{"title":"Asymmetry of 2-step transit probabilities in 2-coloured regular graphs","authors":"Ron Gray, J. Robert Johnson","doi":"10.1016/j.disc.2025.114645","DOIUrl":"10.1016/j.disc.2025.114645","url":null,"abstract":"<div><div>Suppose that the vertices of a regular graph are coloured red and blue with an equal number of each (we call this a balanced colouring). Since the graph is undirected, the number of edges from a red vertex to a blue vertex is clearly the same as the number of edges from a blue vertex to a red vertex. However, if instead of edges we count walks of length 2 which do not stay within their starting colour class, then this symmetry disappears. Our aim in this paper is to investigate how extreme this asymmetry can be.</div><div>Our main question is: Given a <em>d</em>-regular graph, for which pairs <span><math><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo><mo>∈</mo><msup><mrow><mo>[</mo><mn>0</mn><mo>,</mo><mn>1</mn><mo>]</mo></mrow><mrow><mn>2</mn></mrow></msup></math></span> is there a balanced colouring for which the probability that a random walk starting from a red vertex stays within the red class for at least 2 steps is <em>x</em>, and the corresponding probability for blue is <em>y</em>?</div><div>Our most general result is that for any <em>d</em>-regular graph, these pairs lie within the convex hull of the 2<em>d</em> points <span><math><mo>{</mo><mrow><mo>(</mo><mfrac><mrow><mi>l</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>,</mo><mfrac><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mfrac><mrow><msup><mrow><mi>l</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow><mrow><msup><mrow><mi>d</mi></mrow><mrow><mn>2</mn></mrow></msup></mrow></mfrac><mo>,</mo><mfrac><mrow><mi>l</mi></mrow><mrow><mi>d</mi></mrow></mfrac><mo>)</mo></mrow><mo>:</mo><mn>0</mn><mo>≤</mo><mi>l</mi><mo>≤</mo><mi>d</mi><mo>}</mo></math></span>.</div><div>Our main focus is the torus for which we prove both sharper bounds and existence results via constructions. In particular, for the 2-dimensional torus we show that asymptotically the region in which these pairs of probabilities can lie is exactly the convex hull of:<span><span><span><math><mrow><mo>{</mo><mrow><mo>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mn>16</mn></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mfrac><mrow><mn>9</mn></mrow><mrow><mn>16</mn></mrow></mfrac><mo>,</mo><mfrac><mrow><mn>3</mn></mrow><mrow><mn>4</mn></mrow></mfrac><mo>)</mo></mrow><mo>,</mo><mrow><mo>(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>)</mo></mrow><mo>}</mo></mrow><mo>.</mo></math></span><","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114645"},"PeriodicalIF":0.7,"publicationDate":"2025-06-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313529","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":"Several classes of wide minimal binary linear codes based on general Maiorana-McFarland class","authors":"Xiaoni Du , Siqi Gao , Wenping Yuan , Xingbin Qiao","doi":"10.1016/j.disc.2025.114642","DOIUrl":"10.1016/j.disc.2025.114642","url":null,"abstract":"<div><div>Minimal linear codes have important applications in secret sharing schemes and secure two-party computation. In this paper, we extend the construction of wide minimal binary linear codes presented by Ding et al. <span><span>[8]</span></span> (2018) to a more general case. More specifically, we first construct a class of Boolean functions belonging to the general Maiorana-McFarland class with more flexible parameters. Then we provide a framework for examining the Walsh transform of the new functions via the Krawtchouk polynomial. Finally, we obtain several classes of wide minimal binary linear codes with a few weights and determine their weight distribution explicitly. Our results cover all the related existing ones.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114642"},"PeriodicalIF":0.7,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144313530","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":"Enumeration of sets of equiangular lines with common angle arccos(1/3)","authors":"Kiyoto Yoshino","doi":"10.1016/j.disc.2025.114647","DOIUrl":"10.1016/j.disc.2025.114647","url":null,"abstract":"<div><div>In 2018, Szöllősi and Östergård used a computer to enumerate sets of equiangular lines with common angle <span><math><mi>arccos</mi><mo></mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math></span> in dimension 7. They observed that the numbers <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of sets of <em>n</em> equiangular lines with common angle <span><math><mi>arccos</mi><mo></mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math></span> in dimension 7 are almost symmetric around <span><math><mi>n</mi><mo>=</mo><mn>14</mn></math></span>. In this paper, we prove without a computer that the numbers <span><math><mi>ω</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> are indeed almost symmetric by considering isometries from root lattices of rank at most 8 to the root lattice <span><math><msub><mrow><mi>E</mi></mrow><mrow><mn>8</mn></mrow></msub></math></span> of rank 8 and type <em>E</em>. Also, they determined the number <span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> of sets of <em>n</em> equiangular lines with common angle <span><math><mi>arccos</mi><mo></mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≤</mo><mn>13</mn></math></span>. We