{"title":"Many triangles in C5-free graphs","authors":"Zequn Lv , Zhen He , Mei Lu","doi":"10.1016/j.aam.2024.102740","DOIUrl":"10.1016/j.aam.2024.102740","url":null,"abstract":"<div><p>In the present paper, we introduce a new approach and use it to prove that the maximum number of triangles in a <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free graph on <em>n</em> vertices is at most <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn><msqrt><mrow><mn>6</mn></mrow></msqrt></mrow></mfrac><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>, improving an estimate of Ergemlidze and Methuku <span><span>[4]</span></span>. We also show that the maximum size of an induced-<span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub></math></span>-free and <span><math><msub><mrow><mi>C</mi></mrow><mrow><mn>5</mn></mrow></msub></math></span>-free graph on <em>n</em> vertices is at most <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mn>6</mn></mrow></msqrt></mrow></mfrac><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>o</mi><mo>(</mo><mn>1</mn><mo>)</mo><mo>)</mo></mrow><msup><mrow><mi>n</mi></mrow><mrow><mn>3</mn><mo>/</mo><mn>2</mn></mrow></msup></math></span>, also improving an estimate of Ergemlidze and Methuku <span><span>[4]</span></span>.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102740"},"PeriodicalIF":1.0,"publicationDate":"2024-07-15","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141622872","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":"Non-central sections of the l1-ball","authors":"Hermann König","doi":"10.1016/j.aam.2024.102737","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102737","url":null,"abstract":"<div><p>We determine the maximal non-central hyperplane sections of the <span><math><msubsup><mrow><mi>l</mi></mrow><mrow><mn>1</mn></mrow><mrow><mi>n</mi></mrow></msubsup></math></span>-ball if the fixed distance of the hyperplane to the origin is between <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mn>3</mn></mrow></msqrt></mrow></mfrac></math></span> and <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></mfrac></math></span>. This adds to a result of Liu and Tkocz who considered the distance range between <span><math><mfrac><mrow><mn>1</mn></mrow><mrow><msqrt><mrow><mn>2</mn></mrow></msqrt></mrow></mfrac></math></span> and 1. For <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span>, the maximal sections are parallel to the <span><math><mo>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-dimensional coordinate planes. We also study non-central sections of the complex <span><math><msubsup><mrow><mi>l</mi></mrow><mrow><mo>∞</mo></mrow><mrow><mn>2</mn></mrow></msubsup></math></span>-ball, where the formulas are more complicated than in the real case. Also, the extrema are partially different compared to the real case.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102737"},"PeriodicalIF":1.0,"publicationDate":"2024-07-09","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0196885824000691/pdfft?md5=4ac52765c27da32cc7db516354fb66e2&pid=1-s2.0-S0196885824000691-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141596392","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":"(p,q,t)-Catalan continued fractions, gamma expansions and pattern avoidances","authors":"Bin Han , Qiongqiong Pan","doi":"10.1016/j.aam.2024.102735","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102735","url":null,"abstract":"<div><p>We introduce a kind of <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-Catalan numbers of Type A by generalizing the J-type continued fraction formula, we prove that the corresponding expansions could be expressed by the polynomials counting permutations on <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mn>321</mn><mo>)</mo></math></span> by various descent statistics. Moreover, we introduce a kind of <span><math><mo>(</mo><mi>p</mi><mo>,</mo><mi>q</mi><mo>,</mo><mi>t</mi><mo>)</mo></math></span>-Catalan numbers of Type B by generalizing the J-type continued fraction formula, we prove that the Taylor coefficients and their <em>γ</em>-coefficients could be expressed by the polynomials counting permutations on <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mn>3124</mn><mo>,</mo><mn>4123</mn><mo>,</mo><mn>3142</mn><mo>,</mo><mn>4132</mn><mo>)</mo></math></span> by various descent statistics. Our methods include permutation enumeration techniques involving variations of bijections from permutation patterns to labeled Motzkin paths and modified Foata-Strehl action.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102735"},"PeriodicalIF":1.