A sharp bound for the reconstruction of partitions

TL;DR

This paper improves the bounds for reconstructing integer partitions from their k-deletions, providing a more efficient criterion and demonstrating its optimality with examples.

Contribution

The authors introduce a new reconstruction algorithm that reduces the minimum size requirement for partitions, improving upon previous bounds and establishing their optimality.

Findings

New reconstruction bound: n ≥ k² + 2k

Algorithm reduces the size threshold for reconstructibility

Examples confirm the bound's optimality

Abstract

Answering a question of Cameron, Pretzel and Siemons proved that every integer partition of $n\ge 2(k+3)(k+1)$ can be reconstructed from its set of $k$-deletions. We describe a new reconstruction algorithm that lowers this bound to $n\ge k^2+2k$ and present examples showing that this bound is best possible.