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Distinguishing and Reconstructing Directed Graphs by their (pmb {B})-Polynomials 通过$$pmb {B}$$ -多项式区分和重构有向图
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-06-18 DOI: 10.1007/s00026-024-00702-5
Sagar S. Sawant
{"title":"Distinguishing and Reconstructing Directed Graphs by their (pmb {B})-Polynomials","authors":"Sagar S. Sawant","doi":"10.1007/s00026-024-00702-5","DOIUrl":"10.1007/s00026-024-00702-5","url":null,"abstract":"<div><p>The <i>B</i>-polynomial and quasisymmetric <i>B</i>-function, introduced by Awan and Bernardi, extends the widely studied Tutte polynomial and Tutte symmetric function to digraphs. In this article, we address one of the fundamental questions concerning these digraph invariants, which is, the determination of the classes of digraphs uniquely characterized by them. We solve an open question originally posed by Awan and Bernardi, regarding the identification of digraphs that result from replacing every edge of a graph with a pair of opposite arcs. Further, we address the more challenging problem of reconstructing digraphs using their quasisymmetric functions. In particular, we show that the quasisymmetric <i>B</i>-function reconstructs <i>partially symmetric</i> orientations of <i>proper</i> caterpillars. As a consequence, we establish that all orientations of paths and asymmetric proper caterpillars can be reconstructed from their quasisymmetric <i>B</i>-functions. These results enhance the pool of oriented trees distinguishable through quasisymmetric functions.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"143 - 165"},"PeriodicalIF":0.6,"publicationDate":"2024-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509567","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a General Approach to Bessenrodt–Ono Type Inequalities and Log-Concavity Properties 关于贝森罗德-奥诺型不等式和对数凹凸特性的一般方法
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-05-21 DOI: 10.1007/s00026-024-00700-7
Krystian Gajdzica, Piotr Miska, Maciej Ulas
{"title":"On a General Approach to Bessenrodt–Ono Type Inequalities and Log-Concavity Properties","authors":"Krystian Gajdzica,&nbsp;Piotr Miska,&nbsp;Maciej Ulas","doi":"10.1007/s00026-024-00700-7","DOIUrl":"10.1007/s00026-024-00700-7","url":null,"abstract":"<div><p>In recent literature concerning integer partitions one can find many results related to both the Bessenrodt–Ono type inequalities and log-concavity properties. In this note, we offer some general approach to this type of problems. More precisely, we prove that under some mild conditions on an increasing function <i>F</i> of at most exponential growth satisfying the condition <span>(F(mathbb {N})subset mathbb {R}_{+})</span>, we have <span>(F(a)F(b)&gt;F(a+b))</span> for sufficiently large positive integers <i>a</i>, <i>b</i>. Moreover, we show that if the sequence <span>((F(n))_{nge n_{0}})</span> is log-concave and <span>(limsup _{nrightarrow +infty }F(n+n_{0})/F(n)&lt;F(n_{0}))</span>, then <i>F</i> satisfies the Bessenrodt–Ono type inequality.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"211 - 225"},"PeriodicalIF":0.6,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00700-7.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141116908","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On d-Permutations and Pattern Avoidance Classes 关于 d-Permutations 和模式规避类
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-05-18 DOI: 10.1007/s00026-024-00695-1
Nathan Sun
{"title":"On d-Permutations and Pattern Avoidance Classes","authors":"Nathan Sun","doi":"10.1007/s00026-024-00695-1","DOIUrl":"10.1007/s00026-024-00695-1","url":null,"abstract":"<div><p>Multidimensional permutations, or <i>d</i>-permutations, are represented by their diagrams on <span>([n]^d)</span> such that there exists exactly one point per hyperplane <span>(x_i)</span> that satisfies <span>(x_i= j)</span> for <span>(i in [d])</span> and <span>(j in [n])</span>. Bonichon and Morel previously enumerated 3-permutations avoiding small patterns, and we extend their results by first proving four conjectures, which exhaustively enumerate 3-permutations avoiding any two fixed patterns of size 3. We further provide a enumerative result relating 3-permutation avoidance classes with their respective recurrence relations. In particular, we show a recurrence relation for 3-permutations avoiding the patterns 132 and 213, which contributes a new sequence to the OEIS database. We then extend our results to completely enumerate 3-permutations avoiding three patterns of size 3.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 3","pages":"701 - 732"},"PeriodicalIF":0.6,"publicationDate":"2024-05-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141064188","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Proof of the Plethystic Murnaghan–Nakayama Rule Using Loehr’s Labelled Abacus 使用罗尔标记算盘证明褶皱穆纳汉-中山规则
