A Sharp Bound for the Reconstruction of Partitions

Analogues and variations of Ulam’s notorious graph reconstruction conjecture have been studied for a variety of combinatorial objects, for instance words (see Schützenberger and Simon [2, Theorem 6.2.16]), permutations (see Raykova [4] and Smith [5]), and compositions (see Vatter [6]), to name a few. In answer to Cameron’s query [1] about the partition context, Pretzel and Siemons [3] proved that every partition of n ¥ 2pk 3qpk 1q can be reconstructed from its set of k-deletions. Herein we describe a new reconstruction algorithm that lowers this bound, establishing the following result, which Negative Example 2 shows is best possible.