We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on small rank-metric lattices whose characteristic polynomial we cannot compute, and provide a formula for them under a polynomiality assumption on their Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.
{"title":"Rank-Metric Lattices","authors":"Giuseppe Cotardo, Alberto Ravagnani","doi":"10.37236/11373","DOIUrl":"https://doi.org/10.37236/11373","url":null,"abstract":"We introduce the class of rank-metric geometric lattices and initiate the study of their structural properties. Rank-metric lattices can be seen as the $q$-analogues of higher-weight Dowling lattices, defined by Dowling himself in 1971. We fully characterize the supersolvable rank-metric lattices and compute their characteristic polynomials. We then concentrate on small rank-metric lattices whose characteristic polynomial we cannot compute, and provide a formula for them under a polynomiality assumption on their Whitney numbers of the first kind. The proof relies on computational results and on the theory of vector rank-metric codes, which we review in this paper from the perspective of rank-metric lattices. More precisely, we introduce the notion of lattice-rank weights of a rank-metric code and investigate their properties as combinatorial invariants and as code distinguishers for inequivalent codes.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"4 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-01-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"135897990","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}
Pub Date : 2023-01-05DOI: 10.48550/arXiv.2301.02020
N. Bousquet, Bastien Durain, Théo Pierron, St'ephan Thomass'e
The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent. Extremal problems on independent sets are widely studied: for example, it is well known that an $n$-vertex graph has at most $3^{n/3}$ maximum independent sets (and this is tight). This paper investigates the asymptotic behavior of maximum possible length of a shortest reconfiguration sequence for independent sets of size $k$ among all $n$-vertex graphs. We give a tight bound for $k=2$. We also provide a subquadratic upper bound (using the hypergraph removal lemma) as well as an almost tight construction for $k=3$. We generalize our results for larger values of $k$ by proving an $n^{2lfloor k/3 rfloor}$ lower bound.
{"title":"Extremal Independent Set Reconfiguration","authors":"N. Bousquet, Bastien Durain, Théo Pierron, St'ephan Thomass'e","doi":"10.48550/arXiv.2301.02020","DOIUrl":"https://doi.org/10.48550/arXiv.2301.02020","url":null,"abstract":"The independent set reconfiguration problem asks whether one can transform one given independent set of a graph into another, by changing vertices one by one in such a way the intermediate sets remain independent. Extremal problems on independent sets are widely studied: for example, it is well known that an $n$-vertex graph has at most $3^{n/3}$ maximum independent sets (and this is tight). This paper investigates the asymptotic behavior of maximum possible length of a shortest reconfiguration sequence for independent sets of size $k$ among all $n$-vertex graphs. We give a tight bound for $k=2$. We also provide a subquadratic upper bound (using the hypergraph removal lemma) as well as an almost tight construction for $k=3$. We generalize our results for larger values of $k$ by proving an $n^{2lfloor k/3 rfloor}$ lower bound.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"PP 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2023-01-05","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"84163349","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}
We give a natural definition of rowmotion for $321$-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne--Kreweras involution, the analogous notion for antichains of the positive root poset of type $A$. We prove that some permutation statistics, such as the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. Our setting also provides a more natural description of the celebrated Armstrong--Stump--Thomas equivariant bijection between antichains and non-crossing matchings in types $A$ and $B$, by showing that it is equivalent to the Robinson--Schensted--Knuth correspondence on $321$-avoiding permutations permutations.
通过引入Dyck路径的双射和Lalanne—Kreweras对合，我们给出了$321$-避免置换的行运动的一个自然定义，这与类型$ a $的正根序集的反链的类似概念是等价的。我们证明了一些排列统计量，如不动点的数目，在行运动下是同调的，这意味着它们在其轨道上有一个常数平均值。我们的设置也提供了一个更自然的描述著名的Armstrong- Stump- Thomas等变双射之间的反链和非交叉匹配类型$ a $和$B$，通过表明它是等价于$321$-避免排列排列的Robinson- Schensted- Knuth对应。
{"title":"Rowmotion on 321-Avoiding Permutations","authors":"Ben Adenbaum, S. Elizalde","doi":"10.37236/11792","DOIUrl":"https://doi.org/10.37236/11792","url":null,"abstract":"We give a natural definition of rowmotion for $321$-avoiding permutations, by translating, through bijections involving Dyck paths and the Lalanne--Kreweras involution, the analogous notion for antichains of the positive root poset of type $A$. We prove that some permutation statistics, such as the number of fixed points, are homomesic under rowmotion, meaning that they have a constant average over its orbits. \u0000Our setting also provides a more natural description of the celebrated Armstrong--Stump--Thomas equivariant bijection between antichains and non-crossing matchings in types $A$ and $B$, by showing that it is equivalent to the Robinson--Schensted--Knuth correspondence on $321$-avoiding permutations permutations.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"389 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"80796629","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}
An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of $r$-uniform hypergraphs $mathcal{F}$, the linear Turán number ex$_r^{lin}(n,mathcal{F})$ is the maximum number of edges of a linear $r$-uniform hypergraph on $n$ vertices that does not contain any member of $mathcal{F}$ as a subgraph. Let $K_l$ be a complete graph with $l$ vertices and $rgeq 2$. The $r$-expansion of $K_l$ is the $r$-graph $K_l^+$ obtained from $K_l$ by enlarging each edge of $K_l$ with $r-2$ new vertices disjoint from $V(K_l)$ such that distinct edges of $K_l$ are enlarged by distinct vertices. When $lgeq r geq 3$ and $n$ is sufficiently large, we prove the following extension of Turán's Theorem $$ex_{r}^{lin}left(n, K_{l+1}^{+}right)leq |TD_r(n,l)|,$$ with equality achieved only by the Turán design $TD_r(n,l)$, where the Turán design $TD_r(n,l)$ is an almost balanced $l$-partite $r$-graph such that each pair of vertices from distinct parts are contained in one edge exactly. Moreover, some results on linear Turán number of general configurations are also presented.
