Andrews George E, Ballantine Cristina
Department of Mathematics, Pennsylvania State University, University Park, PA 16802;
Department of Mathematics and Computer Science, College of the Holy Cross, Worcester, MA 01610
Proc Natl Acad Sci U S A. 2019 Mar 19;116(12):5428-5436. doi: 10.1073/pnas.1820945116. Epub 2019 Mar 4.
An almost partition identity is an identity for partition numbers that is true asymptotically [Formula: see text] of the time and fails infinitely often. We prove a kind of almost partition identity, namely that the number of parts in all self-conjugate partitions of n is almost always equal to the number of partitions of n in which no odd part is repeated and there is exactly one even part (possibly repeated). Not only does the identity fail infinitely often, but also, the error grows without bound. In addition, we prove several identities involving the number of parts in restricted partitions. We show that the difference in the number of parts in all self-conjugate partitions of n and the number of parts in all partitions of n into distinct odd parts equals the number of partitions of n in which no odd part is repeated, the smallest part is odd, and there is exactly one even part (possibly repeated). We provide both analytic and combinatorial proofs of this identity.
一个几乎成立的分拆恒等式是关于分拆数的一个恒等式，它在渐近意义下（公式：见原文）几乎总是成立，但会有无穷多次不成立。我们证明了一种几乎成立的分拆恒等式，即(n)的所有自共轭分拆中的分部数几乎总是等于(n)的分拆数，其中没有奇数部分重复且恰好有一个偶数部分（可能重复）。不仅该恒等式会有无穷多次不成立，而且误差会无界增长。此外，我们证明了几个涉及受限分拆中分部数的恒等式。我们表明，(n)的所有自共轭分拆中的分部数与(n)的所有分成不同奇数部分的分拆中的分部数之差，等于(n)的分拆数，其中没有奇数部分重复，最小部分是奇数，且恰好有一个偶数部分（可能重复）。我们给出了这个恒等式的解析证明和组合证明。
