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Durfee-Square Decomposition

Abstract

Partition diagrams split at their maximal square and produce the normalized pentagonal inverse series.

Theorem 1.1 (Durfee-square normalization).

Lean statement: D5/S3/Combinatorics/InversionSeq/InversionSeq207Durfee.durfee_square_normalization

Proof. Machine-checked in Lean as D5/S3/Combinatorics/InversionSeq/InversionSeq207Durfee.durfee_square_normalization (✓ std3). ∎

Source. Repository-derived.

Acknowledgement. George E. Andrews, Mohamed El Bachraoui (2025). Certain positive q-series and inequalities for two-color partitions. DOI: 10.48550/arXiv.2507.09276. URL: https://arxiv.org/abs/2507.09276v1.

Commentary.

With the discrete topology on the rationals, the sum over natural sizes of q raised to the square of the size times the square of the inverse Euler denominator converges to the inverse of the pentagonal power series. The decomposition is obtained by removing the maximal square from each partition diagram and separating the right and bottom pieces.

References