Abstract In 1956, Alder conjectured that q d ( n ) − Q d ( n ) ≧ 0 , where q d ( n ) and Q d ( n ) are the number of partitions of n into parts differing by at least d and the number of partitions of n into parts which are congruent to ± 1 (mod d +3), respectively. It took more than 50 years to complete the proof and the first breakthrough was made by Andrews in 1971, who proved that the conjecture holds for d = 2 r − 1 ( r ≧ 4 ). In this paper, we prove two analogous partition inequalities following Andrew’s method. One of them generalizes the second Rogers–Ramanujan identity, which is the number of partitions of n into parts differing by at least d with the smallest part at least 2 is greater than or equal to that of partitions of n into parts congruent to ≡ ± 2 ( mod d + 3 ) excluding d + 1 when d = 2 r − 2 ( r ≧ 2 , r ≠ 3 , 4 ) .

中文翻译：

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推广第二罗杰斯-拉马努金恒等式的阿尔德-安德鲁斯猜想的类似物

摘要 1956 年，Alder 推测 qd ( n ) − Q d ( n ) ≧ 0 ，其中 qd ( n ) 和 Q d ( n ) 是将 n 划分为至少相差 d 的部分的数量和划分的数量分别将 n 分成与 ± 1 (mod d +3) 一致的部分。用了 50 多年的时间来完成证明，安德鲁斯在 1971 年取得了第一个突破，他证明了该猜想对于 d = 2 r − 1 ( r ≧ 4 ) 成立。在本文中，我们按照安德鲁的方法证明了两个类似的划分不等式。其中之一概括了第二个 Rogers-Ramanujan 恒等式，即 n 的分割数至少相差 d 且最小的部分至少有 2 大于或等于 n 的分割数等于 ≡ ± 2 ( mod d + 3 ) 不包括 d + 1 当 d = 2 r − 2 (r ≧ 2 , r ≠ 3, 4) 。