{"title":"Cells of fixed height in Catalan words and restricted growth functions","authors":"Aubrey Blecher, Arnold Knopfmacher","doi":"10.1016/j.aam.2024.102835","DOIUrl":"10.1016/j.aam.2024.102835","url":null,"abstract":"<div><div>A word <span><math><mi>w</mi><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of length <em>n</em> over the set of positive integers is called a Catalan word whenever <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>+</mo><mn>1</mn></math></span> for <span><math><mi>k</mi><mo>=</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi></math></span>. A restricted growth function is defined as a word <span><math><mi>w</mi><mo>=</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>⋯</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of length <em>n</em> over the set of positive integers where <span><math><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>=</mo><mn>1</mn></math></span> and for <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> we have <span><math><mn>1</mn><mo>≤</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>≤</mo><mi>max</mi><mo></mo><mo>{</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>,</mo><mo>⋯</mo><mo>,</mo><msub><mrow><mi>w</mi></mrow><mrow><mi>k</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>}</mo><mo>+</mo><mn>1</mn></math></span>. We also define cells and heights of cells and we represent such words as bargraphs (otherwise known as polyominoes) where the <em>i</em>th column contains <span><math><msub><mrow><mi>w</mi></mrow><mrow><mi>i</mi></mrow></msub></math></span> cells for <span><math><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi></math></span> and where all columns have their bottom cell on the <em>x</em>-axis. In the case of Catalan words, we prove a relationship between the number of cells at different heights and first terms of the expanded polynomial <span><math><msup><mrow><mo>(</mo><mn>1</mn><mo>+</mo><mi>x</mi><mo>)</mo></mrow><mrow><mn>2</mn><mi>n</mi></mrow></msup></math></span>. In the case of restricted growth functions we find polynomials <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>)</mo></math></span> where the coefficient of <span><math><msup><mrow><mi>x</mi></mrow><mrow><mi>j</mi></mrow></msup></math></span> counts the number of cells of height <em>j</em> across all rgfs with <em>n</em> parts. In this case we also find bivariate generating functions for rgfs with <em>k</em> blocks, where the generating functions tracks the number of cells at a given height as well as the number of parts.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102835"},"PeriodicalIF":1.0,"publicationDate":"2024-12-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154472","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":"Conditions for virtually Cohen–Macaulay simplicial complexes","authors":"Adam Van Tuyl , Jay Yang","doi":"10.1016/j.aam.2024.102830","DOIUrl":"10.1016/j.aam.2024.102830","url":null,"abstract":"<div><div>A simplicial complex Δ is a virtually Cohen–Macaulay simplicial complex if its associated Stanley-Reisner ring <em>S</em> has a virtual resolution, as defined by Berkesch, Erman, and Smith, of length <span><math><mrow><mi>codim</mi></mrow><mo>(</mo><mi>S</mi><mo>)</mo></math></span>. We provide a sufficient condition on Δ to be a virtually Cohen–Macaulay simplicial complex. We also introduce virtually shellable simplicial complexes, a generalization of shellable simplicial complexes. Virtually shellable complexes have the property that they are virtually Cohen–Macaulay, generalizing the well-known fact that shellable simplicial complexes are Cohen–Macaulay.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102830"},"PeriodicalIF":1.0,"publicationDate":"2024-12-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154474","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":"Dynamics of pop-tsack torsing","authors":"Anqi Li","doi":"10.1016/j.aam.2024.102826","DOIUrl":"10.1016/j.aam.2024.102826","url":null,"abstract":"<div><div>For a finite irreducible Coxeter group <span><math><mo>(</mo><mi>W</mi><mo>,</mo><mi>S</mi><mo>)</mo></math></span> with a fixed Coxeter element <em>c</em> and set of reflections <em>T</em>, Defant and Williams define a pop-tsack torsing operation <span><math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>:</mo><mi>W</mi><mo>→</mo><mi>W</mi></math></span> given by <span><math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><mi>w</mi><mo>⋅</mo><mi>π</mi><msup><mrow><mo>(</mo><mi>w</mi><mo>)</mo></mrow><mrow><mo>−</mo><mn>1</mn></mrow></msup></math></span> where <span><math><mi>π</mi><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><msubsup><mrow><mo>⋁</mo></mrow><mrow><mi>t</mi><msub><mrow><mo>≤</mo></mrow><mrow><mi>T</mi></mrow></msub><mi>w</mi><mo>,</mo><mspace></mspace><mi>t</mi><mo>∈</mo><mi>T</mi></mrow><mrow><mi>N</mi><mi>C</mi><mo>(</mo><mi>w</mi><mo>,</mo><mi>c</mi><mo>)</mo></mrow></msubsup><mi>t</mi></math></span> is the join of all reflections lying below <em>w</em> in the absolute order in the non-crossing partition lattice <span><math><mi>N</mi><mi>C</mi><mo>(</mo><mi>w</mi><mo>,</mo><mi>c</mi><mo>)</mo></math></span>. This is a “dual” notion of the pop-stack sorting operator <span><math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></math></span>; <span><math><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>S</mi></mrow></msub></mrow></math></span> was introduced by Defant as a way to generalize the pop-stack sorting operator on <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> to general Coxeter groups. Define the forward orbit of an element <span><math><mi>w</mi><mo>∈</mo><mi>W</mi></math></span> to be <span><math><msub><mrow><mi>O</mi></mrow><mrow><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow></mrow></msub><mo>(</mo><mi>w</mi><mo>)</mo><mo>=</mo><mo>{</mo><mi>w</mi><mo>,</mo><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mo>(</mo><mi>w</mi><mo>)</mo><mo>,</mo><msup><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow><mrow><mn>2</mn></mrow></msup><mo>(</mo><mi>w</mi><mo>)</mo><mo>,</mo><mo>…</mo><mo>}</mo></math></span>. Defant and Williams established the length of the longest possible forward orbits <span><math><msub><mrow><mi>max</mi></mrow><mrow><mi>w</mi><mo>∈</mo><mi>W</mi></mrow></msub><mo></mo><mo>|</mo><msub><mrow><mi>O</mi></mrow><mrow><mrow><mi>Po</mi><msub><mrow><mi>p</mi></mrow><mrow><mi>T</mi></mrow></msub></mrow></mrow></msub><mo>(</mo><mi>w</mi><mo>)</mo><mo>|</mo></math></span> for Coxeter groups of coincidental types and Type D in terms of the corresponding Coxeter number of the group. In their paper, they also proposed multiple conjectures about enumerating elements with near maximal orbit length. We resolve all the ","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102826"},"PeriodicalIF":1.0,"publicationDate":"2024-12-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154477","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":"Letter frequency vs factor frequency in pure morphic words","authors":"Shuo Li","doi":"10.1016/j.aam.2024.102834","DOIUrl":"10.1016/j.aam.2024.102834","url":null,"abstract":"<div><div>We prove that, for any pure morphic word <em>w</em>, if the frequencies of all letters in <em>w</em> exist, then the frequencies of all factors in <em>w</em> exist as well. This result answers a question of Saari in his doctoral thesis.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102834"},"PeriodicalIF":1.0,"publicationDate":"2024-12-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154470","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":"Initial values of ML-degree polynomials","authors":"Maciej Gałązka","doi":"10.1016/j.aam.2024.102831","DOIUrl":"10.1016/j.aam.2024.102831","url":null,"abstract":"<div><div>We prove a conjecture about the initial values of ML-degree polynomials stated by Michałek, Monin, and Wiśniewski.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102831"},"PeriodicalIF":1.0,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154473","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":"Chordal matroids arising from generalized parallel connections II","authors":"James Dylan Douthitt, James Oxley","doi":"10.1016/j.aam.2024.102833","DOIUrl":"10.1016/j.aam.2024.102833","url":null,"abstract":"<div><div>In 1961, Dirac showed that chordal graphs are exactly the graphs that can be constructed from complete graphs by a sequence of clique-sums. In an earlier paper, by analogy with Dirac's result, we introduced the class of <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-chordal matroids as those matroids that can be constructed from projective geometries over <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span> by a sequence of generalized parallel connections across projective geometries over <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>. Our main result showed that when <span><math><mi>q</mi><mo>=</mo><mn>2</mn></math></span>, such matroids have no induced minor in <span><math><mo>{</mo><mi>M</mi><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>,</mo><mi>M</mi><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo><mo>}</mo></math></span>. In this paper, we show that the class of <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mn>2</mn><mo>)</mo></math></span>-chordal matroids coincides with the class of binary matroids that have none of <span><math><mi>M</mi><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>4</mn></mrow></msub><mo>)</mo></math></span>, <span><math><msup><mrow><mi>M</mi></mrow><mrow><mo>⁎</mo></mrow></msup><mo>(</mo><msub><mrow><mi>K</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>3</mn></mrow></msub><mo>)</mo></math></span>, or <span><math><mi>M</mi><mo>(</mo><msub><mrow><mi>C</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>)</mo></math></span> for <span><math><mi>n</mi><mo>≥</mo><mn>4</mn></math></span> as a flat. We also show that <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-chordal matroids can be characterized by an analogous result to Rose's 1970 characterization of chordal graphs as those that have a perfect elimination ordering of vertices.