Fractional partitions and conjectures of Chern–Fu–Tang and Heim–Neuhauser

Abstract

Many papers have studied inequalities for partition functions. Recently, a number of papers have considered mixtures between additive and multiplicative behavior in such inequalities. In particular, Chern–Fu–Tang and Heim–Neuhauser gave conjectures on inequalities for coefficients of powers of the generating partition function. These conjectures were posed in the context of colored partitions and the Nekrasov–Okounkov formula. Here, we study the precise size of differences of products of two such coefficients. This allows us to prove the Chern–Fu–Tang conjecture and to show the Heim–Neuhauser conjecture in a certain range. The explicit error terms provided will also be useful in the future study of partition inequalities. These are laid out in a user-friendly way for the researcher in combinatorics interested in such analytic questions.

Inequalities for Plane Partitions

“…Let k be a positive integer. We would like to call P n (k) the k-colored plane partitions (compare [4]), but at the moment there is no combinatorial interpretation available, as in the case of partitions [7,19]. The topic is quite complicated, since MacMahon's result is already non-trivial and pp(n) can also be identified with the number of all partitions of n = n 1 + n 2 + • • • , where each part n j is allowed to have n j colors.…”

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]. Recently, a second conjecture [HNT21] was proposed for ∆ σ 2 n (x).…”

The second shifted difference of partitions and its applications

“…Next we want to estimate the functions f (j, n). The first step in the proof is to obtain estimates of p(n−j) p(n) analogous to those of [2], proving Theorem 1.1 en-route.…”