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<p>In a recent paper (<i>J. Math. Anal. Appl.</i> <b>549</b> (2025), 129537), Andrews and El Bachraoui studied the number of partitions whose smallest part is repeated exactly <i>k</i> times and the remaining parts are not repeated. They expressed the generating functions of these partition numbers as linear combinations of the <i>q</i>-Pochhammer symbols. They also found similar expressions for the differences in partition numbers between subclasses of their partitions. As corollaries, for cases <InlineEquation ID="IEq1"> <EquationSource Format="TEX">\(k=1,2\)</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>k</mi> <mo>=</mo> <mn>1</mn> <mo>,</mo> <mn>2</mn> </mrow> </math></EquationSource> </InlineEquation>, and 3, they analytically derived new identities and inequalities for the partitions into distinct parts and sought combinatorial proofs. D. Chen, R. Chen and Zhao established alternative combinatorial proofs of the results of Andrews and El Bachraoui. In this paper, we present alternative bijective proofs of some of the results.</p>

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Bijective proofs of some results on partitions with repeated smallest parts

  • 期刊论文

  • 发表日期：

    2025年12月17日

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  • Nayandeep Deka Baruah,
  • Pankaj Jyoti Mahanta

摘要

In a recent paper (J. Math. Anal. Appl. 549 (2025), 129537), Andrews and El Bachraoui studied the number of partitions whose smallest part is repeated exactly k times and the remaining parts are not repeated. They expressed the generating functions of these partition numbers as linear combinations of the q-Pochhammer symbols. They also found similar expressions for the differences in partition numbers between subclasses of their partitions. As corollaries, for cases \(k=1,2\) k = 1 , 2 , and 3, they analytically derived new identities and inequalities for the partitions into distinct parts and sought combinatorial proofs. D. Chen, R. Chen and Zhao established alternative combinatorial proofs of the results of Andrews and El Bachraoui. In this paper, we present alternative bijective proofs of some of the results.