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-05-17 DOI: 10.1007/s00026-024-00698-y
Pavel Turek
{"title":"Proof of the Plethystic Murnaghan–Nakayama Rule Using Loehr’s Labelled Abacus","authors":"Pavel Turek","doi":"10.1007/s00026-024-00698-y","DOIUrl":"10.1007/s00026-024-00698-y","url":null,"abstract":"<div><p>The plethystic Murnaghan–Nakayama rule describes how to decompose the product of a Schur function and a plethysm of the form <span>(p_rcirc h_m)</span> as a sum of Schur functions. We provide a short, entirely combinatorial proof of this rule using the labelled abaci introduced in Loehr (SIAM J Discrete Math 24(4):1356–1370, 2010).</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"183 - 195"},"PeriodicalIF":0.6,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00698-y.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141062847","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On the Number of Neighborly Simplices in (mathbb {R}^d) 论$$mathbb {R}^d$$ 中的邻简数目
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-05-06 DOI: 10.1007/s00026-024-00694-2
Andrzej P. Kisielewicz
{"title":"On the Number of Neighborly Simplices in (mathbb {R}^d)","authors":"Andrzej P. Kisielewicz","doi":"10.1007/s00026-024-00694-2","DOIUrl":"10.1007/s00026-024-00694-2","url":null,"abstract":"<div><p>Two <i>d</i>-dimensional simplices in <span>(mathbb {R}^d)</span> are neighborly if its intersection is a <span>((d-1))</span>-dimensional set. A family of <i>d</i>-dimensional simplices in <span>(mathbb {R}^d)</span> is called neighborly if every two simplices of the family are neighborly. Let <span>(S_d)</span> be the maximal cardinality of a neighborly family of <i>d</i>-dimensional simplices in <span>(mathbb {R}^d)</span>. Based on the structure of some codes <span>(Vsubset {0,1,*}^n)</span> it is shown that <span>(lim _{drightarrow infty }(2^{d+1}-S_d)=infty )</span>. Moreover, a result on the structure of codes <span>(Vsubset {0,1,*}^n)</span> is given.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 3","pages":"733 - 748"},"PeriodicalIF":0.6,"publicationDate":"2024-05-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00694-2.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140885752","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
On a Conjecture on Pattern-Avoiding Machines 关于模式规避机的猜想
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-04-18 DOI: 10.1007/s00026-024-00693-3
Christopher Bao, Giulio Cerbai, Yunseo Choi, Katelyn Gan, Owen Zhang
{"title":"On a Conjecture on Pattern-Avoiding Machines","authors":"Christopher Bao,&nbsp;Giulio Cerbai,&nbsp;Yunseo Choi,&nbsp;Katelyn Gan,&nbsp;Owen Zhang","doi":"10.1007/s00026-024-00693-3","DOIUrl":"10.1007/s00026-024-00693-3","url":null,"abstract":"<div><p>Let <i>s</i> be West’s stack-sorting map, and let <span>(s_{T})</span> be the generalized stack-sorting map, where instead of being required to increase, the stack avoids subpermutations that are order-isomorphic to any permutation in the set <i>T</i>. In 2020, Cerbai, Claesson, and Ferrari introduced the <span>(sigma )</span>-machine <span>(s circ s_{sigma })</span> as a generalization of West’s 2-stack-sorting-map <span>(s circ s)</span>. As a further generalization, in 2021, Baril, Cerbai, Khalil, and Vajnovski introduced the <span>((sigma , tau ))</span>-machine <span>(s circ s_{sigma , tau })</span> and enumerated <span>(textrm{Sort}_{n}(sigma ,tau ))</span>—the number of permutations in <span>(S_n)</span> that are mapped to the identity by the <span>((sigma , tau ))</span>-machine—for six pairs of length 3 permutations <span>((sigma , tau ))</span>. In this work, we settle a conjecture by Baril, Cerbai, Khalil, and Vajnovski on the only remaining pair of length 3 patterns <span>((sigma , tau ) = (132, 321))</span> for which <span>(|textrm{Sort}_{n}(sigma , tau )|)</span> appears in the OEIS. In addition, we enumerate <span>(textrm{Sort}_n(123, 321))</span>, which does not appear in the OEIS, but has a simple closed form.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"123 - 141"},"PeriodicalIF":0.6,"publicationDate":"2024-04-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140623223","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Fully Complementary Higher Dimensional Partitions 完全互补的高维分区
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-04-06 DOI: 10.1007/s00026-024-00691-5
Florian Schreier-Aigner
