Triangle of partition numbers

Summary

In the number theory of integer partitions, the numbers $p_k(n)$ denote both the number of partitions of $n$ into exactly $k$ parts (that is, sums of $k$ positive integers that add to $n$), and the number of partitions of $n$ into parts of maximum size exactly $k$. These two types of partition are in bijection with each other, by a diagonal reflection of their Young diagrams. Their numbers can be arranged into a triangle, the triangle of partition numbers, in which the $n$th row gives the partition numbers $p_1(n), p_2(n), \dots, p_n(n)$: