European Journal of Combinatorics最新文献

{"title":"Asymptotics of local face distributions and the face distribution of the complete graph","authors":"Jesse Campion Loth","doi":"10.1016/j.ejc.2024.104019","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104019","url":null,"abstract":"<div><p>We are interested in the distribution of the number of faces across all the <span><math><mrow><mn>2</mn><mo>−</mo></mrow></math></span>cell embeddings of a graph, which is equivalent to the distribution of genus by Euler’s formula. In order to study this distribution, we consider the local distribution of faces at a single vertex. We show an asymptotic uniformity on this local face distribution which holds for any graph with large vertex degrees.</p><p>We use this to study the usual face distribution of the complete graph. We show that in this case, the local face distribution determines the face distribution for almost all of the whole graph. We use this result to show that a portion of the complete graph of size <span><math><mrow><mrow><mo>(</mo><mn>1</mn><mo>−</mo><mi>o</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>)</mo></mrow><mrow><mo>|</mo><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>|</mo></mrow></mrow></math></span> has the same face distribution as the set of all permutations, up to parity. Along the way, we prove new character bounds and an asymptotic uniformity on conjugacy class products.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104019"},"PeriodicalIF":1.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001045/pdfft?md5=fc1ded09367fd9c944d4c7c3dd000f19&pid=1-s2.0-S0195669824001045-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141479937","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Full asymptotic expansion for orbit-summable quadrant walks and discrete polyharmonic functions","authors":"Andreas Nessmann","doi":"10.1016/j.ejc.2024.104015","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104015","url":null,"abstract":"<div><p>Enumeration of walks with small steps in the quadrant has been a topic of great interest in combinatorics over the last few years. In this article, it is shown how to compute exact asymptotics of the number of such walks with fixed start- and endpoints for orbit-summable models with finite group, up to arbitrary precision. The resulting representation greatly resembles one conjectured for walks starting from the origin in 2020 by Chapon, Fusy and Raschel, differing only in terms appearing due to the periodicity of the model. We will see that the dependency on start- and endpoint is given by discrete polyharmonic functions, which are solutions of <span><math><mrow><msup><mrow><mo>△</mo></mrow><mrow><mi>n</mi></mrow></msup><mi>v</mi><mo>=</mo><mn>0</mn></mrow></math></span> for a discretization <span><math><mo>△</mo></math></span> of a Laplace–Beltrami operator. They can be decomposed into a sum of products of lower order polyharmonic functions of either the start- or the endpoint only, which leads to a partial extension of a theorem by Denisov and Wachtel.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104015"},"PeriodicalIF":1.0,"publicationDate":"2024-06-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824001008/pdfft?md5=e67c218ef49f6659d9d12b5f0c3f5a77&pid=1-s2.0-S0195669824001008-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141479936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Two problems on subset sums","authors":"Xing-Wang Jiang ,&nbsp;Bing-Ling Wu","doi":"10.1016/j.ejc.2024.104016","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104016","url":null,"abstract":"<div><p>For a set <span><math><mi>A</mi></math></span> of positive integers, let <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow></mrow></math></span> denote the set of all finite subset sums of <span><math><mi>A</mi></math></span>. In this paper, we completely solve a problem of Chen and Wu by proving that if <span><math><mrow><mi>B</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mspace></mspace><mo>}</mo></mrow></mrow></math></span> is a sequence of integers with <span><math><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>≥</mo><mn>11</mn></mrow></math></span>, <span><math><mrow><mn>3</mn><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>5</mn><mo>≤</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>≤</mo><mn>4</mn><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span>, <span><math><mrow><mn>3</mn><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mn>2</mn><mo>≤</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>3</mn></mrow></msub><mo>≤</mo><mn>3</mn><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></math></span> and <span><math><mrow><mn>3</mn><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>−</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub><mo>≤</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≤</mo><mn>3</mn><msub><mrow><mi>b</mi></mrow><mrow><mi>n</mi></mrow></msub><mspace></mspace><mrow><mo>(</mo><mi>n</mi><mo>≥</mo><mn>3</mn><mo>)</mo></mrow></mrow></math></span>, then there exists a set of positive integers <span><math><mi>A</mi></math></span> for which <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>A</mi><mo>)</mo></mrow><mo>=</mo><mi>N</mi><mo>∖</mo><mi>B</mi></mrow></math></span>. We also partially answer a problem of Wu by determining the structure of <span><math><mrow><mi>B</mi><mo>=</mo><mrow><mo>{</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&lt;</mo><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&lt;</mo><mo>⋯</mo><mspace></mspace><mo>}</mo></mrow></mrow></math></span> with <span><math><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>&gt;</mo><mn>10</mn></mrow></math></span> and <span><math><mrow><msub><mrow><mi>b</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>&gt;</mo><mn>3</mn><msub><mrow><mi>b</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><mn>4</mn></mrow></math></span>, for which there exists a set of positive integers <span><math><mi>A</mi></math></span> such that <span><math><mrow><mi>P</mi><mrow><mo>(</mo><mi>A</mi><mo>∩</mo><mrow><mo>[</mo><mn>0</mn><mo>,</mo><msub><mrow><mi>b</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>]</mo></mrow><mo>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104016"},"PeriodicalIF":1.0,"publicationDate":"2024-06-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141444154","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Corrigendum to “Describing quasi-graphic matroids” [European J. Combin. 85 (2020) 103062]","authors":"","doi":"10.1016/j.ejc.2024.104004","DOIUrl":"10.1016/j.ejc.2024.104004","url":null,"abstract":"","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104004"},"PeriodicalIF":1.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000891/pdfft?md5=8f6dba2471a3e7b0d7d1cb3d4b1a88bf&pid=1-s2.0-S0195669824000891-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141396945","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"t-sails and sparse hereditary classes of unbounded tree-width","authors":"D. Cocks","doi":"10.1016/j.ejc.2024.104005","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104005","url":null,"abstract":"<div><p>It has long been known that the following basic objects are obstructions to bounded tree-width: for arbitrarily large <span><math><mi>t</mi></math></span>, <span><math><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow></math></span> the complete graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi></mrow></msub></math></span>, <span><math><mrow><mo>(</mo><mn>2</mn><mo>)</mo></mrow></math></span> the complete bipartite graph <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>t</mi><mo>,</mo><mi>t</mi></mrow></msub></math></span>, <span><math><mrow><mo>(</mo><mn>3</mn><mo>)</mo></mrow></math></span> a subdivision of the <span><math><mrow><mo>(</mo><mi>t</mi><mo>×</mo><mi>t</mi><mo>)</mo></mrow></math></span>-wall and <span><math><mrow><mo>(</mo><mn>4</mn><mo>)</mo></mrow></math></span> the line graph of a subdivision of the <span><math><mrow><mo>(</mo><mi>t</mi><mo>×</mo><mi>t</mi><mo>)</mo></mrow></math></span>-wall. We now add a further <em>boundary object</em> to this list, a <span><math><mi>t</mi></math></span>-<em>sail</em>. These results have been obtained by studying sparse hereditary <em>path-star</em> graph classes, each of which consists of the finite induced subgraphs of a single infinite graph whose edges can be partitioned into a path (or forest of paths) with a forest of stars, characterised by an infinite word over a possibly infinite alphabet. We show that a path-star class whose infinite graph has an unbounded number of stars, each of which connects an unbounded number of times to the path, has unbounded tree-width. In addition, we show that such a class is not a subclass of the hereditary class of circle graphs. We identify a collection of <em>nested</em> words with a recursive structure that exhibit interesting characteristics when used to define a path-star graph class. These graph classes do not contain any of the four basic obstructions but instead contain graphs that have large tree-width if and only if they contain arbitrarily large <span><math><mi>t</mi></math></span>-sails. We show that these classes are infinitely defined and, like classes of bounded degree or classes excluding a fixed minor, do not contain a minimal class of unbounded tree-width.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"122 ","pages":"Article 104005"},"PeriodicalIF":1.0,"publicationDate":"2024-06-10","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0195669824000908/pdfft?md5=e4d9091488efe1ad037850e52d6372a3&pid=1-s2.0-S0195669824000908-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141298105","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Special Issue dedicated to the memory of Pierre Rosenstiehl","authors":"Robert Cori (Editors),&nbsp;Jaroslav Nešetřil,&nbsp;Patrice Ossona de Mendez","doi":"10.1016/j.ejc.2023.103800","DOIUrl":"10.1016/j.ejc.2023.103800","url":null,"abstract":"","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103800"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135388042","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Permutation Tutte polynomial","authors":"Csongor Beke ,&nbsp;Gergely Kál Csáji ,&nbsp;Péter Csikvári ,&nbsp;Sára Pituk","doi":"10.1016/j.ejc.2024.104003","DOIUrl":"https://doi.org/10.1016/j.ejc.2024.104003","url":null,"abstract":"<div><p>The classical Tutte polynomial is a two-variate polynomial <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> associated to graphs or more generally, matroids. In this paper, we introduce a polynomial <span><math><mrow><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> associated to a bipartite