论文标题
富兰克林身份的类似物引起的最小排除物的新概括
A new generalization of the minimal excludant arising from an analogue of Franklin's identity
论文作者
论文摘要
Euler的经典身份指出，整数分配给奇数部分和不同部分的分区数量是平等的。富兰克林（Franklin）通过考虑$ j $不同的$ r $的分区来进行概括，这是一个正整数$ r $。我们通过研究总共$ r $ $ j $倍数的分区数量来证明富兰克林的身份，在此过程中，发现了最小排除物（MEX）的自然概括，我们称之为$ r $ $ - 链MEX。此外，我们得出了$σ_{rc} \ textup {mex}（n）$的生成函数，$ r $ chain mex的总和接管了$ n $的所有分区，从而为$σ_{rc} {rc} \ textup {mex}（mex} $ for n n n n n n eate and for n n eate and for and for n n eate and for and for $ n $的组合标识$σ\ textup {mex}（n）$，$ n $的所有分区的MEX和。
Euler's classical identity states that the number of partitions of an integer into odd parts and distinct parts are equinumerous. Franklin gave a generalization by considering partitions with exactly $j$ different multiples of $r$, for a positive integer $r$. We prove an analogue of Franklin's identity by studying the number of partitions with $j$ multiples of $r$ in total and in the process, discover a natural generalization of the minimal excludant (mex) which we call the $r$-chain mex. Further, we derive the generating function for $σ_{rc} \textup{mex}(n)$, the sum of $r$-chain mex taken over all partitions of $n$, thereby deducing a combinatorial identity for $σ_{rc} \textup{mex}(n)$, which neatly generalizes the result of Andrews and Newman for $σ\textup{mex}(n)$, the sum of mex over all partitions of $n$.
