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.

Dyson famously provided combinatorial explanations for Ramanujan's partition congruences modulo 5 and 7 via his rank function, and postulated that an invariant explaining all of Ramanujan's congruences modulo 5, 7, and 11 should exist. Garvan and Andrews-Garvan later discovered such an invariant called the crank, fulfilling Dyson's goal. Many further examples of congruences of partition functions are known in the literature. In this paper, we provide a framework for discovering and proving the existence of such invariants for families of congruences and partition functions. As a first example, we find a family of crank functions that simultaneously explains most known congruences for colored partition functions. The key insight is to utilize a powerful recent theory of theta blocks due to Gritsenko, Skoruppa, and Zagier. The method used here should be useful in the study of other combinatorial functions.

In this paper, we find the ω-value of the generators of any numerical semigroup with embedding dimension three. This allows us to determine all possible orderings of the ω-values of the generators. In addition, we relate the ω-value of the numerical semigroup to its catenary degree.

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