{"title":"Truncated theta series from the Bailey lattice","authors":"Xiangyu Ding, Lisa Hui Sun","doi":"10.1016/j.aam.2025.102884","DOIUrl":"10.1016/j.aam.2025.102884","url":null,"abstract":"<div><div>In 2012, Andrews and Merca obtained a truncated version of Euler's pentagonal number theorem and showed the nonnegativity related to partition functions. Meanwhile, Andrews and Merca, Guo and Zeng independently conjectured that the truncated Jacobi triple product series has nonnegative coefficients, which has been confirmed analytically and also combinatorially. In 2022, Merca proposed a stronger version for this conjecture. In this paper, by applying Agarwal, Andrews and Bressoud's identity derived from the Bailey lattice, we obtain a truncated version for the Jacobi triple product series with odd basis, which reduces to the Andrews–Gordon identity as a special instance. As consequences, we obtain new truncated forms for Euler's pentagonal number theorem, Gauss' theta series on triangular numbers and square numbers, which lead to inequalities for certain partition functions. Moreover, by considering a truncated theta series involving <em>ℓ</em>-regular partitions, we confirm a conjecture proposed by Ballantine and Merca about 6-regular partitions and show that Merca's stronger conjecture on truncated Jacobi triple product series holds when <span><math><mi>R</mi><mo>=</mo><mn>3</mn><mi>S</mi></math></span> for <span><math><mi>S</mi><mo>≥</mo><mn>1</mn></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102884"},"PeriodicalIF":1.0,"publicationDate":"2025-03-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143704752","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":"Further results on r-Euler-Mahonian statistics","authors":"Kaimei Huang, Sherry H.F. Yan","doi":"10.1016/j.aam.2025.102882","DOIUrl":"10.1016/j.aam.2025.102882","url":null,"abstract":"<div><div>As natural generalizations of the descent number (<span><math><mi>des</mi></math></span>) and the major index (<span><math><mi>maj</mi></math></span>), Rawlings introduced the notions of the <em>r</em>-descent number (<span><math><mi>r</mi><mrow><mi>des</mi></mrow></math></span>) and the <em>r</em>-major index (<span><math><mi>r</mi><mrow><mi>maj</mi></mrow></math></span>) for a given positive integer <em>r</em>. A pair <span><math><mo>(</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>1</mn></mrow></msub><mo>,</mo><msub><mrow><mi>st</mi></mrow><mrow><mn>2</mn></mrow></msub><mo>)</mo></math></span> of permutation statistics is said to be <em>r</em>-Euler-Mahonian if <span><math><mo>(</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>1</mn></mrow></msub></mrow><mo>,</mo><mrow><mi>s</mi><msub><mrow><mi>t</mi></mrow><mrow><mn>2</mn></mrow></msub></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> are equidistributed over the set <span><math><msub><mrow><mi>S</mi></mrow><mrow><mi>n</mi></mrow></msub></math></span> of all permutations of <span><math><mo>{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo>,</mo><mo>…</mo><mo>,</mo><mi>n</mi><mo>}</mo></math></span>. The main objective of this paper is to confirm a recent conjecture posed by Liu which asserts that <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> is <span><math><mo>(</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn><mo>)</mo></math></span>-Euler-Mahonian for all positive integers <em>g</em> and <em>ℓ</em>, where <span><math><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level excedance number and <span><math><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> denotes the <em>g</em>-gap <em>ℓ</em>-level Denert's statistic. This is accomplished via a bijective proof of the equidistribution of <span><math><mo>(</mo><mi>g</mi><msub><mrow><mi>exc</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>,</mo><mi>g</mi><msub><mrow><mi>den</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>)</mo></math></span> and <span><math><mo>(</mo><mi>r</mi><mrow><mi>des</mi></mrow><mo>,</mo><mi>r</mi><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> where <span><math><mi>r</mi><mo>=</mo><mi>g</mi><mo>+</mo><mi>ℓ</mi><mo>−</mo><mn>1</mn></math></span>. Setting <span><math><mi>g</mi><mo>=</mo><mi>ℓ</mi><mo>=</mo><mn>1</mn></math></span>, our result recovers the equidistribution of <span><math><mo>(</mo><mrow><mi>des</mi></mrow><mo>,</mo><mrow><mi>maj</mi></mrow><mo>)</mo></math></span> and <span><math><mo>(</mo><mrow><mi>exc</mi></mrow><mo>,</mo><mrow><mi>den</mi></mrow><mo>)</mo></math></span>, which was first conjectured by Denert and proved by Foata ","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102882"},"PeriodicalIF":1.0,"publicationDate":"2025-03-20","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143686514","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":"k-loose elements and k-paving matroids","authors":"Jagdeep Singh","doi":"10.1016/j.aam.2025.102885","DOIUrl":"10.1016/j.aam.2025.102885","url":null,"abstract":"<div><div>For a matroid of rank <em>r</em> and a non-negative integer <em>k</em>, an element is called <em>k</em>-loose if every circuit containing it has size greater than <span><math><mi>r</mi><mo>−</mo><mi>k</mi></math></span>. Zaslavsky and the author characterized all binary matroids with a 1-loose element. In this paper, we establish a sharp linear bound on the size of a binary matroid, in terms of its rank, that contains a <em>k</em>-loose element. A matroid is called <em>k</em>-paving if all its elements are <em>k</em>-loose. Rajpal showed that for a prime power <em>q</em>, the rank of a <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroid that is <em>k</em>-paving is bounded. We provide a bound on the rank of <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroids that are cosimple and have two <em>k</em>-loose elements. Consequently, we strengthen the result of Rajpal by providing a bound on the rank of <span><math><mi>G</mi><mi>F</mi><mo>(</mo><mi>q</mi><mo>)</mo></math></span>-matroids that are <em>k</em>-paving. Additionally, we provide a bound on the size of binary matroids that are <em>k</em>-paving.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102885"},"PeriodicalIF":1.0,"publicationDate":"2025-03-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143644876","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":"Polynomial resultants and Ramsey numbers of a theta graph","authors":"Meng Liu , Ye Wang","doi":"10.1016/j.aam.2025.102881","DOIUrl":"10.1016/j.aam.2025.102881","url":null,"abstract":"<div><div>Let <span><math><msub><mrow><mi>θ</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow></msub></math></span> be the graph consisting of three internally disjoint paths of length four sharing common endpoints. It is shown <span><math><msub><mrow><mi>R</mi></mrow><mrow><mi>k</mi></mrow></msub><mo>(</mo><msub><mrow><mi>θ</mi></mrow><mrow><mn>3</mn><mo>,</mo><mn>4</mn></mrow></msub><mo>)</mo><mo>=</mo><mi>Θ</mi><mo>(</mo><msup><mrow><mi>k</mi></mrow><mrow><mn>4</mn><mo>/</mo><mn>3</mn></mrow></msup><mo>)</mo></math></span> as <span><math><mi>k</mi><mo>→</mo><mo>∞</mo></math></span> by computing polynomial resultants.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102881"},"PeriodicalIF":1.0,"publicationDate":"2025-03-18","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143642896","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":"Linear orbits of smooth quadric surfaces","authors":"Franquiz Caraballo Alba","doi":"10.1016/j.aam.2025.102880","DOIUrl":"10.1016/j.aam.2025.102880","url":null,"abstract":"<div><div>The <em>linear orbit</em> of a degree <em>d</em> hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> is its orbit under the natural action of <span><math><mi>P</mi><mi>GL</mi><mo>(</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>, in the projective space of dimension <span><math><mi>N</mi><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>d</mi></mrow></mtd></mtr><mtr><mtd><mi>d</mi></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><mn>1</mn></math></span> parameterizing such hypersurfaces. This action restricted to a specific hypersurface <em>X</em> extends to a rational map from the projectivization of the space of matrices to <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>N</mi></mrow></msup></math></span>. The class of the graph of this map is the <em>predegree polynomial</em> of its corresponding hypersurface. The objective of this paper is threefold. First, we formally define the predegree polynomial of a hypersurface in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>, introduced in the case of plane curves by Aluffi and Faber, and prove some results in the general case. A key result in the general setting is that a partial resolution of said rational map can contain enough information to compute the predegree polynomial of a hypersurface. Second, we compute the leading term of the predegree polynomial of a smooth quadric in <span><math><msup><mrow><mi>P</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span> over an algebraically closed field with characteristic 0, and compute the other coefficients in the specific case <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>. In analogy to Aluffi and Faber's work, the tool for computing this invariant is producing a (partial) resolution of the previously mentioned rational map which contains enough information to obtain the invariant. Third, we provide a complete resolution of the rational map in the case <span><math><mi>n</mi><mo>=</mo><mn>3</mn></math></span>, which in principle could be used to compute more refined invariants.