Determinants of Partition Matrices

✍ Scribed by Georg Martin Reinhart

• Publisher: Elsevier Science
• Year: 1996
• Tongue: English
• Weight: 532 KB
• Volume: 56
• Category: Article
• ISSN: 0022-314X
• DOI: 10.1006/jnth.1996.0018

✦ Synopsis

Let : 1 , ..., : k be partitions of 2n with at least n 1's and ; 1 , ..., ; k be partitions of 2n with exactly n parts. By M n we denote the matrix whose entries m ij are the number of ways to refine ; j into : i . It is shown that det M n =1 for all n.

1996 Academic Press, Inc.

1. Introduction

Let p(n) denote the partition function, i.e., the number of distinct ways to write n as a sum of positive integers. Partitions of n will be denoted by tuples of positive integers arranged in non-increasing order. For reasons that will become apparent later, we do not allow 1's in the tuples. The 1's of a partition can be determined by n which is considered as known, e.g. the partition of n=10: 10=3+3+2+1+1 will be written as (3, 3, 2).