We compute the set of facets of the polytope which is the convex hull of the Coxeter groups F 4 or H 4 : • For the group F 4 we found 2 orbits of facets which contradicts previous results published in [19]. • For the group H 4 we found 1063 orbits of facets which provides a coun-terexample to the conjecture of [19].
{"title":"The Birkhoff polytope of the groups F4 and H4","authors":"Mathieu Dutour Sikirić","doi":"10.5592/co/ccd.2022.03","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.03","url":null,"abstract":"We compute the set of facets of the polytope which is the convex hull of the Coxeter groups F 4 or H 4 : • For the group F 4 we found 2 orbits of facets which contradicts previous results published in [19]. • For the group H 4 we found 1063 orbits of facets which provides a coun-terexample to the conjecture of [19].","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"9 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"125553361","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The probability that a randomly and uniformly chosen point from the circumball of a tetrahedron lies outside of the inscribed ball of the tetrahedron can be bounded very sharply from below in terms of the edge lengths of the tetrahedron. One can imagine four stars in the Universe (vertices) with known mutual distances and a small (exo-) planet orbiting between them within the circumsphere. The least probability that the planet is outside of the insphere is given in terms of the distances of the stars. The least probability occurs for the regular tetrahedron and it is 0.962962. . . . Geometrically, this is a tricky corollary of (refinements of) the famous Euler inequality: circumradius is at least three times bigger than the inradius of a tetrahedron with equality for a regular tetrahedron. The Euler inequality can be extended to Euclidean sim-plices in all dimensions and to non-Euclidean planes. The most relevant cases of 3D and 4D being in accordance with the relativity theory are considered.
{"title":"Planets are (very likely) in orbits of stars","authors":"D. Veljan","doi":"10.5592/co/ccd.2022.10","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.10","url":null,"abstract":"The probability that a randomly and uniformly chosen point from the circumball of a tetrahedron lies outside of the inscribed ball of the tetrahedron can be bounded very sharply from below in terms of the edge lengths of the tetrahedron. One can imagine four stars in the Universe (vertices) with known mutual distances and a small (exo-) planet orbiting between them within the circumsphere. The least probability that the planet is outside of the insphere is given in terms of the distances of the stars. The least probability occurs for the regular tetrahedron and it is 0.962962. . . . Geometrically, this is a tricky corollary of (refinements of) the famous Euler inequality: circumradius is at least three times bigger than the inradius of a tetrahedron with equality for a regular tetrahedron. The Euler inequality can be extended to Euclidean sim-plices in all dimensions and to non-Euclidean planes. The most relevant cases of 3D and 4D being in accordance with the relativity theory are considered.","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"46 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122156776","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Balanced simplicial complexes are important objects in combinatorics and commutative algebra. A d -dimensional simplicial complex is balanced if its vertices can be coloured into d +1 colors, so there is no monochromatic edge. In this article, we establish two results concerning balanced simplicial complexes assigned to tilings of m × n board in a plane and a torus by I p -omino tile
平衡简复是组合学和交换代数中的重要对象。如果一个d维简单复合体的顶点可以被染成d +1种颜色，那么它就是平衡的，所以没有单色的边。在本文中，我们建立了两个关于m × n板在平面和环面上由I p -omino瓷砖分配的平衡简单配合物的结果
{"title":"Balanced simplicial complex associated with 1 × p polyomino","authors":"Đorđe Barlić, Edin Liđan","doi":"10.5592/co/ccd.2022.01","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.01","url":null,"abstract":"Balanced simplicial complexes are important objects in combinatorics and commutative algebra. A d -dimensional simplicial complex is balanced if its vertices can be coloured into d +1 colors, so there is no monochromatic edge. In this article, we establish two results concerning balanced simplicial complexes assigned to tilings of m × n board in a plane and a torus by I p -omino tile","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"10 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133392135","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We show that an alternating binomial sum which is connected with the Catalan numbers is divisible by n . A natural generalization of this sum is connected with the generalized Catalan numbers and also divisible by n . A new class of binomial sum is used. In Appendix A, we consider a positive binomial sum connected with Fibonacci and Lucas numbers. In Appendix B, we consider an alternating binomial sum which is also connected with Catalan numbers and divisible by ( a + 1) n + 1. Similar reasoning was already used by the author to reprove more simply Calkin’s result for divisibility of the alternating sum of powers of binomials coefficients by the central binomial coefficient.
