Graphical condensation, overlapping Pfaffians and superpositions of matchings

TL;DR

This paper clarifies how graphical condensation identities relate to classical Pfaffian identities using the Kasteleyn-Percus method, and introduces a general overlapping Pfaffian identity, correcting previous errors.

Contribution

It provides a unified presentation linking graphical condensation to Pfaffian identities and introduces a new general overlapping Pfaffian identity.

Findings

Derivation of graphical condensation identities from Pfaffian identities

Introduction of a general overlapping Pfaffian identity

Correction of previous methodological errors

Abstract

The purpose of this note is to exhibit clearly how the "graphical condensation" identities of Kuo, Yan, Yeh and Zhang follow from classical Pfaffian identities by the Kasteleyn-Percus method for the enumeration of matchings. Knuth termed the relevant identities "overlapping Pfaffian" identities and the key concept of proof "superpositions of matchings". In our uniform presentation of the material, we also give an apparently unpublished general "overlapping Pfaffian" identity of Krattenthaler. A previous version of this paper contained an erroneous application of the Kasteleyn-Percus method, which is now corrected.