Biases in Non-Unitary Partitions

Recently, the concept of parity bias in integer partitions has been studied by several authors. We continue this study here, but for non-unitary partitions (namely, partitions with parts greater than $1$). We prove analogous results for these restricted partitions to those that have been obtained by Kim, Kim, and Lovejoy (2020) and Kim and Kim (2021). We also look at inequalities between two classes of partitions studied by Andrews (2019), where the parts are separated by parity (either all odd parts are smaller than all even parts or vice versa).

💡 Research Summary

The paper investigates parity bias in integer partitions when the part 1 is excluded, i.e., in non‑unitary partitions, and also studies size comparisons between families of partitions whose even and odd parts are separated in size, a topic introduced by Andrews. The authors aim to re‑prove several known inequalities using analytic q‑series techniques rather than the combinatorial arguments that appeared in earlier works.

Section 1 reviews the background: parity bias was first observed by Kim, Kim and Lovejoy (2020) who showed that for most n the number of partitions with more odd parts exceeds those with more even parts. Subsequent work by Kim and Kim (2021) and by Banerjee et al. (2022) extended the phenomenon to distinct parts and to partitions with various restrictions. The present work focuses on the case where the part 1 never appears, calling such objects “non‑unitary partitions”.

Section 2 collects the q‑series machinery needed for the proofs. Standard notation ((a;q)_n) is introduced, together with Heine’s transformation (4), its iterated form (5) – often called the q‑analogue of Euler’s transformation – and several Euler product identities (6)–(10). These identities allow the authors to translate combinatorial conditions on partitions into explicit generating functions.

In Section 3 the main analytic results for non‑unitary partitions are proved.

• Theorem 3.1 states that for every integer (n\ge8) the number (q_o(n)) of non‑unitary partitions of n having more odd parts than even parts is strictly smaller than the number (q_e(n)) of partitions with more even parts than odd parts. The proof constructs the generating functions \