Two congruences involving Andrews-Paule's broken 3-diamond partitions and 5-diamond partitions
Two congruences involving Andrews-Paule's broken 3-diamond partitions and 5-diamond partitions
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DOI：
10.3792/pjaa.87.65
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发表时间：
2010-06
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影响因子：
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通讯作者：
Xinhua Xiong
中科院分区：
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作者：
Xinhua Xiong
摘要本文给出了Peter Paule和SilviuRadu猜想的两个涉及破3菱形分区和破5菱形分区的同余证明。2007年，Gorges e . andrewand peter Paule b[1]引入了一类新的组合对象，称为破碎k-钻石。设∆k (n)表示n的k-菱形分割破碎数，则表明x∞n=0∆k (n)q n= Y∞n=1(1−q 2n)(1−q (2k+1)n)(1−q n) 3（1−q (4k+2)n）。2008年Song Heng Chan[3]证明了k = 2时的无限同余族，2009年peter Paule和Silviu Radu[10]给出了两个非标准的无限破2-对角同余族。此外，他们还提出了与破碎的3菱形分区和5菱形分区有关的四个猜想。在本文中，我们证明了他们的第一个猜想和第三个猜想为真：定理1.1（[10]，猜想3.1）。Y∞n=1(1−q n) 4(1−q 2n) 6≡6X∞n=0∆3 (7n +5)q n （mod 7）。定理1.2（[10]，猜想3.3）。e2 (q2
AbstractIn this note, we will give proofs of two congruences involving broken 3-diamond parti-tions and broken 5-diamond partitions which were conjectured by Peter Paule and SilviuRadu. 1 Introduction In 2007 Gorges E.AndrewsandPeter Paule [1] introduced a newclass of combinatorial objectscalled broken k-diamonds. Let ∆ k (n) denote the number of broken k-diamond partitions ofn, then they showed thatX ∞n=0 ∆ k (n)q n =Y ∞n=1 (1 −q 2n )(1 −q (2k+1)n )(1 −q n ) 3 (1 −q (4k+2)n ).In 2008 Song Heng Chan [3] proved an infinite family of congruences when k = 2, in 2009Peter Paule and Silviu Radu [10] gave two non-standard infinite families of broken 2-diamondcongruences. Moreover they stated four conjectures related to broken 3-diamond partitionsand 5-diamond partitions. In this note we show that their first conjecture and the thirdconjecture are true:Theorem 1.1 ( [10], Conjecture 3.1).Y ∞n=1 (1 −q n ) 4 (1 −q 2n ) 6 ≡ 6X ∞n=0 ∆ 3 (7n +5)q n (mod 7).Theorem 1.2 ( [10], Conjecture 3.3).E 4 (q 2