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102833"},"PeriodicalIF":1.0,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143154471","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":"Consecutive pattern containment and c-Wilf equivalence","authors":"Reza Rastegar","doi":"10.1016/j.aam.2024.102829","DOIUrl":"10.1016/j.aam.2024.102829","url":null,"abstract":"<div><div>We offer elementary proofs for several results in consecutive pattern containment that were previously demonstrated using ideas from cluster method and analytical combinatorics. Furthermore, we establish new general bounds on the growth rates of consecutive pattern avoidance in permutations.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102829"},"PeriodicalIF":1.0,"publicationDate":"2024-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143153922","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":"Simultaneous orderings and combinatorial interpretations of generalized factorials and binomial coefficients","authors":"Lan Nguyen","doi":"10.1016/j.aam.2024.102810","DOIUrl":"10.1016/j.aam.2024.102810","url":null,"abstract":"<div><div>In this paper, we answer several questions raised by Manjul Bhargava concerning <em>p</em>-orderings, <em>p</em>-sequences, and combinatorial interpretations of generalized factorials and binomial coefficients associated with subsets <em>S</em> of <span><math><mi>Z</mi></math></span>. First, we prove some results of Bhargava concerning <em>p</em>-orderings, <em>p</em>-sequences and the integrality of generalized binomial coefficients directly from definitions, answering two questions of Bhargava. As a result, we also obtain some explicit descriptions of the <em>p</em>-sequences associated with <em>p</em>-orderings which do not exist in Bhargava's proofs. Second, we provide some combinatorial interpretations for the generalized factorials and the generalized binomial coefficients associated with subsets of <span><math><mi>Z</mi></math></span> which possess simultaneous <em>p</em>-orderings, answering two other questions raised by Bhargava.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"164 ","pages":"Article 102810"},"PeriodicalIF":1.0,"publicationDate":"2024-12-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143153923","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":"A direct proof of well-definedness for the polymatroid Tutte polynomial","authors":"Xiaxia Guan , Xian'an Jin","doi":"10.1016/j.aam.2024.102809","DOIUrl":"10.1016/j.aam.2024.102809","url":null,"abstract":"<div><div>For a polymatroid <em>P</em> over <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, Bernardi et al. (2022) <span><span>[1]</span></span> introduced the polymatroid Tutte polynomial <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> relying on the order <span><math><mn>1</mn><mo><</mo><mn>2</mn><mo><</mo><mo>⋯</mo><mo><</mo><mi>n</mi></math></span> of <span><math><mo>[</mo><mi>n</mi><mo>]</mo></math></span>, which generalizes the classical Tutte polynomial from matroids to polymatroids. They proved the independence of this order by the fact that <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>P</mi></mrow></msub></math></span> is equivalent to another polynomial that only depends on <em>P</em>. In this paper, similar to the Tutte's original proof of the well-definedness of the Tutte polynomial defined by the summation over all spanning trees using activities depending on the order of edges, we give a direct and elementary proof of the well-definedness of the polymatroid Tutte polynomial.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102809"},"PeriodicalIF":1.0,"publicationDate":"2024-11-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142722289","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":"Existence of solutions to the even Gaussian dual Minkowski problem","authors":"Yibin Feng , Shengnan Hu , Lei Xu","doi":"10.1016/j.aam.2024.102808","DOIUrl":"10.1016/j.aam.2024.102808","url":null,"abstract":"<div><div>In this paper, we consider the Gaussian dual Minkowski problem. The problem involves a new type of fully nonlinear partial differential equations on the unit sphere. Our main purpose is to show the existence of solutions to the even Gaussian dual Minkowski problem for <span><math><mi>q</mi><mo>></mo><mn>0</mn></math></span>. More precisely, we will show that there exists an origin-symmetric convex body <em>K</em> in <span><math><msup><mrow><mi>R</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> such that its Gaussian dual curvature measure <span><math><msub><mrow><mover><mrow><mi>C</mi></mrow><mrow><mo>˜</mo></mrow></mover></mrow><mrow><msub><mrow><mi>γ</mi></mrow><mrow><mi>n</mi></mrow></msub><mo>,</mo><mi>q</mi></mrow></msub><mo>(</mo><mi>K</mi><mo>,</mo><mo>⋅</mo><mo>)</mo></math></span> has density <em>f</em> (up to a constant) on the unit sphere when <span><math><mi>q</mi><mo>></mo><mn>0</mn></math></span> and <em>f</em> has positive upper and lower bounds. Note that if <em>f</em> is smooth then <em>K</em> is also smooth. As the application of smooth solutions, we completely solve the even Gaussian dual Minkowski problem for <span><math><mi>q</mi><mo>></mo><mn>0</mn></math></span> based on an approximation argument.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"163 ","pages":"Article 102808"},"PeriodicalIF":1.0,"publicationDate":"2024-11-25","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"142703820","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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