construct all the sets of equiangular lines with common angle <span><math><mi>arccos</mi><mo></mo><mo>(</mo><mn>1</mn><mo>/</mo><mn>3</mn><mo>)</mo></math></span> in dimension greater than 7 from root lattices of type <em>A</em> or <em>D</em> with the aid of switching roots. As an application, we determine the number <span><math><mi>s</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span> for every positive integer <em>n</em>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114647"},"PeriodicalIF":0.7,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307727","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":"Packing large balanced trees into bipartite graphs","authors":"Cristina G. Fernandes , Tássio Naia , Giovanne Santos , Maya Stein","doi":"10.1016/j.disc.2025.114641","DOIUrl":"10.1016/j.disc.2025.114641","url":null,"abstract":"<div><div>We prove that for every <span><math><mi>γ</mi><mo>></mo><mn>0</mn></math></span> there exists <span><math><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub><mo>∈</mo><mi>N</mi></math></span> such that for every <span><math><mi>n</mi><mo>≥</mo><msub><mrow><mi>n</mi></mrow><mrow><mn>0</mn></mrow></msub></math></span> any family of up to <span><math><msup><mrow><mi>n</mi></mrow><mrow><mfrac><mrow><mn>1</mn></mrow><mrow><mn>2</mn></mrow></mfrac><mo>−</mo><mi>γ</mi></mrow></msup></math></span> trees having at most <span><math><mo>(</mo><mn>1</mn><mo>−</mo><mi>γ</mi><mo>)</mo><mi>n</mi></math></span> vertices in each bipartition class can be packed into <span><math><msub><mrow><mi>K</mi></mrow><mrow><mi>n</mi><mo>,</mo><mi>n</mi></mrow></msub></math></span>. As a tool for our proof, we show an approximate bipartite version of the Komlós–Sárközy–Szemerédi Theorem, which we believe to be of independent interest.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114641"},"PeriodicalIF":0.7,"publicationDate":"2025-06-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144307726","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 finding the eigenvalues of the matrix of rotation symmetric Boolean functions","authors":"Manuel Albrizzio","doi":"10.1016/j.disc.2025.114627","DOIUrl":"10.1016/j.disc.2025.114627","url":null,"abstract":"<div><div>Cryptographic properties of rotation symmetric Boolean functions can be efficiently computed using a particular square matrix <span><math><mmultiscripts><mrow><mi>A</mi></mrow><mprescripts></mprescripts><mrow><mi>n</mi></mrow><none></none></mmultiscripts></math></span>, the construction of which uses orbit representatives of the cyclic shifting action. In 2018, Ciungu and Iovanov proved that <span><math><mmultiscripts><mrow><mi>A</mi></mrow><none></none><mrow><mn>2</mn></mrow><mprescripts></mprescripts><mrow><mi>n</mi></mrow><none></none></mmultiscripts><mo>=</mo><msup><mrow><mn>2</mn></mrow><mrow><mi>n</mi></mrow></msup><mo>⋅</mo><mi>I</mi></math></span>, the identity matrix of dimension <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>×</mo><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> where <span><math><msub><mrow><mi>g</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> is the number of orbits. In this paper, we answer the open question of the precise number of positive and negative eigenvalues of <span><math><mmultiscripts><mrow><mi>A</mi></mrow><mprescripts></mprescripts><mrow><mi>n</mi></mrow><none></none></mmultiscripts></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114627"},"PeriodicalIF":0.7,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144271203","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":"Hopping forcing number in random d-regular graphs","authors":"Paweł Prałat, Harjas Singh","doi":"10.1016/j.disc.2025.114644","DOIUrl":"10.1016/j.disc.2025.114644","url":null,"abstract":"<div><div>Hopping forcing is a single player combinatorial game in which the player is presented a graph on <em>n</em> vertices, some of which are initially blue with the remaining vertices being white. In each round <em>t</em>, a blue vertex <em>v</em> with all neighbours blue may hop and colour a white vertex blue in the second neighbourhood, provided that <em>v</em> has not performed a hop in the previous <span><math><mi>t</mi><mo>−</mo><mn>1</mn></math></span> rounds. The objective of the game is to eventually colour every vertex blue by repeatedly applying the hopping forcing rule. Subsequently, for a given graph <em>G</em>, the hopping forcing number is the minimum number of initial blue vertices that are required to achieve the objective.</div><div>In this paper, we study the hopping forcing number for random <em>d</em>-regular graphs. Specifically, we aim to derive asymptotic upper and lower bounds for the hopping forcing number for various values of <span><math><mi>d</mi><mo>≥</mo><mn>2</mn></math></span>.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114644"},"PeriodicalIF":0.7,"publicationDate":"2025-06-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144271204","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":"Chip-firing on graphs of groups","authors":"Margaret Meyer, Dmitry Zakharov","doi":"10.1016/j.disc.2025.114631","DOIUrl":"10.1016/j.disc.2025.114631","url":null,"abstract":"<div><div>We define the Laplacian matrix and the Jacobian group of a finite graph of groups. We prove analogues of the matrix tree theorem and the class number formula for the order of the Jacobian of a graph of groups. Given a group <em>G</em> acting on a graph <em>X</em>, we define natural pushforward and pullback maps between the Jacobian groups of <em>X</em> and the quotient graph of groups <span><math><mi>X</mi><mo>/</mo><mo>/</mo><mi>G</mi></math></span>. For the case <span><math><mi>G</mi><mo>=</mo><mi>Z</mi><mo>/</mo><mn>2</mn><mi>Z</mi></math></span>, we also prove a combinatorial formula for the order of the kernel of the pushforward map.