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141594846","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":"Strongly unimodal sequences and Hecke-type identities","authors":"Su-Ping Cui , Hai-Xing Du , Nancy S.S. Gu","doi":"10.1016/j.aam.2024.102738","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102738","url":null,"abstract":"<div><p>A strongly unimodal sequence of size <em>n</em> is a sequence of integers <span><math><msubsup><mrow><mo>{</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>j</mi></mrow></msub><mo>}</mo></mrow><mrow><mi>j</mi><mo>=</mo><mn>1</mn></mrow><mrow><mi>s</mi></mrow></msubsup></math></span> satisfying the following conditions:<span><span><span><math><mn>0</mn><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo><</mo><mo>⋯</mo><mo><</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>></mo><mo>⋯</mo><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>></mo><mn>0</mn><mspace></mspace><mtext>and</mtext><mspace></mspace><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>+</mo><mo>⋯</mo><mo>+</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>s</mi></mrow></msub><mo>=</mo><mi>n</mi><mo>,</mo></math></span></span></span> for a certain index <em>k</em>, and we usually define its rank as <span><math><mi>s</mi><mo>−</mo><mn>2</mn><mi>k</mi><mo>+</mo><mn>1</mn></math></span>. Let <span><math><mi>u</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> be the number of strongly unimodal sequences of size <em>n</em> with rank <em>m</em>, and the generating function for <span><math><mi>u</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo></math></span> is written as<span><span><span><math><mi>U</mi><mo>(</mo><mi>z</mi><mo>;</mo><mi>q</mi><mo>)</mo><mo>:</mo><mo>=</mo><munder><mo>∑</mo><mrow><mi>m</mi><mo>,</mo><mi>n</mi></mrow></munder><mi>u</mi><mo>(</mo><mi>m</mi><mo>,</mo><mi>n</mi><mo>)</mo><msup><mrow><mi>z</mi></mrow><mrow><mi>m</mi></mrow></msup><msup><mrow><mi>q</mi></mrow><mrow><mi>n</mi></mrow></msup><mo>.</mo></math></span></span></span> Recently, Chen and Garvan established some Hecke-type identities for the third order mock theta function <span><math><mi>ψ</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><mi>U</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>, which are the specializations of <span><math><mi>U</mi><mo>(</mo><mi>z</mi><mo>;</mo><mi>q</mi><mo>)</mo></math></span>, as advocated by <span><math><mi>ψ</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><mi>U</mi><mo>(</mo><mo>±</mo><mi>i</mi><mo>;</mo><mi>q</mi><mo>)</mo></math></span> and <span><math><mi>U</mi><mo>(</mo><mi>q</mi><mo>)</mo><mo>=</mo><mi>U</mi><mo>(</mo><mn>1</mn><mo>;</mo><mi>q</mi><mo>)</mo></math></span>. Meanwhile, they inquired whether these Hecke-type identities could be proved via the Bailey pair machinery. In this paper, we not only answer the inquiry of Chen and Garvan in the affirmative, but offer more instances in a broader setting, with, for example, some classical third order mock theta functions due to Ramanujan involved. Furthermo","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102738"},"PeriodicalIF":1.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141594848","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":"Prism permutations in the Bruhat order","authors":"Bridget Eileen Tenner","doi":"10.1016/j.aam.2024.102734","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102734","url":null,"abstract":"<div><p>The boolean elements of a Coxeter group have been characterized and shown to possess many interesting properties and applications. Here we introduce “prism permutations,” a generalization of those elements, characterizing the prism permutations equivalently in terms of their reduced words and in terms of pattern containment. As part of this work, we introduce the notion of “calibration” to permutation patterns.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102734"},"PeriodicalIF":1.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141594847","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":"Difference ascent sequences","authors":"Mark Dukes , Bruce E. Sagan","doi":"10.1016/j.aam.2024.102736","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102736","url":null,"abstract":"<div><p>Let <span><math><mi>α</mi><mo>=</mo><msub><mrow><mi>a</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>a</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>…</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> be a sequence of nonnegative integers. The ascent set of <em>α</em>, Asc <em>α</em>, consists of all indices <em>k</em> where <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>. An ascent sequence is <em>α</em> where the growth of the <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> is bounded by the elements of Asc <em>α</em>. These sequences were introduced by Bousquet-Mélou, Claesson, Dukes and Kitaev and have many wonderful properties. In particular, they are in bijection with unlabeled <span><math><mo>(</mo><mn>2</mn><mo>+</mo><mn>2</mn><mo>)</mo></math></span>-free posets, permutations avoiding a particular bivincular pattern, certain upper-triangular nonnegative integer matrices, and a class of matchings. A weak ascent of <em>α</em> is an index <em>k</em> with <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>≥</mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> and weak ascent sequences are defined analogously to ascent sequences. These were studied by Bényi, Claesson and Dukes and shown to have analogous equinumerous sets. Given a nonnegative integer <em>d</em>, we define a difference <em>d</em> ascent to be an index <em>k</em> such that <span><math><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>></mo><msub><mrow><mi>a</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>−</mo><mi>d</mi></math></span>. We study the properties of the corresponding <em>d</em>-ascent sequences, showing that some of the maps from the weak case can be extended to bijections for general <em>d</em> while the extensions of others continue to be injective (but not surjective). We also make connections with other combinatorial objects such as rooted duplication trees and restricted growth functions.