{"title":"SOME COUNTING FORMULAE FOR -QUIDDITIES OVER THE RINGS","authors":"FLAVIEN MABILAT","doi":"10.1017/s0004972724000340","DOIUrl":null,"url":null,"abstract":"\n\t <jats:p>The <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline3.png\"/>\n\t\t<jats:tex-math>\n$\\lambda $\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>-quiddities of size <jats:italic>n</jats:italic> are <jats:italic>n</jats:italic>-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter’s friezes. Their number and properties are closely linked to the structure and the cardinality of the chosen set. Our main objective is an explicit formula giving the number of <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline4.png\"/>\n\t\t<jats:tex-math>\n$\\lambda $\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>-quiddities of odd size, and a lower and upper bound for the number of <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline5.png\"/>\n\t\t<jats:tex-math>\n$\\lambda $\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>-quiddities of even size, over the rings <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline6.png\"/>\n\t\t<jats:tex-math>\n${\\mathbb {Z}}/2^{m}{\\mathbb {Z}}$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula> (<jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline7.png\"/>\n\t\t<jats:tex-math>\n$m \\geq 2$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>). We also give explicit formulae for the number of <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline8.png\"/>\n\t\t<jats:tex-math>\n$\\lambda $\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>-quiddities of size <jats:italic>n</jats:italic> over <jats:inline-formula>\n\t <jats:alternatives>\n\t\t<jats:inline-graphic xmlns:xlink=\"http://www.w3.org/1999/xlink\" mime-subtype=\"png\" xlink:href=\"S0004972724000340_inline9.png\"/>\n\t\t<jats:tex-math>\n${\\mathbb {Z}}/8{\\mathbb {Z}}$\n</jats:tex-math>\n\t </jats:alternatives>\n\t </jats:inline-formula>.</jats:p>","PeriodicalId":50720,"journal":{"name":"Bulletin of the Australian Mathematical Society","volume":null,"pages":null},"PeriodicalIF":0.6000,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Bulletin of the Australian Mathematical Society","FirstCategoryId":"100","ListUrlMain":"https://doi.org/10.1017/s0004972724000340","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
The
$\lambda $
-quiddities of size n are n-tuples of elements of a fixed set, solutions of a matrix equation appearing in the study of Coxeter’s friezes. Their number and properties are closely linked to the structure and the cardinality of the chosen set. Our main objective is an explicit formula giving the number of
$\lambda $
-quiddities of odd size, and a lower and upper bound for the number of
$\lambda $
-quiddities of even size, over the rings
${\mathbb {Z}}/2^{m}{\mathbb {Z}}$
(
$m \geq 2$
). We also give explicit formulae for the number of
$\lambda $
-quiddities of size n over
${\mathbb {Z}}/8{\mathbb {Z}}$
.
期刊介绍:
Bulletin of the Australian Mathematical Society aims at quick publication of original research in all branches of mathematics. Papers are accepted only after peer review but editorial decisions on acceptance or otherwise are taken quickly, normally within a month of receipt of the paper. The Bulletin concentrates on presenting new and interesting results in a clear and attractive way.
Published Bi-monthly
Published for the Australian Mathematical Society