A sign-reversing involution for an extension of Torelli's Pfaffian identity

Richard Ehrenborg; N. Bradley Fox

doi:10.1016/j.disc.2014.06.002

A sign-reversing involution for an extension of Torelli's Pfaffian identity

Richard Ehrenborg
, N. Bradley Fox

Mathematics

Research output: Contribution to journal › Article › peer-review

Abstract

We evaluate the hyperpfaffian of a skew-symmetric k-ary polynomial f of degree k/2·(n-1). The result is a product of the Vandermonde product and a certain expression involving the coefficients of the polynomial f. The proof utilizes a sign reversing involution on a set of weighted, oriented partitions. When restricting to the classical case when k=2 and the polynomial is (xj ^-xi)n-1, we obtain an identity due to Torelli.

Original language	English
Pages (from-to)	69-74
Number of pages	6
Journal	Discrete Mathematics
Volume	332
DOIs	https://doi.org/10.1016/j.disc.2014.06.002
State	Published - Oct 6 2014

Bibliographical note

Funding Information:
The authors are grateful to the referees for their comments on an earlier version of this note. The first author was partially supported by National Science Foundation grant DMS 0902063 .

Funding

The authors are grateful to the referees for their comments on an earlier version of this note. The first author was partially supported by National Science Foundation grant DMS 0902063 .

Funders	Funder number
National Science Foundation Arctic Social Science Program	DMS 0902063
Directorate for Mathematical and Physical Sciences	0902063

Keywords

Exterior algebra
Hyperpfaffian
Pfaffian
Sign-reversing involution
The Vandermonde identity
Torelli's identity

ASJC Scopus subject areas

Theoretical Computer Science
Discrete Mathematics and Combinatorics

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10.1016/j.disc.2014.06.002

Cite this

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title = "A sign-reversing involution for an extension of Torelli's Pfaffian identity",

abstract = "We evaluate the hyperpfaffian of a skew-symmetric k-ary polynomial f of degree k/2·(n-1). The result is a product of the Vandermonde product and a certain expression involving the coefficients of the polynomial f. The proof utilizes a sign reversing involution on a set of weighted, oriented partitions. When restricting to the classical case when k=2 and the polynomial is (xj -xi)n-1, we obtain an identity due to Torelli.",

keywords = "Exterior algebra, Hyperpfaffian, Pfaffian, Sign-reversing involution, The Vandermonde identity, Torelli's identity",

author = "Richard Ehrenborg and Fox, \{N. Bradley\}",

note = "Funding Information: The authors are grateful to the referees for their comments on an earlier version of this note. The first author was partially supported by National Science Foundation grant DMS 0902063 . ",

year = "2014",

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doi = "10.1016/j.disc.2014.06.002",

language = "English",

volume = "332",

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T1 - A sign-reversing involution for an extension of Torelli's Pfaffian identity

AU - Ehrenborg, Richard

AU - Fox, N. Bradley

N1 - Funding Information: The authors are grateful to the referees for their comments on an earlier version of this note. The first author was partially supported by National Science Foundation grant DMS 0902063 .

PY - 2014/10/6

Y1 - 2014/10/6

N2 - We evaluate the hyperpfaffian of a skew-symmetric k-ary polynomial f of degree k/2·(n-1). The result is a product of the Vandermonde product and a certain expression involving the coefficients of the polynomial f. The proof utilizes a sign reversing involution on a set of weighted, oriented partitions. When restricting to the classical case when k=2 and the polynomial is (xj -xi)n-1, we obtain an identity due to Torelli.

AB - We evaluate the hyperpfaffian of a skew-symmetric k-ary polynomial f of degree k/2·(n-1). The result is a product of the Vandermonde product and a certain expression involving the coefficients of the polynomial f. The proof utilizes a sign reversing involution on a set of weighted, oriented partitions. When restricting to the classical case when k=2 and the polynomial is (xj -xi)n-1, we obtain an identity due to Torelli.

KW - Exterior algebra

KW - Hyperpfaffian

KW - Pfaffian

KW - Sign-reversing involution

KW - The Vandermonde identity

KW - Torelli's identity

UR - https://www.scopus.com/pages/publications/84903178091

UR - https://www.scopus.com/pages/publications/84903178091#tab=citedBy

U2 - 10.1016/j.disc.2014.06.002

DO - 10.1016/j.disc.2014.06.002

M3 - Article

AN - SCOPUS:84903178091

SN - 0012-365X

VL - 332

SP - 69

EP - 74

JO - Discrete Mathematics

JF - Discrete Mathematics

ER -

A sign-reversing involution for an extension of Torelli's Pfaffian identity

Abstract

Bibliographical note

Funding

Keywords

ASJC Scopus subject areas

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