Closed-form expression for restricted partition counts q(m,n)

Determine whether an exact closed-form expression exists for q(m,n), the number of restricted partitions of the integer m into at most n non-zero parts used in the prior P(η|D) for degree sequences, and, if it exists, derive such an expression to enable exact evaluation of the prior without approximation.

Background

In constructing nonparametric priors for atomic degree sequences, the authors use a hyper-prior that conditions on the degree distribution η, where one factor requires computing q(m,n), the number of restricted partitions of an integer m into at most n non-zero parts.

The paper notes that no exact closed-form expression is currently known for q(m,n) and relies on approximations to evaluate the prior efficiently. Establishing a closed-form would improve analytic tractability and potentially enhance inference accuracy and computational efficiency.

References

Although no exact closed form expression is known for q(m,n) in practice it can be efficiently approximated to a high degree of accuracy, as described in .

— Nonparametric inference of higher order interaction patterns in networks (2403.15635 - Wegner et al., 2024) in Methods, Priors for atomic degree sequences (one-dimensional prior P2)