TAILIEUCHUNG - Báo cáo toán học: " Reconstruction of Partitions Oliver Pretzel"

Tuyển tập các báo cáo nghiên cứu khoa học trên tạp chí toán học quốc tế đề tài: Reconstruction of Partitions Oliver Pretzel. | Reconstruction of Partitions Oliver Pretzel Department of Mathematics Imperial College London SW7 2AZ United Kingdom and Johannes Siemons School of Mathematics University of East Anglia Norwich NR4 7TJ United Kingdom Submitted Mar 22 2005 Accepted Sep 19 2005 Published Sep 28 2005 Mathematics Subject Classification 05A17 05C60 Abstract For the partition x x1 x2 xk of the integer n Pi xi a t-deletion is a partition y yi y yk with xi yi 0 and pi xi - yi t. We prove that all partitions of n are reconstructible from their t-deletions if n is sufficiently large in relation to t. KEywORDS Partitions reconstruction. Introduction and Notation The question that we address in this note is when a partition of an integer is reconstructible from certain of its subpartitions. Such reconstruction problems arise naturally for instance in the representation theory of symmetric and Lie groups where partitions index the irreducible representations of such groups. This will be explained in the next section. The theorem we intend to prove was announced in 3 to which we refer the reader for a detailed discussion of partition reconstruction and further results. It states that for all t and all n sufficiently large in relation to t any partition of n is fully determined by those of its subpartitions whose sum is n t. In contrast to some other reconstruction problems we do not consider the multiplicity with which subpartitions occur. In order to formulate the result precisely we make a few definitions. Research carried out as part of the project Reconstruction Indices of Permutation Groups funded by the Leverhulme foundation. THE ELECTRONIC JOURNAL OF COMBINATORICS 11 2 2005 N5 1 Definitions. a partition of the integer n is a sequence x x1 x2 xk of integers xi 0 with x Pk I xi n. To avoid cumbersome distinctions we identify two partitions if they differ only in the number of zero terms. If y y1 y2 . ye is a partition then y is a subpartition of x denoted y x if yi xi for all i 1 . max k .

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