{"title":"作品的场地周界","authors":"A. Blecher, C. Brennan, A. Knopfmacher","doi":"10.1515/dma-2022-0007","DOIUrl":null,"url":null,"abstract":"Abstract Compositions of n are finite sequences of positive integers (σi)i=1k $\\begin{array}{} (\\sigma_i)_{i = 1}^k \\end{array} $ such that σ1+σ2+⋯+σk=n. $$\\begin{array}{} \\sigma_1+\\sigma_2+\\cdots +\\sigma_k = n. \\end{array} $$ We represent a composition of n as a bargraph with area n such that the height of the i-th column of the bargraph equals the size of the i-th part of the composition. We consider the site-perimeter which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the total site-perimeter of compositions is obtained. In addition, we rederive the average site-perimeter of a composition by direct counting. Finally we determine the average site-perimeter of a bargraph with a given semi-perimeter.","PeriodicalId":11287,"journal":{"name":"Discrete Mathematics and Applications","volume":"32 1","pages":"75 - 89"},"PeriodicalIF":0.3000,"publicationDate":"2022-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"The site-perimeter of compositions\",\"authors\":\"A. Blecher, C. Brennan, A. Knopfmacher\",\"doi\":\"10.1515/dma-2022-0007\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"Abstract Compositions of n are finite sequences of positive integers (σi)i=1k $\\\\begin{array}{} (\\\\sigma_i)_{i = 1}^k \\\\end{array} $ such that σ1+σ2+⋯+σk=n. $$\\\\begin{array}{} \\\\sigma_1+\\\\sigma_2+\\\\cdots +\\\\sigma_k = n. \\\\end{array} $$ We represent a composition of n as a bargraph with area n such that the height of the i-th column of the bargraph equals the size of the i-th part of the composition. We consider the site-perimeter which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the total site-perimeter of compositions is obtained. In addition, we rederive the average site-perimeter of a composition by direct counting. Finally we determine the average site-perimeter of a bargraph with a given semi-perimeter.\",\"PeriodicalId\":11287,\"journal\":{\"name\":\"Discrete Mathematics and Applications\",\"volume\":\"32 1\",\"pages\":\"75 - 89\"},\"PeriodicalIF\":0.3000,\"publicationDate\":\"2022-04-01\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Discrete Mathematics and Applications\",\"FirstCategoryId\":\"1085\",\"ListUrlMain\":\"https://doi.org/10.1515/dma-2022-0007\",\"RegionNum\":0,\"RegionCategory\":null,\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q4\",\"JCRName\":\"MATHEMATICS, APPLIED\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Discrete Mathematics and Applications","FirstCategoryId":"1085","ListUrlMain":"https://doi.org/10.1515/dma-2022-0007","RegionNum":0,"RegionCategory":null,"ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q4","JCRName":"MATHEMATICS, APPLIED","Score":null,"Total":0}
Abstract Compositions of n are finite sequences of positive integers (σi)i=1k $\begin{array}{} (\sigma_i)_{i = 1}^k \end{array} $ such that σ1+σ2+⋯+σk=n. $$\begin{array}{} \sigma_1+\sigma_2+\cdots +\sigma_k = n. \end{array} $$ We represent a composition of n as a bargraph with area n such that the height of the i-th column of the bargraph equals the size of the i-th part of the composition. We consider the site-perimeter which is the number of nearest-neighbour cells outside the boundary of the polyomino. The generating function that counts the total site-perimeter of compositions is obtained. In addition, we rederive the average site-perimeter of a composition by direct counting. Finally we determine the average site-perimeter of a bargraph with a given semi-perimeter.
期刊介绍：
The aim of this journal is to provide the latest information on the development of discrete mathematics in the former USSR to a world-wide readership. The journal will contain papers from the Russian-language journal Diskretnaya Matematika, the only journal of the Russian Academy of Sciences devoted to this field of mathematics. Discrete Mathematics and Applications will cover various subjects in the fields such as combinatorial analysis, graph theory, functional systems theory, cryptology, coding, probabilistic problems of discrete mathematics, algorithms and their complexity, combinatorial and computational problems of number theory and of algebra.
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