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114631"},"PeriodicalIF":0.7,"publicationDate":"2025-06-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144262090","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":"An (F2,F6)-partition of planar graphs without cycles of length 4 and 6","authors":"Ziwen Huang , Xiangwen Li , Lin Niu","doi":"10.1016/j.disc.2025.114626","DOIUrl":"10.1016/j.disc.2025.114626","url":null,"abstract":"<div><div>Let <em>G</em> be a graph with the vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span>, an <span><math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><msub><mrow><mi>d</mi></mrow><mrow><mi>k</mi></mrow></msub></mrow></msub><mo>)</mo></math></span>-partition of <em>G</em> is a partition of its vertex set <span><math><mi>V</mi><mo>(</mo><mi>G</mi><mo>)</mo></math></span> into <em>k</em> sets <span><math><msub><mrow><mi>V</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><mo>…</mo><mo>,</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> such that the graph <span><math><mi>G</mi><mo>[</mo><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub><mo>]</mo></math></span> induced by <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> is a forest with maximum degree at most <span><math><msub><mrow><mi>d</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> for each <span><math><mi>i</mi><mo>∈</mo><mo>{</mo><mn>1</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>k</mi><mo>}</mo></math></span>. Huang et al. (2023) <span><span>[16]</span></span> and Sittitrai and Nakprasit, <span><span>arXiv:2203.06466</span><svg><path></path></svg></span> <span><span>[24]</span></span>, independently, showed that every planar graph without cycles of length 4 and 6 has an <span><math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mi>F</mi><mo>)</mo></math></span>-partition. Huang et al. (2023) <span><span>[16]</span></span> posed a question whether there is a positive integer <em>d</em> such that every planar graph without cycles of length 4 and 6 has an <span><math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mi>d</mi></mrow></msub><mo>)</mo></math></span>-partition. In this paper, we answer affirmatively this question and prove that every planar graph without cycles of length 4 and 6 has an <span><math><mo>(</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><msub><mrow><mi>F</mi></mrow><mrow><mn>6</mn></mrow></msub><mo>)</mo></math></span>-partition, which strengthens the earlier results of Huang, Huang and Lv, and Sittitrai and Nakprasit.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114626"},"PeriodicalIF":0.7,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254159","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":"Bijections in weakly increasing trees via binary trees","authors":"Yang Li, Zhicong Lin","doi":"10.1016/j.disc.2025.114632","DOIUrl":"10.1016/j.disc.2025.114632","url":null,"abstract":"<div><div>As a unification of increasing trees and plane trees, the weakly increasing trees labeled by a multiset was introduced by Lin-Ma-Ma-Zhou in 2021. Motived by some symmetries in plane trees proved recently by Dong, Du, Ji and Zhang, we construct four bijections on weakly increasing trees in the same flavor via switching the role of left child and right child of some specified nodes in their corresponding binary trees. Consequently, bijective proofs of the aforementioned symmetries found by Dong et al. and a non-recursive construction of a bijection on plane trees of Deutsch are provided. Applications of some symmetries in weakly increasing trees to permutation patterns and statistics will also be discussed.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114632"},"PeriodicalIF":0.7,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254157","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":"Packing of the k-power of Hamilton cycles","authors":"Wanfang Chen, Changhong Lu, Qi Wu, Long-Tu Yuan","doi":"10.1016/j.disc.2025.114630","DOIUrl":"10.1016/j.disc.2025.114630","url":null,"abstract":"<div><div>The <em>k</em>-power of a Hamilton cycle is obtained from it by adding edges between all two vertices whose distance in it is at most <em>k</em>. For sufficiently large <em>n</em>, we determine the maximum number of edges of an <em>n</em>-vertex graph without containing the <em>k</em>-power of a Hamilton cycle, and identify all <em>n</em>-vertex graphs with at most <span><math><mi>n</mi><mo>−</mo><mn>2</mn><mi>k</mi><mo>+</mo><mi>ℓ</mi></math></span> edges which do not pack with the <em>k</em>-power of a Hamilton cycle.</div></div>","PeriodicalId":50572,"journal":{"name":"Discrete Mathematics","volume":"348 12","pages":"Article 114630"},"PeriodicalIF":0.7,"publicationDate":"2025-06-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"144254158","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}
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