Malvina VamvakariDept. Informatics and Telematics, Harokopio University of Athens, Greece
In this work we establish local limit theorems for q-multinomial and multiple Heine distributions. Specifically, the pointwise convergence of the q-multinomial distribution of the first kind, as well as for its discrete limit, the multiple Heine distribution, to a multivariate Stieltjes-Wigert type distribution, are provided.
{"title":"Local Limit Theorems for $q$-Multinomial and Multiple Heine Distributions","authors":"Malvina VamvakariDept. Informatics and Telematics, Harokopio University of Athens, Greece","doi":"arxiv-2406.16420","DOIUrl":"https://doi.org/arxiv-2406.16420","url":null,"abstract":"In this work we establish local limit theorems for q-multinomial and multiple\u0000Heine distributions. Specifically, the pointwise convergence of the\u0000q-multinomial distribution of the first kind, as well as for its discrete\u0000limit, the multiple Heine distribution, to a multivariate Stieltjes-Wigert type\u0000distribution, are provided.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141507053","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}
Benjamin BuckleySimon Fraser University, Marni MishnaSimon Fraser University
A family of walks confined to the first orthant whose defining stepset has drift outside of the region can be challenging to sample uniformly at random for large lengths. We address this by generalizing the 2D walk sampler of Lumbroso et al. to handle 3D walks restricted to the first orthant. The sampler includes a visualizer and means to animate the walks.
{"title":"Uniform Sampling and Visualization of 3D Reluctant Walks","authors":"Benjamin BuckleySimon Fraser University, Marni MishnaSimon Fraser University","doi":"arxiv-2406.16397","DOIUrl":"https://doi.org/arxiv-2406.16397","url":null,"abstract":"A family of walks confined to the first orthant whose defining stepset has\u0000drift outside of the region can be challenging to sample uniformly at random\u0000for large lengths. We address this by generalizing the 2D walk sampler of\u0000Lumbroso et al. to handle 3D walks restricted to the first orthant. The sampler\u0000includes a visualizer and means to animate the walks.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"38 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526090","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}
Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, Jay Pantone
Permutations are usually enumerated by size, but new results can be found by enumerating them by inversions instead, in which case one must restrict one's attention to indecomposable permutations. In the style of the seminal paper by Simion and Schmidt, we investigate all combinations of permutation patterns of length at most 3.
{"title":"Restricted Permutations Enumerated by Inversions","authors":"Atli Fannar Franklín, Anders Claesson, Christian Bean, Henning Úlfarsson, Jay Pantone","doi":"arxiv-2406.16403","DOIUrl":"https://doi.org/arxiv-2406.16403","url":null,"abstract":"Permutations are usually enumerated by size, but new results can be found by\u0000enumerating them by inversions instead, in which case one must restrict one's\u0000attention to indecomposable permutations. In the style of the seminal paper by\u0000Simion and Schmidt, we investigate all combinations of permutation patterns of\u0000length at most 3.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"141 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526049","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}
Olivier BodiniLIPN, Francis DurandLIPN, Philippe MarchalLAGA
This article presents two novel algorithms for generating random increasing trees. The first algorithm efficiently generates strictly increasing binary trees using an ad hoc method. The second algorithm improves the recursive method for weighted strictly increasing unary-binary increasing trees, optimizing randomness usage.
