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A Bump Statistic on Permutations Resulting from the Robinson–Schensted Correspondence 罗宾逊-申斯泰德对应关系产生的排列的凹凸统计量
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2024-07-08 DOI: 10.1007/s00026-024-00708-z
Mark Dukes, Andrew Mullins
{"title":"A Bump Statistic on Permutations Resulting from the Robinson–Schensted Correspondence","authors":"Mark Dukes, Andrew Mullins","doi":"10.1007/s00026-024-00708-z","DOIUrl":"https://doi.org/10.1007/s00026-024-00708-z","url":null,"abstract":"<p>In this paper, we investigate a permutation statistic that was independently introduced by Romik (Funct Anal Appl 39(2):52–155, 2005). This statistic counts the number of bumps that occur during the execution of the Robinson–Schensted procedure when applied to a given permutation. We provide several interpretations of this bump statistic that include the tableaux shape and also as an extremal problem concerning permutations and increasing subsequences. Several aspects of this bump statistic are investigated from both structural and enumerative viewpoints.</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141570583","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 the Positivity of Infinite Products Connected to Partitions with Even Parts Below Odd Parts and Copartitions 论与偶数部分低于奇数部分的分区和共分区相连的无限积的实在性
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2024-07-01 DOI: 10.1007/s00026-024-00704-3
Hannah E. Burson, Dennis Eichhorn
{"title":"On the Positivity of Infinite Products Connected to Partitions with Even Parts Below Odd Parts and Copartitions","authors":"Hannah E. Burson, Dennis Eichhorn","doi":"10.1007/s00026-024-00704-3","DOIUrl":"https://doi.org/10.1007/s00026-024-00704-3","url":null,"abstract":"<p>In this paper, we give a combinatorial proof of a positivity result of Chern related to Andrews’s <span>(mathcal{E}mathcal{O}^*)</span>-type partitions. This combinatorial proof comes after reframing Chern’s result in terms of copartitions.Using this new perspective, we also reprove an overpartition result of Chern by showing that it comes essentially “for free” from our combinatorial proof and some basic properties of copartitions. Finally, the application of copartitions leads us to more general positivity conjectures for families of both infinite and finite products, with a proof in one special case.</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-07-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509563","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
Insertion Algorithms for Gelfand $$S_n$$ -Graphs 格尔凡$S_n$$图的插入算法
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2024-06-28 DOI: 10.1007/s00026-024-00701-6
Eric Marberg, Yifeng Zhang
{"title":"Insertion Algorithms for Gelfand $$S_n$$ -Graphs","authors":"Eric Marberg, Yifeng Zhang","doi":"10.1007/s00026-024-00701-6","DOIUrl":"https://doi.org/10.1007/s00026-024-00701-6","url":null,"abstract":"<p>The two tableaux assigned by the Robinson–Schensted correspondence are equal if and only if the input permutation is an involution, so the RS algorithm restricts to a bijection between involutions in the symmetric group and standard tableaux. Beissinger found a concise way of formulating this restricted map, which involves adding an extra cell at the end of a row after a Schensted insertion process. We show that by changing this algorithm slightly to add cells at the end of columns rather than rows, one obtains a different bijection from involutions to standard tableaux. Both maps have an interesting connection to representation theory. Specifically, our insertion algorithms classify the molecules (and conjecturally the cells) in the pair of <i>W</i>-graphs associated with the unique equivalence class of perfect models for a generic symmetric group.</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509564","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
Legendre Theorems for a Class of Partitions with Initial Repetitions 具有初始重复的一类分区的勒让德定理
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2024-06-25 DOI: 10.1007/s00026-024-00706-1
Darlison Nyirenda, Beaullah Mugwangwavari
{"title":"Legendre Theorems for a Class of Partitions with Initial Repetitions","authors":"Darlison Nyirenda, Beaullah Mugwangwavari","doi":"10.1007/s00026-024-00706-1","DOIUrl":"https://doi.org/10.1007/s00026-024-00706-1","url":null,"abstract":"<p>Partitions with initial repetitions were introduced by George Andrews. We consider a subclass of these partitions and find Legendre theorems associated with their respective partition functions. The results in turn provide partition-theoretic interpretations of some Rogers–Ramanujan identities due to Lucy J. Slater.</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509565","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
Fan Valuations and Spherical Intrinsic Volumes 扇形估值和球形内在体积
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2024-06-24 DOI: 10.1007/s00026-024-00699-x
