An Investigation on Partitions with Equal Products

2018/11/19 by Cha, Byungchul, Claman, Adam, Harrington, Joshua +6
#05A17 #11B75 #11B83 #Combinatorics (math.CO) #FOS: Mathematics #Number Theory (math.NT)

paper · doi:10.48550/arxiv.1811.07451

Abstract

An ordered triple (s,p,n) is called admissible if there exist two different multisets X=\x₁,x₂,\dotsc,x_n\ and Y=\y₁,y₂,\dotsc,y_n\ such that X and Y share the same sum s, the same product p, and the same size n. We first count the number of n such that (s,p,n) are admissible for a fixed s. We also fully characterize the values p such that (s,p,n) is admissible. Finally, we consider the situation where r different multisets are needed, instead of just two. This project is also related to John Conway's wizard puzzle from the 1960s.

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