitly characterizes the number of partitions with designated summands whose parts are not divisible by $2^\ell$, $2$, and $3^\ell$ working modulo $2,\ 4,$ and $3$, respectively, greatly extending previous results on the subject. We provide a few applications of our characterizations throughout in the form of congruences and a computationally fast recurrence. Moreover, we illustrate a previously undocumented connection between the number of partitions with designated summands and the number of partitions with odd multiplicities.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
