Graphical condensation, overlapping Pfaffians and superpositions of matchings

Details Graphical condensation, overlapping Pfaffians and superpositions of matchings Graphical condensation, overlapping Pfaffians and superpositions of matchings

2009-06-23

arxiv.org/abs/0906.4236v3

Mathematics

Combinatorics

Th error in the application of the Kasteleyn-Percus method has been corrected

Scientific paper

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 In our uniform presentation of the material, we also give an apparently unpublished general "overlapping Pfaffian" identity of Krattenthaler.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.

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