我们证明了一个与加泰罗尼亚数有关的交替二项式和能被n整除。这个和的一个自然推广与广义加泰罗尼亚数有关，也能被n整除。使用了一类新的二项式和。在附录A中，我们考虑一个与Fibonacci数和Lucas数相关的正二项式和。在附录B中，我们考虑一个交替的二项式和，它也与加泰罗尼亚数有关，并且可以被(a + 1) n + 1整除。类似的推理已经被作者用来更简单地反驳卡尔金关于二项式系数的交替幂和可被中心二项式系数整除的结果。
{"title":"On divisibility properties of some binomial sums connected with the Catalan and Fibonacci numbers","authors":"Jovan Mikić","doi":"10.5592/co/ccd.2022.05","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.05","url":null,"abstract":"We show that an alternating binomial sum which is connected with the Catalan numbers is divisible by n . A natural generalization of this sum is connected with the generalized Catalan numbers and also divisible by n . A new class of binomial sum is used. In Appendix A, we consider a positive binomial sum connected with Fibonacci and Lucas numbers. In Appendix B, we consider an alternating binomial sum which is also connected with Catalan numbers and divisible by ( a + 1) n + 1. Similar reasoning was already used by the author to reprove more simply Calkin’s result for divisibility of the alternating sum of powers of binomials coefficients by the central binomial coefficient.","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"42 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124013471","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
For any three circles in the plane where each circle is tangent to the other two, the Descartes’ theorem yields the existence of a fourth circle tangent to the starting three. Continuing this process by adding a new circle between any three tangent circles leads to Apollonian packings. The fractal structures resulting from infinite continuation of such processes are known as Apollonian gaskets. Close-packed dimer configurations on such structures are well modeled by perfect matchings in the corresponding graphs. We consider Apollonian gaskets for several types of initial configurations and present explicit expressions for the number of perfect matchings in such graphs
{"title":"Solving the dimer problem on Apollonian gasket","authors":"T. Došlić, Luka Podrug","doi":"10.5592/co/ccd.2022.02","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.02","url":null,"abstract":"For any three circles in the plane where each circle is tangent to the other two, the Descartes’ theorem yields the existence of a fourth circle tangent to the starting three. Continuing this process by adding a new circle between any three tangent circles leads to Apollonian packings. The fractal structures resulting from infinite continuation of such processes are known as Apollonian gaskets. Close-packed dimer configurations on such structures are well modeled by perfect matchings in the corresponding graphs. We consider Apollonian gaskets for several types of initial configurations and present explicit expressions for the number of perfect matchings in such graphs","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"5 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126551127","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Extended abstract on Local Irregularity Conjecture and cactus graphs","authors":"J. Sedlar, R. Škrekovski","doi":"10.5592/co/ccd.2022.08","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.08","url":null,"abstract":",","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133896890","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In this article, we compare the famous Flory model with its variant that was recently introduced by the authors. Instead of looking at these models on a one-dimensional lattice, we consider a two-row ladder. For both models we compute the complexity function, and we analyze the differences of static and dynamic versions of these models.
{"title":"Complexity function for a variant of Flory model on a ladder","authors":"Mate Puljiz, Stjepan Šebek, Josip Žubrinić","doi":"10.5592/co/ccd.2022.07","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.07","url":null,"abstract":"In this article, we compare the famous Flory model with its variant that was recently introduced by the authors. Instead of looking at these models on a one-dimensional lattice, we consider a two-row ladder. For both models we compute the complexity function, and we analyze the differences of static and dynamic versions of these models.","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"73 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116069559","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"Grids of equidistant walks","authors":"Biserka Kolarec","doi":"10.5592/co/ccd.2022.04","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.04","url":null,"abstract":",","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"29 5","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"132237718","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On circumradius equations of cyclic polygons","authors":"D. Svrtan","doi":"10.5592/co/ccd.2022.09","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.09","url":null,"abstract":",","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"197 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122369773","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
{"title":"On a linear recurrence relation of the divide-and-conquer type","authors":"Daniele Parisse","doi":"10.5592/co/ccd.2022.06","DOIUrl":"https://doi.org/10.5592/co/ccd.2022.06","url":null,"abstract":",","PeriodicalId":306191,"journal":{"name":"Proceedings of the 4th Croatian Combinatorial Days","volume":"28 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-06-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"129072020","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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