2026/04/30 by AMAN DEVNANI, PRAMOD EYYUNNI, Pramod Eyyunni
Mathematics · #Advanced Mathematical Identities #Advanced Combinatorial Mathematics #Mathematical functions and polynomials
paper · doi:10.1017/s0004972726101567
Abstract Ballantine et al. [‘Partitions and elementary symmetric polynomials: an experimental approach’, Ramanujan J. 66 (2) (2025), Article no. 34] proposed two conjectures on the injectivity of a class of maps <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:msub> <mml:mrow> <mml:mi>p</mml:mi> <mml:mi>r</mml:mi> <mml:mi>e</mml:mi> </mml:mrow> <mml:mi>k</mml:mi> </mml:msub> </mml:math> prek p r e Subscript k defined on integer partitions. These maps arise from applying the sequence of elementary symmetric polynomials to integer partitions. We provide an infinite family of examples to disprove the conjecture for <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>k</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>3</mml:mn> </mml:math> k≥ 3 k greater than or equals 3 and state a modified version of it. Throwing fresh light on this class of maps, we study the inter-relationships between them, deviating from the approaches so far, which study these maps one at a time. While the conjecture for <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>k</mml:mi> <mml:mo>=</mml:mo> <mml:mn>2</mml:mn> </mml:math> k=2 k equals 2 has now been settled, we provide alternate proofs of three subcases. We also discuss lower bounds for the number of partitions of n that are in the image of the map <mml:math xmlns:xlink="http://www.w3.org/1999/xlink" xmlns:mnf="http://cambridge.org/core/manifest" xmlns:cup="http://contentservices.cambridge.org" xmlns:mml="http://www.w3.org/1998/Math/MathML" xmlns:m="http://cambridge.org/core/metadata" xmlns:core="http://cambridge.org/core" xmlns:c="http://cambridge.org/core/content" display="inline"> <mml:mi>p</mml:mi> <mml:mi>r</mml:mi> <mml:msub> <mml:mi>e</mml:mi> <mml:mn>2</mml:mn> </mml:msub> </mml:math> pre2 p r e 2 .