Combinatorics / CONCEPT

Partitions with the fewest extensions are boxes

In every dimension d, the d-dimensional partitions of n that extend in exactly d ways to a partition of n + 1, and the d-dimensional partitions of n + 1 that contain exactly one partition of n, are the boxes, so both are counted by the number of ordered factorizations of n, respectively n + 1, into d factors; for d = 4 these are the first columns of A098052 and A098530 on solid partitions.