### Abstract

In the open problem session of the FPSAC'03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285-305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist's Pfaffian identities (J. Algebr. Comb. 5:135-148, 1996). Then we give a proof of Stanley's open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley's open problem.

Original language | English |
---|---|

Pages (from-to) | 211-234 |

Number of pages | 24 |

Journal | Ramanujan Journal |

Volume | 16 |

Issue number | 2 |

DOIs | |

Publication status | Published - Jul 2008 |

Externally published | Yes |

### Fingerprint

### Keywords

- Determinants
- Minor summation formula of Pfaffians
- Pfaffians
- Schur functions

### ASJC Scopus subject areas

- Algebra and Number Theory

### Cite this

**Minor summation formula and a proof of Stanley's open problem.** / Ishikawa, Masao.

Research output: Contribution to journal › Article

*Ramanujan Journal*, vol. 16, no. 2, pp. 211-234. https://doi.org/10.1007/s11139-007-9106-9

}

TY - JOUR

T1 - Minor summation formula and a proof of Stanley's open problem

AU - Ishikawa, Masao

PY - 2008/7

Y1 - 2008/7

N2 - In the open problem session of the FPSAC'03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285-305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist's Pfaffian identities (J. Algebr. Comb. 5:135-148, 1996). Then we give a proof of Stanley's open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley's open problem.

AB - In the open problem session of the FPSAC'03, R.P. Stanley gave an open problem about a certain sum of the Schur functions. The purpose of this paper is to give a proof of this open problem. The proof consists of three steps. At the first step we express the sum by a Pfaffian as an application of our minor summation formula (Ishikawa and Wakayama in Linear Multilinear Algebra 39:285-305, 1995). In the second step we prove a Pfaffian analogue of a Cauchy type identity which generalizes Sundquist's Pfaffian identities (J. Algebr. Comb. 5:135-148, 1996). Then we give a proof of Stanley's open problem in Sect. 4. At the end of this paper we present certain corollaries obtained from this identity involving the Big Schur functions and some polynomials arising from the Macdonald polynomials, which generalize Stanley's open problem.

KW - Determinants

KW - Minor summation formula of Pfaffians

KW - Pfaffians

KW - Schur functions

UR - http://www.scopus.com/inward/record.url?scp=48349102352&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=48349102352&partnerID=8YFLogxK

U2 - 10.1007/s11139-007-9106-9

DO - 10.1007/s11139-007-9106-9

M3 - Article

VL - 16

SP - 211

EP - 234

JO - Ramanujan Journal

JF - Ramanujan Journal

SN - 1382-4090

IS - 2

ER -