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102736"},"PeriodicalIF":1.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141539589","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":"Simplicial Kirchhoff index of random complexes","authors":"Woong Kook , Kang-Ju Lee","doi":"10.1016/j.aam.2024.102733","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102733","url":null,"abstract":"<div><p>Kirchhoff index is an electrical network-theoretic invariant which is defined as the sum of effective resistances between all pairs of vertices. As a robustness measure of simplicial networks, a simplicial analogue of the Kirchhoff index is defined to be the sum of simplicial effective resistances for all subsets of vertices of size dimension plus one. In this paper, we investigate the Kirchhoff index of random simplicial complexes as a generalization of random graphs. We present a formula for the expectation of the random variable and show how it concentrates around the expectation. We also perform numerical experiments revealing that the expectation and the fluctuation are still valid for realizations of the random simplicial Kirchhoff index.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102733"},"PeriodicalIF":1.0,"publicationDate":"2024-07-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141539590","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":"On the maximum value of the stairs2 index","authors":"Bryan Currie, Kristina Wicke","doi":"10.1016/j.aam.2024.102732","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102732","url":null,"abstract":"<div><p>Measures of tree balance play an important role in different research areas such as mathematical phylogenetics or theoretical computer science. The balance of a tree is usually quantified in a single number, called a balance or imbalance index, and several such indices exist in the literature. Here, we focus on the stairs2 balance index for rooted binary trees, which was first introduced in the context of viral phylogenetics but has not been fully analyzed from a mathematical viewpoint yet. While it is known that the caterpillar tree uniquely minimizes the stairs2 index for all leaf numbers and the fully balanced tree uniquely maximizes the stairs2 index for leaf numbers that are powers of two, understanding the maximum value and maximal trees for arbitrary leaf numbers has been an open problem in the literature. In this note, we fill this gap by showing that for all leaf numbers, there is a unique rooted binary tree maximizing the stairs2 index. Additionally, we obtain recursive and closed expressions for the maximum value of the stairs2 index of a rooted binary tree with <em>n</em> leaves.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"159 ","pages":"Article 102732"},"PeriodicalIF":1.0,"publicationDate":"2024-06-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141484958","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}
Qianghui Guo , Yinglie Jin , Lisa H. Sun , Shina Xu
{"title":"Bijective enumeration of general stacks","authors":"Qianghui Guo , Yinglie Jin , Lisa H. Sun , Shina Xu","doi":"10.1016/j.aam.2024.102722","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102722","url":null,"abstract":"<div><p>Combinatorial enumeration of various RNA secondary structures and protein contact maps is of great interest for both combinatorists and computational biologists. Counting protein contact maps is much more difficult than that of RNA secondary structures due to the significant higher vertex degree. The state of art upper bound for vertex degree in previous works is two. This paper proposes a solution for counting general stacks with arbitrary vertex degree upper bound. By establishing a bijection between such general stacks and <em>m</em>-regular Λ-avoiding <em>DLU</em>-paths, and counting these pattern avoiding lattice paths, we obtain a unified system of equations for the generating functions of the number of general stacks. We further show that previous enumeration results for RNA secondary structures and linear stacks of protein contact maps can be derived from the equations for general stacks as special cases.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"158 ","pages":"Article 102722"},"PeriodicalIF":1.1,"publicationDate":"2024-05-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141240721","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 “Alternatives for the q-matroid axioms of independent spaces, bases, and spanning spaces” [Adv. Appl. Math. 153 (2024) 102632]","authors":"Michela Ceria , Relinde Jurrius","doi":"10.1016/j.aam.2024.102708","DOIUrl":"https://doi.org/10.1016/j.aam.2024.102708","url":null,"abstract":"<div><p>The authors regret that there was a mistake in <span>[1, Definition 26]</span> with our new basis axiom (nB3). We explain and correct this mistake here.</p></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"158 ","pages":"Article 102708"},"PeriodicalIF":1.1,"publicationDate":"2024-05-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S019688582400040X/pdfft?md5=9b50047bfc6da0a9025d496c33117ce3&pid=1-s2.0-S019688582400040X-main.pdf","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"140948756","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}
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