The number of maximal unrefinable partitions

This paper completes the classification of maximal unrefinable partitions, extending a previous work of Aragona et al. devoted only to the case of triangular numbers. We show that the number of maximal unrefinable partitions of an integer coincides with the number of suitable partitions into distinct parts, depending on the distance from the successive triangular number.

💡 Research Summary

The paper completes the classification of maximal unrefinable partitions, extending the earlier work that dealt only with triangular numbers. An unrefinable partition of a positive integer N is a partition into distinct parts such that no missing part pair sums to an existing part; a maximal unrefinable partition is one whose largest part λ_t is as large as possible among all unrefinable partitions of N.

The authors first recall the known result for triangular numbers T_n = n(n+1)/2: for any unrefinable partition λ of T_n the largest part satisfies λ_t ≤ 2n‑4, and this bound is sharp. Moreover, the number of maximal unrefinable partitions of T_n is 1 when n is even, while for odd n it equals the number of partitions of (n+1)/2 into distinct parts.

The main contribution is to treat non‑triangular integers, which can be uniquely written as T_{n,d}=T_n‑d with 1 ≤ d ≤ n‑1. The paper establishes a sharp upper bound for the largest part of any unrefinable partition of T_{n,d}. The bound depends on the parity of the distance n‑d:

• If 3 < d ≤ n‑1 and n‑d is even, then λ_t ≤ 2n‑5.
• If 3 < d ≤ n‑1 and n‑d is odd, then λ_t ≤ 2n‑4.
• For the small distances d = 1, 2, 3 the bounds improve to λ_t ≤ 2n‑2, 2n‑3, and 2n‑4 respectively.

These bounds are proved by a combinatorial argument based on Lemma 2.2, which shows that the number m of missing parts in an unrefinable partition satisfies m ≤ ⌊λ_t/2⌋. The authors start from a canonical unrefinable partition \