Spencer Backman, Sebastian Manecke, Raman Sanyal
{"title":"Fan Valuations and Spherical Intrinsic Volumes","authors":"Spencer Backman, Sebastian Manecke, Raman Sanyal","doi":"10.1007/s00026-024-00699-x","DOIUrl":"https://doi.org/10.1007/s00026-024-00699-x","url":null,"abstract":"<p>We generalize valuations on polyhedral cones to valuations on (plane) fans. For fans induced by hyperplane arrangements, we show a correspondence between rotation-invariant valuations and deletion– restriction invariants. In particular, we define a characteristic polynomial for fans in terms of spherical intrinsic volumes and show that it coincides with the usual characteristic polynomial in the case of hyperplane arrangements. This gives a simple deletion–restriction proof of a result of Klivans–Swartz. The metric projection of a cone is a piecewise-linear map, whose underlying fan prompts a generalization of spherical intrinsic volumes to indicator functions. We show that these <i>intrinsic indicators</i> yield valuations that separate polyhedral cones. Applied to hyperplane arrangements, this generalizes a result of Kabluchko on projection volumes.</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509566","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
Infinite Families of Vertex-Transitive Graphs with Prescribed Hamilton Compression 具有规定汉密尔顿压缩的顶点-传递图无穷族
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2024-06-20 DOI: 10.1007/s00026-024-00703-4
Klavdija Kutnar, Dragan Marušič, Andriaherimanana Sarobidy Razafimahatratra
{"title":"Infinite Families of Vertex-Transitive Graphs with Prescribed Hamilton Compression","authors":"Klavdija Kutnar, Dragan Marušič, Andriaherimanana Sarobidy Razafimahatratra","doi":"10.1007/s00026-024-00703-4","DOIUrl":"https://doi.org/10.1007/s00026-024-00703-4","url":null,"abstract":"<p>Given a graph <i>X</i> with a Hamilton cycle <i>C</i>, the <i>compression factor </i><span>(kappa (X,C))</span> <i>of </i><i>C</i> is the order of the largest cyclic subgroup of <span>({textrm{Aut}},(C)cap {textrm{Aut}},(X))</span>, and the <i>Hamilton compression </i><span>(kappa (X))</span> <i>of </i><i>X</i> is the maximum of <span>(kappa (X,C))</span> where <i>C</i> runs over all Hamilton cycles in <i>X</i>. Generalizing the well-known open problem regarding the existence of vertex-transitive graphs without Hamilton paths/cycles, it was asked by Gregor et al. (Ann Comb, arXiv:2205.08126v1, https://doi.org/10.1007/s00026-023-00674-y, 2023) whether for every positive integer <i>k</i>, there exists infinitely many vertex-transitive graphs (Cayley graphs) with Hamilton compression equal to <i>k</i>. Since an infinite family of Cayley graphs with Hamilton compression equal to 1 was given there, the question is completely resolved in this paper in the case of Cayley graphs with a construction of Cayley graphs of semidirect products <span>(mathbb {Z}_prtimes mathbb {Z}_k)</span> where <i>p</i> is a prime and <span>(k ge 2)</span> a divisor of <span>(p-1)</span>. Further, infinite families of non-Cayley vertex-transitive graphs with Hamilton compression equal to 1 are given. All of these graphs being metacirculants, some additional results on Hamilton compression of metacirculants of specific orders are also given.\u0000</p>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-06-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141509568","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
Distinguishing and Reconstructing Directed Graphs by their $$pmb {B}$$ -Polynomials 通过$$pmb {B}$$ -多项式区分和重构有向图
IF 0.5 4区 数学
Annals of Combinatorics Pub Date : 2024-06-18 DOI: 10.1007/s00026-024-00702-5
Sagar S. Sawant
{"title":"Distinguishing and Reconstructing Directed Graphs by their $$pmb {B}$$ -Polynomials","authors":"Sagar S. Sawant","doi":"10.1007/s00026-024-00702-5","DOIUrl":"https://doi.org/10.1007/s00026-024-00702-5","url":null,"abstract":"<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>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"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 0.5 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, Piotr Miska, Maciej Ulas","doi":"10.1007/s00026-024-00700-7","DOIUrl":"https://doi.org/10.1007/s00026-024-00700-7","url":null,"abstract":"","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","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":null,"pages":null},"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 0.5 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":"https://doi.org/10.1007/s00026-024-00698-y","url":null,"abstract":"<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>","PeriodicalId":50769,"journal":{"name":"Annals of Combinatorics","volume":null,"pages":null},"PeriodicalIF":0.5,"publicationDate":"2024-05-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","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":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
引用次数: 0
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