{"title":"Optimal Generation of Strictly Increasing Binary Trees and Beyond","authors":"Olivier BodiniLIPN, Francis DurandLIPN, Philippe MarchalLAGA","doi":"arxiv-2406.16396","DOIUrl":"https://doi.org/arxiv-2406.16396","url":null,"abstract":"This article presents two novel algorithms for generating random increasing\u0000trees. The first algorithm efficiently generates strictly increasing binary\u0000trees using an ad hoc method. The second algorithm improves the recursive\u0000method for weighted strictly increasing unary-binary increasing trees,\u0000optimizing randomness usage.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"18 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526092","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}
Andrea SportielloLIPN, and CNRS, Université Sorbonne Paris Nord
We introduce a natural Boltzmann measure over polyominoes induced by boundary avalanches in the Abelian Sandpile Model. Through the study of a suitable associated process, we give an argument suggesting that the probability distribution of the avalnche sizes has a power-law decay with exponent 3/2, in contrast with the present understanding of bulk avalanches in the model (which has some exponent between 1 and 5/4), and to the ordinary generating function of polyominoes (which is conjectured to have a logarithmic singularity, i.e. exponent 1). We provide some numerical evidence for our claims, and evaluate some other statistical observables on our process, most notably the density of triple points.
{"title":"Natural Measures on Polyominoes Induced by the Abelian Sandpile Model","authors":"Andrea SportielloLIPN, and CNRS, Université Sorbonne Paris Nord","doi":"arxiv-2406.16418","DOIUrl":"https://doi.org/arxiv-2406.16418","url":null,"abstract":"We introduce a natural Boltzmann measure over polyominoes induced by boundary\u0000avalanches in the Abelian Sandpile Model. Through the study of a suitable\u0000associated process, we give an argument suggesting that the probability\u0000distribution of the avalnche sizes has a power-law decay with exponent 3/2, in\u0000contrast with the present understanding of bulk avalanches in the model (which\u0000has some exponent between 1 and 5/4), and to the ordinary generating function\u0000of polyominoes (which is conjectured to have a logarithmic singularity, i.e.\u0000exponent 1). We provide some numerical evidence for our claims, and evaluate\u0000some other statistical observables on our process, most notably the density of\u0000triple points.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"25 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526046","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}
Eli Bagno, Estrella Eisenberg, Shulamit Reches, Moriah Sigron
The Interval poset of a permutation is an effective way of capturing all the intervals of the permutation and the inclusions between them and was introduced recently by Tenner. Thi paper explores the geometric interpretation of interval posets of permutations. We present a bijection between tree interval posets and convex polygons with non-crossing diagonals, offering a novel geometric perspective on this purely combinatorial concept. Additionally, we provide an enumeration of interval posets using this bijection and demonstrate its application to block-wise simple permutations.
{"title":"Interval Posets and Polygon Dissections","authors":"Eli Bagno, Estrella Eisenberg, Shulamit Reches, Moriah Sigron","doi":"arxiv-2406.16392","DOIUrl":"https://doi.org/arxiv-2406.16392","url":null,"abstract":"The Interval poset of a permutation is an effective way of capturing all the\u0000intervals of the permutation and the inclusions between them and was introduced\u0000recently by Tenner. Thi paper explores the geometric interpretation of interval\u0000posets of permutations. We present a bijection between tree interval posets and\u0000convex polygons with non-crossing diagonals, offering a novel geometric\u0000perspective on this purely combinatorial concept. Additionally, we provide an\u0000enumeration of interval posets using this bijection and demonstrate its\u0000application to block-wise simple permutations.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"59 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526091","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}
Dylan Laplace MermoudUMA, ENSTA Paris, Institut Polytechnique de Paris, Pierre PopoliDepartment of Mathematics, Uliège
In cooperative game theory, the social configurations of players are modeled by balanced collections. The Bondareva-Shapley theorem, perhaps the most fundamental theorem in cooperative game theory, characterizes the existence of solutions to the game that benefit everyone using balanced collections. Roughly speaking, if the trivial set system of all players is one of the most efficient balanced collections for the game, then the set of solutions from which each coalition benefits, the so-called core, is non-empty. In this paper, we discuss some interactions between combinatorics and cooperative game theory that are still relatively unexplored. Indeed, the similarity between balanced collections and uniform hypergraphs seems to be a relevant point of view to obtain new properties on those collections through the theory of combinatorial species.
