Gaussian Enumeration of Rectangles

Abstract

Gaussian polynomials enumerate weakly decreasing tuples by their total size.

Theorem 1.1 (The rectangle enumerator).

Lean statement: D5/S3/Combinatorics/CylindricPartition/LiUncuRectangle.gauss_rectangle

Proof. Machine-checked in Lean as D5/S3/Combinatorics/CylindricPartition/LiUncuRectangle.gauss_rectangle (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. Runqiao Li, Ali K. Uncu (2025). A MacMahon Analysis View of Cylindric Partitions. DOI: 10.48550/arXiv.2501.19272. URL: https://arxiv.org/abs/2501.19272v1.

Commentary.

For all nonnegative integers M and m, G(M+m,m) is the sum of q raised to the sum of the entries over all weakly decreasing tuples of m entries between zero and M. Thus it enumerates partitions contained in a rectangle of width M and height m.

References

• Truth anchor: D5/S3/Combinatorics/CylindricPartition/LiUncuRectangle.gauss_rectangle
• Dependency: D5/S3/Combinatorics/CylindricPartition/LiUncuGaussian