### 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 |
---|---|

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 |

### Fingerprint

### 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

### Cite this

**Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities.** / Ishikawa, Masao; Wakayama, Masato.

Research output: Contribution to journal › Article

*Journal of Combinatorial Theory - Series A*, vol. 113, no. 1, pp. 113-155. https://doi.org/10.1016/j.jcta.2005.05.008

}

TY - JOUR

T1 - Applications of minor summation formula III, Plücker relations, lattice paths and Pfaffian identities

AU - Ishikawa, Masao

AU - Wakayama, Masato

PY - 2006/1

Y1 - 2006/1

N2 - 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.

AB - 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.

KW - Cauchy identity

KW - Lattice paths method

KW - Lewis Carroll formula

KW - Minor summation formula

KW - Partition

KW - Pfaffian

KW - Plücker relations

KW - q-series

KW - Schur function

UR - http://www.scopus.com/inward/record.url?scp=30444443612&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=30444443612&partnerID=8YFLogxK

U2 - 10.1016/j.jcta.2005.05.008

DO - 10.1016/j.jcta.2005.05.008

M3 - Article

VL - 113

SP - 113

EP - 155

JO - Journal of Combinatorial Theory - Series A

JF - Journal of Combinatorial Theory - Series A

SN - 0097-3165

IS - 1

ER -