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102880"},"PeriodicalIF":1.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636655","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":"Rotational hypersurfaces family satisfying Ln−3G=AG in the n-dimensional Euclidean space","authors":"Erhan Güler , Nurettin Cenk Turgay","doi":"10.1016/j.aam.2025.102879","DOIUrl":"10.1016/j.aam.2025.102879","url":null,"abstract":"<div><div>In this paper, we investigate rotational hypersurfaces family in <em>n</em>-dimensional Euclidean space <span><math><msup><mrow><mi>E</mi></mrow><mrow><mi>n</mi></mrow></msup></math></span>. Our focus is on studying the Gauss map <span><math><mi>G</mi></math></span> of this family with respect to the operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span>, which acts on functions defined on the hypersurfaces. The operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>k</mi></mrow></msub></math></span> can be viewed as a modified Laplacian and is known by various names, including the Cheng–Yau operator in certain cases. Specifically, we focus on the scenario where <span><math><mi>k</mi><mo>=</mo><mi>n</mi><mo>−</mo><mn>3</mn></math></span> and <span><math><mi>n</mi><mo>≥</mo><mn>3</mn></math></span>. By applying the operator <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub></math></span> to the Gauss map <span><math><mi>G</mi></math></span>, we establish a classification theorem. This theorem establishes a connection between the <span><math><mi>n</mi><mo>×</mo><mi>n</mi></math></span> matrix <span><math><mi>A</mi></math></span>, and the Gauss map <span><math><mi>G</mi></math></span> through the equation <span><math><msub><mrow><mi>L</mi></mrow><mrow><mi>n</mi><mo>−</mo><mn>3</mn></mrow></msub><mi>G</mi><mo>=</mo><mi>A</mi><mi>G</mi></math></span>.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102879"},"PeriodicalIF":1.0,"publicationDate":"2025-03-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143636651","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":"Semi-coarse spaces, homotopy and homology","authors":"Antonio Rieser, Jonathan Treviño-Marroquín","doi":"10.1016/j.aam.2025.102870","DOIUrl":"10.1016/j.aam.2025.102870","url":null,"abstract":"<div><div>We begin the study the algebraic topology of semi-coarse spaces, which are generalizations of coarse spaces that enable one to endow non-trivial ‘coarse-like’ structures to compact metric spaces, something which is impossible in coarse geometry. We first study homotopy in this context, and we then construct homology groups which are invariant under semi-coarse homotopy equivalence. We further show that any undirected graph <span><math><mi>G</mi><mo>=</mo><mo>(</mo><mi>V</mi><mo>,</mo><mi>E</mi><mo>)</mo></math></span> induces a semi-coarse structure on its set of vertices <span><math><msub><mrow><mi>V</mi></mrow><mrow><mi>G</mi></mrow></msub></math></span>, and that the respective semi-coarse homology is isomorphic to the Vietoris-Rips homology. This, in turn, leads to a homotopy invariance theorem for the Vietoris-Rips homology of undirected graphs.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"167 ","pages":"Article 102870"},"PeriodicalIF":1.0,"publicationDate":"2025-02-27","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143510565","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":"Automatic sequences and parity of partition functions","authors":"Shi-Chao Chen","doi":"10.1016/j.aam.2025.102869","DOIUrl":"10.1016/j.aam.2025.102869","url":null,"abstract":"<div><div>Let <span><math><mi>k</mi><mo>≥</mo><mn>2</mn></math></span> be an integer, <em>ℓ</em> a prime and <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub></math></span> the finite field with <em>ℓ</em> elements. A sequence <span><math><msub><mrow><mo>(</mo><mi>a</mi><mo>(</mo><mi>n</mi><mo>)</mo><mo>)</mo></mrow><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow></msub></math></span> is called <em>k</em>-automatic if there exists a deterministic finite automaton with output that reads the canonical base-<em>k</em> representation of <em>n</em> and the outputs <span><math><mi>a</mi><mo>(</mo><mi>n</mi><mo>)</mo></math></span>. We apply the properties of automatic sequences to prove the transcendence of a formal power series over <span><math><msub><mrow><mi>F</mi></mrow><mrow><mi>ℓ</mi></mrow></msub><mo>(</mo><mi>X</mi><mo>)</mo></math></span> related to infinite products. As applications, the parity results of various partition functions are obtained, including the root partition function and the prime parts partition function. We also establish the transcendence of the power series associated with holomorphic modular forms with integer coefficients.