Lagmental Vicfred

A Durfee Square Decomposes a Partition into Three Pieces by Vicfred

Last updated: Sun 13 August 2023

tags: mathematics, integer partitions, Young tableaux, q-analogues

Conditioning on the largest square in a Ferrers diagram gives a sum of squared reciprocal products. A small computation will anchor the general statement before the abstraction takes over.

Statement

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.

$$ d(\lambda)=\max\{d:\lambda_d\ge d\} $$

I read the first line as input and the second as output. The symbols \(\forall\) and \(\exists\) are not interchangeable, and neither may be upgraded silently to \(\Longleftrightarrow\).

$$ \frac1{(q;q)_\infty}=\sum_{d\ge0}\frac{q^{d^2}}{(q;q)_d^2} $$

Worked algebra

Here is a concrete symbolic test. Reading it from left to right reveals which transformation is reversible and which is only an implication.

$$ \lambda=\underbrace{d\times d}_{q^{d^2}}+\underbrace{\text{partition right of square}}_{1/(q;q)_d}+\underbrace{\text{partition below square}}_{1/(q;q)_d} $$

Conceptual compression

The aligned summary deliberately puts the datum and conclusion on different rows. Mathematically, this is the distinction between specifying an object and proving a property of it.

$$ \begin{aligned} \mathsf{D}\;&:\quad d(\lambda)=\max\{d:\lambda_d\ge d\},\\[5pt] \mathsf{C}\;&:\quad \frac1{(q;q)_\infty}=\sum_{d\ge0}\frac{q^{d^2}}{(q;q)_d^2}. \end{aligned} $$

Caveat

Partitions forget order, compositions retain it, and tableaux add labels subject to row and column rules. Interchanging these objects changes the count.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \frac1{(q;q)_\infty}=\sum_{d\ge0}\frac{q^{d^2}}{(q;q)_d^2} \end{gathered}} $$

The final box is a summary, not a new assumption; the proof still lives in the definitions and the intervening calculation. The source keeps each scope delimiter visible for later inspection.

This article was posted on Mon 31 August 2015. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.