graph <span><math><mi>H</mi></math></span> that we call the permutation Tutte polynomial of the graph <span><math><mi>H</mi></math></span>. It turns out that <span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> and <span><math><mrow><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><mrow><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></mrow></mrow></math></span> share many properties, and the permutation Tutte polynomial serves as a tool to study the classical Tutte polynomial. We discuss the analogues of Brylawsi’s identities and Conde–Merino–Welsh type inequalities. In particular, we will show that if <span><math><mi>H</mi></math></span> does not contain isolated vertices, then <span><span><span><math><mrow><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>≥</mo><msub><mrow><mover><mrow><mi>T</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><mi>H</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup><mo>,</mo></mrow></math></span></span></span>which gives a short proof of the analogous result of Jackson: <span><span><span><math><mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mn>3</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>3</mn><mo>)</mo></mrow><mo>≥</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow></mrow><mrow><mn>2</mn></mrow></msup></mrow></math></span></span></span>\u0000for graphs without loops and bridges. We also give improvement on the constant 3 in this statement by showing that one can replace it with 2.9243.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"120 ","pages":"Article 104003"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141242539","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Wachs permutations, Bruhat order and weak order","authors":"Francesco Brenti ,&nbsp;Paolo Sentinelli","doi":"10.1016/j.ejc.2023.103804","DOIUrl":"10.1016/j.ejc.2023.103804","url":null,"abstract":"<div><p>We study the partial orders<span><span> induced on Wachs and signed Wachs permutations by the Bruhat and </span>weak orders<span> of the symmetric and hyperoctahedral groups. We show that these orders are graded, determine their rank function, characterize their ordering and covering relations, and compute their characteristic polynomials, when partially ordered by Bruhat order, and determine their structure explicitly when partially ordered by right weak order.</span></span></p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103804"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135849064","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Unambiguously coded shifts","authors":"Marie-Pierre Béal ,&nbsp;Dominique Perrin ,&nbsp;Antonio Restivo","doi":"10.1016/j.ejc.2023.103812","DOIUrl":"10.1016/j.ejc.2023.103812","url":null,"abstract":"<div><p>We study the coded shifts introduced by Blanchard and Hansel (1986). We give several constructions which allow one to represent a coded shift as an unambiguous one.</p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103812"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135894828","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}

{"title":"Theorems and conjectures on some rational generating functions","authors":"Richard P. Stanley","doi":"10.1016/j.ejc.2023.103814","DOIUrl":"10.1016/j.ejc.2023.103814","url":null,"abstract":"<div><p>Let <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> denote the <span><math><mi>i</mi></math></span>th Fibonacci number, and define <span><math><mrow><msubsup><mrow><mo>∏</mo></mrow><mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup><mfenced><mrow><mn>1</mn><mo>+</mo></mrow></mfenced><mfenced><mrow><msup><mrow><mi>x</mi></mrow><mrow><msub><mrow><mi>F</mi></mrow><mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msup></mrow></mfenced><mo>=</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>k</mi></mrow></msup></mrow></math></span>. The paper is concerned primarily with the coefficients <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span>. In particular, for any <span><math><mrow><mi>r</mi><mo>≥</mo><mn>0</mn></mrow></math></span> the generating function <span><math><mrow><msub><mrow><mo>∑</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub><mrow><mo>(</mo><msub><mrow><mo>∑</mo></mrow><mrow><mi>k</mi></mrow></msub><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><msup><mrow><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow><mrow><mi>r</mi></mrow></msup><mo>)</mo></mrow><msup><mrow><mi>x</mi></mrow><mrow><mi>n</mi></mrow></msup></mrow></math></span> is rational. The coefficients <span><math><mrow><msub><mrow><mi>c</mi></mrow><mrow><mi>n</mi></mrow></msub><mrow><mo>(</mo><mi>k</mi><mo>)</mo></mrow></mrow></math></span> can be displayed in an array called the <span><em>Fibonacci triangle </em><em>poset</em></span> <span><math><mi>F</mi></math></span><span> with some interesting further properties, including an encoding of a certain dense linear order on the nonnegative integers. Some generalizations are briefly considered, but there remain many open questions.</span></p></div>","PeriodicalId":50490,"journal":{"name":"European Journal of Combinatorics","volume":"119 ","pages":"Article 103814"},"PeriodicalIF":1.0,"publicationDate":"2024-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135685696","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":3,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}