Binomial coefficients whose products are perfect kth powers (Q1062058)

A \(P_ k\)-set can be defined to be a finite set P of ordered pairs of integers such that for all integers n,r the product of all binomial coefficients \(\binom{n+i}{r+j}\) taken over all elements (i,j) of Pis a perfect kth power. For example, since \(\binom{n}{r} \binom{n+1}{r+2} \binom{n+2}{r+1}= \binom{n}{r+1} \binom{n+1}{r} \binom{n+2}{r+2}\) for all integers n,r it follows that \(\{(0,0),(0,1),(1,0),(1,2),(2,1),(2,2)\}\) is a \(P_ 2\)-set. This example was found in 1971 by \textit{V. E. Hoggatt} and \textit{W. Hansell} [Fibonacci Q. 9, 120-133 (1971; Zbl 0246.05004)], and it served as the inspiration for the concept of a \(P_ k\)-set. (Another way to visualize a \(P_ k\)-set is to think of a template of windows which can be translated to any position over Pascal's triangular array of the binomial coefficients so that the product of the entries appearing in the windows is always a perfect kth power.) The authors prove that a finite set S of ordered pairs of integers is a \(P_ k\)-set if and only if the number of elements (i,j) in S on each line \(i=c\), \(j=c\), \(i-j=c\) is a multiple of k for each integer c. An easy corollary is that if S is a \(P_ k\)-set, then \(| S|\) is a multiple of k. Furthermore, they show that the smallest size a \(P_ k\)-set can have is k(2k-1) elements.