{"title":"Fully Complementary Higher Dimensional Partitions","authors":"Florian Schreier-Aigner","doi":"10.1007/s00026-024-00691-5","DOIUrl":"10.1007/s00026-024-00691-5","url":null,"abstract":"<div><p>We introduce a symmetry class for higher dimensional partitions—<i>fully complementary higher dimensional partitions</i> (FCPs)—and prove a formula for their generating function. By studying symmetry classes of FCPs in dimension 2, we define variations of the classical symmetry classes for plane partitions. As a by-product, we obtain conjectures for three new symmetry classes of plane partitions and prove that another new symmetry class, namely <i>quasi-transpose-complementary plane partitions</i>, are equinumerous to symmetric plane partitions.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"1 - 23"},"PeriodicalIF":0.6,"publicationDate":"2024-04-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://link.springer.com/content/pdf/10.1007/s00026-024-00691-5.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140586291","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"OA","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Runs and RSK Tableaux of Boolean Permutations 布尔排列的运行和 RSK 表法
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-04-04 DOI: 10.1007/s00026-024-00689-z
Emily Gunawan, Jianping Pan, Heather M. Russell, Bridget Eileen Tenner
{"title":"Runs and RSK Tableaux of Boolean Permutations","authors":"Emily Gunawan,&nbsp;Jianping Pan,&nbsp;Heather M. Russell,&nbsp;Bridget Eileen Tenner","doi":"10.1007/s00026-024-00689-z","DOIUrl":"10.1007/s00026-024-00689-z","url":null,"abstract":"<div><p>We define and construct the “canonical reduced word” of a boolean permutation, and show that the RSK tableaux for that permutation can be read off directly from this reduced word. We also describe those tableaux that can correspond to boolean permutations, and enumerate them. In addition, we generalize a result of Mazorchuk and Tenner, showing that the “run” statistic influences the shape of the RSK tableau of arbitrary permutations, not just of those that are boolean.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"65 - 90"},"PeriodicalIF":0.6,"publicationDate":"2024-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140602418","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Polynomization of the Bessenrodt–Ono Type Inequalities for A-Partition Functions A 分区函数的贝森罗德-奥诺型不等式的多项式化
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-04-03 DOI: 10.1007/s00026-024-00692-4
Krystian Gajdzica, Bernhard Heim, Markus Neuhauser
{"title":"Polynomization of the Bessenrodt–Ono Type Inequalities for A-Partition Functions","authors":"Krystian Gajdzica,&nbsp;Bernhard Heim,&nbsp;Markus Neuhauser","doi":"10.1007/s00026-024-00692-4","DOIUrl":"10.1007/s00026-024-00692-4","url":null,"abstract":"<div><p>For an arbitrary set or multiset <i>A</i> of positive integers, we associate the <i>A</i>-partition function <span>(p_A(n))</span> (that is the number of partitions of <i>n</i> whose parts belong to <i>A</i>). We also consider the analogue of the <i>k</i>-colored partition function, namely, <span>(p_{A,-k}(n))</span>. Further, we define a family of polynomials <span>(f_{A,n}(x))</span> which satisfy the equality <span>(f_{A,n}(k)=p_{A,-k}(n))</span> for all <span>(nin mathbb {Z}_{ge 0})</span> and <span>(kin mathbb {N})</span>. This paper concerns a polynomialization of the Bessenrodt–Ono inequality, namely </p><div><div><span>$$begin{aligned} f_{A,a}(x)f_{A,b}(x)&gt;f_{A,a+b}(x), end{aligned}$$</span></div></div><p>where <i>a</i>, <i>b</i> are positive integers. We determine efficient criteria for the solutions of this inequality. Moreover, we also investigate a few basic properties related to both functions <span>(f_{A,n}(x))</span> and <span>(f_{A,n}'(x))</span>.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"28 4","pages":"1323 - 1345"},"PeriodicalIF":0.6,"publicationDate":"2024-04-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140586393","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
Some Infinite-Dimensional Representations of Certain Coxeter Groups 某些柯克西特群的一些无穷维表示
IF 0.6 4区 数学
Annals of Combinatorics Pub Date : 2024-03-25 DOI: 10.1007/s00026-024-00690-6
Hongsheng Hu
{"title":"Some Infinite-Dimensional Representations of Certain Coxeter Groups","authors":"Hongsheng Hu","doi":"10.1007/s00026-024-00690-6","DOIUrl":"10.1007/s00026-024-00690-6","url":null,"abstract":"<div><p>A Coxeter group admits infinite-dimensional irreducible complex representations if and only if it is not finite or affine. In this paper, we provide a construction of some of those representations for certain Coxeter groups using some topological information of the corresponding Coxeter graphs.</p></div>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":"29 1","pages":"101 - 115"},"PeriodicalIF":0.6,"publicationDate":"2024-03-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140301791","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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