2023/05/31 by Jia Xu, Yong Yao, Xu, Jia +1
Mathematics · #05E05 14P99 90C22 #Combinatorics (math.CO) #FOS: Computer and information sciences #FOS: Mathematics #Mathematical Inequalities and Applications #Symbolic Computation (cs.SC)
paper · pdf · doi:10.48550/arxiv.2305.19830
openalex publication_date 2023/05/31 · openalex created_date 2023/06/02 · openalex updated_date 2026/07/28
Inequalities among symmetric polynomial functions are fundamental questions in mathematics and have various applications in science and engineering. This paper investigates a beautiful and inspiring conjecture, proposed by Cuttler, Greene and Skandera in 2011, on inequalities among the complete homogeneous symmetric polynomial function Hn,λ: It states that the inequality Hn,λ≤ Hn,μ implies majorization order λ\preceqμ. The conjecture is a close analogy with other known results on Muirhead-type inequalities. In 2021, Heaton and Shankar disproved the conjecture by showing a counterexample for number of variables n=3 and degree d=8. They then asked whether the conjecture is true when n is sufficiently large. In this paper, we show, by a family of counter-examples, that the conjecture does not hold for any n and any d as long as n≥2 and d≥8. Based on the insights gained from the counter-examples, we propose a new conjecture for the inequality Hn,λ≤ Hn,μ.