### Abstract

The initial purpose of the present paper is to provide a combinatorial proof of the minor summation formula of Pfaffians in [Ishikawa, Wakayama, Minor summation formula of Pfaffians, Linear and Multilinear Algebra 39 (1995) 285-305] based on the lattice path method. The second aim is to study applications of the minor summation formula for obtaining several identities. Especially, a simple proof of Kawanaka's formula concerning a q-series identity involving the Schur functions [Kawanaka, A q-series identity involving Schur functions and related topics, Osaka J. Math. 36 (1999) 157-176] and of the identity in [Kawanaka, A q-Cauchy identity involving Schur functions and imprimitive complex reflection groups, Osaka J. Math. 38 (2001) 775-810] which is regarded as a determinant version of the previous one are given.

Original language | English |
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Pages (from-to) | 113-155 |

Number of pages | 43 |

Journal | Journal of Combinatorial Theory - Series A |

Volume | 113 |

Issue number | 1 |

DOIs | |

Publication status | Published - Jan 2006 |

Externally published | Yes |

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### Keywords

- Cauchy identity
- Lattice paths method
- Lewis Carroll formula
- Minor summation formula
- Partition
- Pfaffian
- Plücker relations
- q-series
- Schur function

### ASJC Scopus subject areas

- Discrete Mathematics and Combinatorics
- Theoretical Computer Science