Joint distributions from MacMahon partition analysis

Prove joint distribution laws for pairs of interesting partition statistics by using multivariate generating functions obtained through MacMahon’s partition analysis.

Background

MacMahon’s partition analysis is presented as a systematic method for constructing multivariate generating functions that track several statistics or impose linear inequalities among parts. The paper develops this framework for the alternating sum and discusses extensions involving the largest part and the length of a partition.

The unresolved question is whether such multivariate generating functions can be combined with asymptotic methods to establish limiting joint laws for pairs of partition statistics, rather than only marginal distributions.

References

Can one prove, from multi-variate generating functions coming from MacMahon's partition analysis, joint distributions laws for pairs of interesting partition statistics?

— Distribution of Alternating Sums of Parts in Partitions  (2501.17065 - Craig et al., 28 Jan 2025) in Question in Section 5.2, “Implications for distributions”