{"title":"Combinatorics on Social Configurations","authors":"Dylan Laplace MermoudUMA, ENSTA Paris, Institut Polytechnique de Paris, Pierre PopoliDepartment of Mathematics, Uliège","doi":"arxiv-2406.16409","DOIUrl":"https://doi.org/arxiv-2406.16409","url":null,"abstract":"In cooperative game theory, the social configurations of players are modeled\u0000by balanced collections. The Bondareva-Shapley theorem, perhaps the most\u0000fundamental theorem in cooperative game theory, characterizes the existence of\u0000solutions to the game that benefit everyone using balanced collections. Roughly\u0000speaking, if the trivial set system of all players is one of the most efficient\u0000balanced collections for the game, then the set of solutions from which each\u0000coalition benefits, the so-called core, is non-empty. In this paper, we discuss some interactions between combinatorics and\u0000cooperative game theory that are still relatively unexplored. Indeed, the\u0000similarity between balanced collections and uniform hypergraphs seems to be a\u0000relevant point of view to obtain new properties on those collections through\u0000the theory of combinatorial species.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"14 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141532101","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 work we recall Pansiot's result on the complexity of pure morphic sequences and we use the tools developed by Devyatov for morphic sequences to prove the decidability of the complexity class of pure morphic sequences.
{"title":"Morphic Sequences: Complexity and Decidability","authors":"Raphael HenryI2M, Aix-Marseille Université","doi":"arxiv-2406.16406","DOIUrl":"https://doi.org/arxiv-2406.16406","url":null,"abstract":"In this work we recall Pansiot's result on the complexity of pure morphic\u0000sequences and we use the tools developed by Devyatov for morphic sequences to\u0000prove the decidability of the complexity class of pure morphic sequences.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"1 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526047","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}
I show how to express the question of whether a polyform tiles the plane isohedrally as a Boolean formula that can be tested using a SAT solver. This approach is adaptable to a wide range of polyforms, requires no special-case code for different isohedral tiling types, and integrates seamlessly with existing software for computing Heesch numbers of polyforms.
我展示了如何用布尔公式来表达多面体是否平铺平面的问题，该布尔公式可以使用 SAT 求解器进行测试。这种方法适用于各种多面体，不需要为不同的等面平铺类型编写特殊的案例代码，而且可以与现有的计算多面体海斯数的软件无缝集成。
{"title":"Detecting Isohedral Polyforms with a SAT Solver","authors":"Craig S. KaplanUniversity of Waterloo","doi":"arxiv-2406.16407","DOIUrl":"https://doi.org/arxiv-2406.16407","url":null,"abstract":"I show how to express the question of whether a polyform tiles the plane\u0000isohedrally as a Boolean formula that can be tested using a SAT solver. This\u0000approach is adaptable to a wide range of polyforms, requires no special-case\u0000code for different isohedral tiling types, and integrates seamlessly with\u0000existing software for computing Heesch numbers of polyforms.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"39 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141525985","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 partial sums of integer sequences that count the occurrences of a specific pattern in the binary expansion of positive integers have been investigated by different authors since the 1950s. In this note, we introduce generalized pattern sequences, which count the occurrences of a finite number of different patterns in the expansion of positive integers in any integer base, and analyze their partial sums.
{"title":"On the Orthogonality of Generalized Pattern Sequences","authors":"Shuo LiThe University of Winnipeg","doi":"arxiv-2406.16411","DOIUrl":"https://doi.org/arxiv-2406.16411","url":null,"abstract":"The partial sums of integer sequences that count the occurrences of a\u0000specific pattern in the binary expansion of positive integers have been\u0000investigated by different authors since the 1950s. In this note, we introduce\u0000generalized pattern sequences, which count the occurrences of a finite number\u0000of different patterns in the expansion of positive integers in any integer\u0000base, and analyze their partial sums.","PeriodicalId":501216,"journal":{"name":"arXiv - CS - Discrete Mathematics","volume":"16 1","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141526045","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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