Tuyển tập các báo cáo nghiên cứu khoa học về toán học trên tạp chí toán học quốc tế đề tài: A sharp bound for the reconstruction of partitions. | A sharp bound for the reconstruction of partitions Vincent Vatter Department of Mathematics Dartmouth College Hanover NH 03755 Submitted May 16 2008 Accepted Jun 22 2008 Published Jun 30 2008 Mathematics Subject Classification 05A17 06A07 Abstract Answering a question of Cameron Pretzel and Siemons proved that every integer partition of n 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 k2 2k and present examples showing that this bound is best possible. Analogues and variations of Ulam s notorious graph reconstruction conjecture have been studied for a variety of combinatorial objects for instance words see Schutzenberger and Simon 2 Theorem 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 2 k 3 k 1 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. Theorem 1. Every partition of n k2 2k can be reconstructed from its set of k--deletions. We begin with notation. Recall that a partition of n A Al . A is a finite sequence of nonincreasing integers whose sum which we denote A is n. The Ferrers diagram of A which we often identify with A consists of left-justified rows where row i contains Ai cells. An inner corner in this diagram is a cell whose removal leaves the diagram of a partition and we refer to all other cells as interior cells. We write p A if pi Ai for all i another way of stating this is that p A if and only if p is contained in A here identifying partitions with their diagrams . If p A we write A p to denote the set of cells which lie in A but not in p. We say that the partition p is a k--deletion of the partition of A if p A and A p k. Recall that this order defines a .
