AI 中文总结
本文引入分拆的配对理论，定义配对指数、配对秩、配对宽度等统计量，证明相关等分布、同余式等结论，关联超分拆与弗罗贝尼乌斯表示，还研究负配对秩的单配对分拆并得到相关恒等式与行列式结果。
AI 中文摘要
本文旨在引入分拆的配对理论。我们从整数分拆的两个统计量开始：配对指数和配对秩。配对指数与分拆的部分数等分布，而其联合细化将这两个组成部分与偶数部分和奇数部分的数量对应起来。我们进一步引入配对宽度，并证明配对指数与配对宽度和部分数、最大部分等联合分布。所得的有限高斯生成函数具有分圆因式分解，从中可导出著名的二项式系数库默尔进位定理。我们还证明了模5同余式：在5n+4的分拆中，模4余1的未配对部分的数量超过模4余3的未配对部分的数量的差值满足该同余式。一个带符号的特化表明，配对秩的奇偶性由自共轭分拆决定。受此启发，我们接着引入第二种图示配对：将德菲方的两个翼折叠在一起后，未配对的单元分裂为连通的对角块。这些块可独立反射，从而对分拆集合进行布尔分解，其中每个分拆对应唯一代表，其所有连续秩均非负。随后我们将该理论与超分拆和弗罗贝尼乌斯表示关联起来，作为推论得到了超分拆的几何实现，即所有主钩均为偶数的分拆。最后，我们研究配对秩为负的单配对分拆，得到涉及奇除数和超分拆的恒等式、配对秩为-2的奇偶性定理，以及一个托普利茨行列式，其按系数的极限是与平面分拆的麦克马洪乘积相关的显式无穷乘积。
英文摘要
The aim of this paper is to introduce pairing theory for partitions. We begin with two statistics on integer partitions, the \emph{pairing index} and the \emph{pairing rank}. The pairing index is equidistributed with the number of parts, while a joint refinement identifies its two constituents with the numbers of even and odd parts. We further introduce the \emph{pairing width} and prove that pairing index and pairing width are jointly equidistributed with the number of parts and the largest part. The resulting finite Gaussian generating function has a cyclotomic factorization from which Kummer's famous carry theorem for binomial coefficients follows. We also prove a mod-$5$ congruence for the excess of unpaired parts congruent to $1$ modulo $4$ over those congruent to $3$ modulo $4$ in the partitions of $5n+4$. A signed specialization exhibits that the parity of the pairing rank is governed by self-conjugate partitions. Motivated by this, we go on to introduce a second, diagrammatic pairing: after the two wings of the Durfee square are folded together, the unpaired cells break into connected \emph{diagonal blocks}. These blocks may be reflected independently, giving a Boolean decomposition of the set of partitions with a unique representative having all successive ranks nonnegative. We then relate our theory to overpartitions and Frobenius representations, obtaining as a corollary a geometric realization of overpartitions in terms of partitions whose principal hooks are all even. Finally, we study simply paired partitions of negative pairing rank, obtaining identities involving odd divisors and overpartitions, a parity theorem for pairing rank $-2$, and a Toeplitz determinant whose coefficientwise limit is an explicit infinite product related to MacMahon's product for plane partitions.
