Hall polynomial

Definition
Suppose $$\mu, \nu, \lambda$$ are unordered integer partitions for integers $$m,n,l$$ respectively such that $$m + n = l$$. The Hall polynomial $$g_{\mu,\nu}^\lambda$$ is a polynomial such that, for any prime number $$p$$, $$g_{\mu,\nu}^\lambda(p)$$ is the number of subgroups $$H$$ of an abelian $$p$$-group $$G$$ of type $$\lambda$$ such that $$H$$ has type $$\nu$$ and $$G/H$$ has type $$\mu$$.

Here, an $$p$$-group $$P$$ of type $$\lambda$$, where $$\lambda$$ is a partition into parts $$a_1, a_2, \dots, a_r$$, is the group $$C_{p^{a_1}} \times C_{p^{a_2}} \times \dots C_{p^{a_r}}$$ where $$C_k$$ denotes the cyclic group of order $$k$$.

The Hall polynomials for various triples of partitions $$(\mu,\nu,\lambda)$$ are related to each other. Specifically, there is an algebra, called the Hall algebra, which is a free module over the integers with generating set indexed by unordered integer partitions, and where the structure constants are the Hall polynomials.

Note that since subgroup lattice and quotient lattice of finite abelian group are isomorphic, the Hall polynomials $$g_{\mu,\nu}^\lambda$$ and $$g_{\nu,\mu}^\lambda$$ are equal.

Particular cases
Note that since $$g_{\mu,\nu}^\lambda = g_{\nu,\mu}^\lambda$$ because subgroup lattice and quotient lattice of finite abelian group are isomorphic, each of our polynomials below also gives another corresponding polynomial obtained by interchanging the roles of $$\mu$$ and $$\nu$$.