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Exact term count of ψn equals cumulative partition count P(n)

Establish that for every integer n ≥ 1, the number of monomials in the polynomial ψn(x1, …, xn) equals P(n), where P(n) denotes the cumulative partition count defined by the paper as the sum of partition numbers up to n.

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Background

Let P(n) be the partition function and define the cumulative count P(n) = \sum_{ν=1}{n} P(ν). The paper proves Theorem 1.2: #ψn ≤ P(n). Computations up to n = 30 suggest the inequality is tight, motivating a conjecture.

This conjecture connects the combinatorial structure of ψn to partition counts, implying that no cancellation of monomials beyond those predicted by partition structures occurs.

References

However, computing the first 30 polynomials $\psi_\nu$ (note that $# \psi_{30} = P(30) = 28\,628$) and verifying the equality in this range, we may state the following conjecture. For $n \geq 1$, we have that $# \psi_n = P(n)$.

— Wilson's theorem modulo higher prime powers I: Fermat and Wilson quotients (2509.05235 - Kellner, 5 Sep 2025) in Introduction (end), following Theorem 1.2 (Theorem \ref{thm:num})