Additive triples of bijections, or the toroidal semiqueens problem

Sean Eberhard; Freddie Manners; Rudi Mrazović

doi:10.4171/JEMS/841

Additive triples of bijections, or the toroidal semiqueens problem

Sean Eberhard, Freddie Manners, Rudi Mrazović

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

11 Citations (Scopus)

Abstract

We prove an asymptotic for the number of additive triples of bijections {1, . . ., n} → Z/nZ, that is, the number of pairs of bijections π₁, π₂ : {1, . . ., n} → Z/nZ such that the pointwise sum π₁ + π₂ is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of Z/nZ, to counting the number of arrangements of n mutually nonattacking semiqueens on an n × n toroidal chessboard, and to counting the number of transversals in a cyclic Latin square.

Original language	English
Pages (from-to)	441-463
Number of pages	23
Journal	Journal of the European Mathematical Society
Volume	21
Issue number	2
DOIs	https://doi.org/10.4171/JEMS/841
Publication status	Published - 25 Oct 2018
Externally published	Yes

Bibliographical note

Publisher Copyright:
© European Mathematical Society 2019

Keywords

Hardy–Littlewood circle method
Permutations
Transversals in Latin squares

ASJC Scopus subject areas

General Mathematics
Applied Mathematics

Access to Document

10.4171/JEMS/841Licence: Unspecified

Cite this

@article{195b4901115e40b2a2328c5a4844d937,

title = "Additive triples of bijections, or the toroidal semiqueens problem",

abstract = "We prove an asymptotic for the number of additive triples of bijections {1, . . ., n} → Z/nZ, that is, the number of pairs of bijections π1, π2 : {1, . . ., n} → Z/nZ such that the pointwise sum π1 + π2 is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of Z/nZ, to counting the number of arrangements of n mutually nonattacking semiqueens on an n × n toroidal chessboard, and to counting the number of transversals in a cyclic Latin square.",

keywords = "Hardy–Littlewood circle method, Permutations, Transversals in Latin squares",

author = "Sean Eberhard and Freddie Manners and Rudi Mrazovi{\'c}",

note = "Publisher Copyright: {\textcopyright} European Mathematical Society 2019",

year = "2018",

month = oct,

day = "25",

doi = "10.4171/JEMS/841",

language = "English",

volume = "21",

pages = "441--463",

journal = "Journal of the European Mathematical Society",

issn = "1435-9855",

publisher = "European Mathematical Society Publishing House",

number = "2",

}

TY - JOUR

T1 - Additive triples of bijections, or the toroidal semiqueens problem

AU - Eberhard, Sean

AU - Manners, Freddie

AU - Mrazović, Rudi

N1 - Publisher Copyright: © European Mathematical Society 2019

PY - 2018/10/25

Y1 - 2018/10/25

N2 - We prove an asymptotic for the number of additive triples of bijections {1, . . ., n} → Z/nZ, that is, the number of pairs of bijections π1, π2 : {1, . . ., n} → Z/nZ such that the pointwise sum π1 + π2 is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of Z/nZ, to counting the number of arrangements of n mutually nonattacking semiqueens on an n × n toroidal chessboard, and to counting the number of transversals in a cyclic Latin square.

AB - We prove an asymptotic for the number of additive triples of bijections {1, . . ., n} → Z/nZ, that is, the number of pairs of bijections π1, π2 : {1, . . ., n} → Z/nZ such that the pointwise sum π1 + π2 is also a bijection. This problem is equivalent to counting the number of orthomorphisms or complete mappings of Z/nZ, to counting the number of arrangements of n mutually nonattacking semiqueens on an n × n toroidal chessboard, and to counting the number of transversals in a cyclic Latin square.

KW - Hardy–Littlewood circle method

KW - Permutations

KW - Transversals in Latin squares

UR - http://www.scopus.com/inward/record.url?scp=85060767270&partnerID=8YFLogxK

U2 - 10.4171/JEMS/841

DO - 10.4171/JEMS/841

M3 - Article

AN - SCOPUS:85060767270

SN - 1435-9855

VL - 21

SP - 441

EP - 463

JO - Journal of the European Mathematical Society

JF - Journal of the European Mathematical Society

IS - 2

ER -

Additive triples of bijections, or the toroidal semiqueens problem

Abstract

Bibliographical note

Keywords

ASJC Scopus subject areas

Access to Document

Other files and links

Fingerprint

Cite this