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102869"},"PeriodicalIF":1.0,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480729","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":"Stable multivariate Narayana polynomials and labeled plane trees","authors":"Harold R.L. Yang , Philip B. Zhang","doi":"10.1016/j.aam.2025.102867","DOIUrl":"10.1016/j.aam.2025.102867","url":null,"abstract":"<div><div>In this paper, we introduce stable multivariate generalizations of Narayana polynomials of types <em>A</em> and <em>B</em>. We give an insertion algorithm for labeled plane trees and introduce the notion of improper edges. Our polynomials are multivariate generating polynomials of labeled plane trees and can be generated by a grammatical labeling based on a context-free grammar. Our proof of real stability uses a characterization of stable-preserving linear operators due to Borcea and Brändén. In particular, we get an alternative multivariate stable refinement of the second-order Eulerian polynomials, which is different from the one given by Haglund and Visontai.</div></div>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102867"},"PeriodicalIF":1.0,"publicationDate":"2025-02-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143480728","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":"Coefficients of the Tutte polynomial and minimal edge cuts of a graph","authors":"Haiyan Chen, Mingxu Guo","doi":"10.1016/j.aam.2025.102868","DOIUrl":"10.1016/j.aam.2025.102868","url":null,"abstract":"<div><div>Let <em>G</em> be a <span><math><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></math></span>-edge connected graph with order <em>n</em> and size <em>m</em>. From a general result on the coefficients of polymatroid Tutte polynomial, Guan et al. (2023) <span><span>[16]</span></span> derived that<span><span><span><math><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>g</mi><mo>−</mo><mi>i</mi></mrow></msup><mo>]</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>,</mo><mspace></mspace><mspace></mspace><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi><mo>,</mo></math></span></span></span> where <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>)</mo></math></span> is the Tutte polynomial of <em>G</em> and <span><math><mi>g</mi><mo>=</mo><mi>m</mi><mo>−</mo><mi>n</mi><mo>+</mo><mn>1</mn></math></span>. Recall that the coefficients of <span><math><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>)</mo></math></span> have many combinatorial explanations, including spanning trees, parking functions, superstable configurations (or recurrent configurations) of the Abelian Sandpile Model (ASM), and so on. Here we find that the above result has a simple and direct proof in terms of the superstable configurations of ASM. Motivated by this, in this paper, by constructing mappings between different sets, we first establish a relationship between non-superstable configurations and minimal edge cuts of <em>G</em>, then we generalize the above result from <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>k</mi></math></span> to <span><math><mn>0</mn><mo>≤</mo><mi>i</mi><mo><</mo><mfrac><mrow><mn>3</mn><mo>(</mo><mi>k</mi><mo>+</mo><mn>1</mn><mo>)</mo></mrow><mrow><mn>2</mn></mrow></mfrac></math></span>. In precise,<span><span><span><math><mo>[</mo><msup><mrow><mi>y</mi></mrow><mrow><mi>g</mi><mo>−</mo><mi>i</mi></mrow></msup><mo>]</mo><msub><mrow><mi>T</mi></mrow><mrow><mi>G</mi></mrow></msub><mo>(</mo><mn>1</mn><mo>,</mo><mi>y</mi><mo>)</mo><mo>=</mo><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>−</mo><munderover><mo>∑</mo><mrow><mi>j</mi><mo>=</mo><mi>k</mi><mo>+</mo><mn>1</mn></mrow><mrow><mi>i</mi></mrow></munderover><mrow><mo>(</mo><mtable><mtr><mtd><mrow><mi>n</mi><mo>+</mo><mi>i</mi><mo>−</mo><mi>j</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr><mtr><mtd><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></mtd></mtr></mtable><mo>)</mo></mrow><mo>|</mo><msub><mrow><mi>EC</mi></mrow><mrow><mi>j</mi></mrow>","PeriodicalId":50877,"journal":{"name":"Advances in Applied Mathematics","volume":"166 ","pages":"Article 102868"},"PeriodicalIF":1.0,"publicationDate":"2025-02-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"143453998","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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