{"title":"n+1$$ 份的分割钻石上的新全等式","authors":"Yongqiang Chen, Eric H. Liu, Olivia X. M. Yao","doi":"10.1007/s11139-024-00934-2","DOIUrl":null,"url":null,"abstract":"<p>Recently, Andrews and Paule introduced a partition function <i>PDN</i>1(<i>N</i>) which counts the number of partition diamonds with <span>\\(n+1\\)</span> copies of <i>n</i> where summing the parts at the links gives <i>N</i>. They also established the generating function of <i>PDN</i>1(<i>n</i>) and proved congruences modulo 5,7,25,49 for <i>PDN</i>1(<i>n</i>). At the end of their paper, Andrews and Paule asked for the existence of other types of congruence relations for <i>PDN</i>1(<i>n</i>). Motivated by their work, we prove some new congruences modulo 125 and 625 for <i>PDN</i>1(<i>n</i>) by using some identities due to Chern and Tang. In particular, we discover a family of strange congruences modulo 625 for <i>PDN</i>1(<i>n</i>). For example, we prove that for <span>\\(k\\ge 0\\)</span>, </p><span>$$\\begin{aligned} PDN1\\left( 5^7 \\cdot 7^{8k}+\\frac{ 19\\cdot 5^7\\cdot 7^{8k}+1 }{24} \\right) \\equiv 5^3 \\pmod {5^4}. \\end{aligned}$$</span>","PeriodicalId":501430,"journal":{"name":"The Ramanujan Journal","volume":"155 1","pages":""},"PeriodicalIF":0.0000,"publicationDate":"2024-08-23","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"New congruences on partition diamonds with $$n+1$$ copies of n\",\"authors\":\"Yongqiang Chen, Eric H. Liu, Olivia X. M. Yao\",\"doi\":\"10.1007/s11139-024-00934-2\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Recently, Andrews and Paule introduced a partition function <i>PDN</i>1(<i>N</i>) which counts the number of partition diamonds with <span>\\\\(n+1\\\\)</span> copies of <i>n</i> where summing the parts at the links gives <i>N</i>. They also established the generating function of <i>PDN</i>1(<i>n</i>) and proved congruences modulo 5,7,25,49 for <i>PDN</i>1(<i>n</i>). At the end of their paper, Andrews and Paule asked for the existence of other types of congruence relations for <i>PDN</i>1(<i>n</i>). Motivated by their work, we prove some new congruences modulo 125 and 625 for <i>PDN</i>1(<i>n</i>) by using some identities due to Chern and Tang. In particular, we discover a family of strange congruences modulo 625 for <i>PDN</i>1(<i>n</i>). For example, we prove that for <span>\\\\(k\\\\ge 0\\\\)</span>, </p><span>$$\\\\begin{aligned} PDN1\\\\left( 5^7 \\\\cdot 7^{8k}+\\\\frac{ 19\\\\cdot 5^7\\\\cdot 7^{8k}+1 }{24} \\\\right) \\\\equiv 5^3 \\\\pmod {5^4}. \\\\end{aligned}$$</span>\",\"PeriodicalId\":501430,\"journal\":{\"name\":\"The Ramanujan Journal\",\"volume\":\"155 1\",\"pages\":\"\"},\"PeriodicalIF\":0.0000,\"publicationDate\":\"2024-08-23\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"The Ramanujan Journal\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1007/s11139-024-00934-2\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"\",\"JCRName\":\"\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"The Ramanujan Journal","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1007/s11139-024-00934-2","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"","JCRName":"","Score":null,"Total":0}
New congruences on partition diamonds with $$n+1$$ copies of n
Recently, Andrews and Paule introduced a partition function PDN1(N) which counts the number of partition diamonds with \(n+1\) copies of n where summing the parts at the links gives N. They also established the generating function of PDN1(n) and proved congruences modulo 5,7,25,49 for PDN1(n). At the end of their paper, Andrews and Paule asked for the existence of other types of congruence relations for PDN1(n). Motivated by their work, we prove some new congruences modulo 125 and 625 for PDN1(n) by using some identities due to Chern and Tang. In particular, we discover a family of strange congruences modulo 625 for PDN1(n). For example, we prove that for \(k\ge 0\),
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