Landau's $g(n)$ from a Complete Partition Search

The order of a permutation is the least common multiple of its cycle lengths, so Landau's function $g(n)$ - the largest order of an element of the symmetric group $S_n$ - is the largest lcm taken over the partitions of $n$. Computing it is therefore a search, and a search returns a theorem about all cycle types only if the enumeration it walks is known to be complete. We give a fuelled generator for the non-increasing partitions, prove both that everything it produces is a partition of $n$ and that every partition of $n$ is produced, and read $g(n)$ off it for $n \\le 10$. The values are $1, 2, 3, 4, 6, 6, 12, 15, 20, 30$. The first departure from intuition is at $n = 7$, where the maximum is carried by the cycle type $4 + 3$ rather than by the $7$-cycle: coprimality buys more than length.