Andrews [‘Binary and semi-Fibonacci partitions’, J. Ramanujan Soc. Math. Math. Sci.7(1) (2019), 1–6] recently proved a new identity between the cardinalities of the set of semi-Fibonacci partitions and the set of partitions into powers of 2 with all parts appearing an odd number of times. We extend the identity to the set of semi-$m$-Fibonacci partitions of $n$ and the set of partitions of $n$ into powers of $m$ in which all parts appear with multiplicity not divisible by $m$. We also give a new characterisation of semi-$m$-Fibonacci partitions and some congruences satisfied by the associated number sequence.

中文翻译：

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电源分区和半--斐波那契分区

Andrews ['二进制和半斐波那契分区'，J. Ramanujan Soc. 数学。数学。科学。7(1) (2019), 1-6] 最近证明了半斐波那契分区集的基数与分区集的 2 次幂之间的新恒等式，所有部分出现奇数次。我们将恒等式扩展到半$m$-斐波那契分区$n$和分区的集合$n$进入权力$m$其中所有部分都以不可被除的多重性出现$m$. 我们还给出了一个新的半$m$- 斐波那契分区和相关数列满足的一些同余。