The infinite product (q;q)_infinity has sparse coefficients at generalized pentagonal numbers. A small computation will anchor the general statement before the abstraction takes over.
Set-up
A partition \(\lambda\vdash n\) is both a decreasing sequence and a Ferrers diagram. Statistics such as hook lengths \(h_{ij}\) turn the diagram into exact product formulas.
There are two layers here: the object \(\mathsf D\) and the law \(\mathsf C\). Writing them separately makes the direction of \(\Longrightarrow\) visible and keeps an accidental converse from slipping in.
The calculation
The computation below is not a second theorem. It is a checksum for the definitions and a place to inspect the difficult LaTeX at full size.
What survives abstraction
The abstraction earns its keep by explaining why the same computation reappears. The notation compresses repeated reasoning without erasing the hypothesis that licenses it.
The boundary
Partitions forget order, compositions retain it, and tableaux add labels subject to row and column rules. Interchanging these objects changes the count.
A symbolic summary is trustworthy only because the example and limitation remain visible beside it. The box compresses the conclusion without hiding its origin.