Polynomization of the Chern–Fu–Tang conjecture

Abstract

Bessenrodt and Ono’s work on additive and multiplicative properties of the partition function and DeSalvo and Pak’s paper on the log-concavity of the partition function have generated many beautiful theorems and conjectures. In January 2020, the first author gave a lecture at the MPIM in Bonn on a conjecture of Chern–Fu–Tang, and presented an extension (joint work with Neuhauser) involving polynomials. Partial results have been announced. Bringmann, Kane, Rolen, and Tripp provided complete proof of the Chern–Fu–Tang conjecture, following advice from Ono to utilize a recently provided exact formula for the fractional partition functions. They also proved a large proportion of Heim–Neuhauser’s conjecture, which is the polynomization of Chern–Fu–Tang’s conjecture. We prove several cases, not covered by Bringmann et. al. Finally, we lay out a general approach for proving the conjecture.

Inequalities for Plane Partitions

“…Actually, in this case all coefficients of ∆ a+1,a−1 (x) are non-negative. The generalization of the former Chern-Fu-Tang conjecture of k-colored partitions and its polynomization [HN21A] We refer to Table 8, where we recorded the largest real zero for ∆ a,b for all pairs (a, b) with a − 1 > b ≥ 0 and 2 ≤ a ≤ 19, which shows that the Conjecture 3 is valid for all admissible pairs (a, b) in this range.…”

“…The conjecture was extended to k ∈ R ≥2 [HN21A], which involves the so-called D'Arcais polynomials or Nekrasov-Okounkov polynomials [NO06,Ha10]. The inequality is symmetric and always fails for…”

“…Log-Concavity. In the spirit of the Chern-Fu-Tang Conjecture [CFT18] on k-colored partitions and their polynomization [HN21A] (see also [BKRT21]), we consider the polynomials ∆ a,b (x) := P a−1 (x)P b+1 (x) − P a (x) P b (x).…”

“…a) The real part of the zeros of ∆ a,1 (x)/ x is negative for 3 ≤ a ≤ 100. b) The method of the proof is similar to the one outlined in [HN21A].…”

Turán Inequalities for Infinite Product Generating Functions

“…In [HN21] a conjecture for ∆ σ 1 n (x) was stated, which generalized a conjecture of Chern-Fu-Tang [CFT18] related to integers x ≥ 2. The Chern-Fu-Tang conjecture was proven by Bringmann, Kane, Rolen, and Tripp [BKRT21].…”