{"title":"A Linear Hypergraph Extension of Turán's Theorem","authors":"Guorong Gao, A. Chang","doi":"10.37236/10525","DOIUrl":"https://doi.org/10.37236/10525","url":null,"abstract":"An $r$-uniform hypergraph is linear if every two edges intersect in at most one vertex. Given a family of $r$-uniform hypergraphs $mathcal{F}$, the linear Turán number ex$_r^{lin}(n,mathcal{F})$ is the maximum number of edges of a linear $r$-uniform hypergraph on $n$ vertices that does not contain any member of $mathcal{F}$ as a subgraph. \u0000Let $K_l$ be a complete graph with $l$ vertices and $rgeq 2$. The $r$-expansion of $K_l$ is the $r$-graph $K_l^+$ obtained from $K_l$ by enlarging each edge of $K_l$ with $r-2$ new vertices disjoint from $V(K_l)$ such that distinct edges of $K_l$ are enlarged by distinct vertices. When $lgeq r geq 3$ and $n$ is sufficiently large, we prove the following extension of Turán's Theorem $$ex_{r}^{lin}left(n, K_{l+1}^{+}right)leq |TD_r(n,l)|,$$ with equality achieved only by the Turán design $TD_r(n,l)$, where the Turán design $TD_r(n,l)$ is an almost balanced $l$-partite $r$-graph such that each pair of vertices from distinct parts are contained in one edge exactly. Moreover, some results on linear Turán number of general configurations are also presented.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"11 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87609445","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}
A finite generalized $2d$-gon of order $(s,t)$ with $d in { 2,3,4 }$ and $s not= 1$ is called extremal if $t$ attains its maximal possible value $s^{e_d}$, where $e_2=e_4=2$ and $e_3=3$. The problem of finding combinatorial conditions that are both necessary and sufficient for a finite generalized $2d$-gon of order $(s,t)$ to be extremal has so far only be solved for the generalized quadrangles. In this paper, we obtain a solution for the generalized hexagons. We also obtain a related combinatorial characterization for extremal regular near hexagons.
{"title":"A Combinatorial Characterization of Extremal Generalized Hexagons","authors":"B. Bruyn","doi":"10.37236/9245","DOIUrl":"https://doi.org/10.37236/9245","url":null,"abstract":"A finite generalized $2d$-gon of order $(s,t)$ with $d in { 2,3,4 }$ and $s not= 1$ is called extremal if $t$ attains its maximal possible value $s^{e_d}$, where $e_2=e_4=2$ and $e_3=3$. The problem of finding combinatorial conditions that are both necessary and sufficient for a finite generalized $2d$-gon of order $(s,t)$ to be extremal has so far only be solved for the generalized quadrangles. In this paper, we obtain a solution for the generalized hexagons. We also obtain a related combinatorial characterization for extremal regular near hexagons.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"5 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75390365","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}
Let $C_k$ be a cycle of order $k$, where $kge 3$. Let ex$(n, n, n, {C_{3}, C_{4}})$ be the maximum number of edges in a balanced $3$-partite graph whose vertex set consists of three parts, each has $n$ vertices that has no subgraph isomorphic to $C_3$ or $C_4$. We construct dense balanced 3-partite graphs without 3-cycles or 4-cycles and show that ex$(n, n, n, {C_{3}, C_{4}})ge (frac{6sqrt{2}-8}{(sqrt{2}-1)^{3/2}}+o(1))n^{3/2}$.
设$C_k$为顺序的一个循环$k$，其中$kge 3$。设ex $(n, n, n, {C_{3}, C_{4}})$为平衡的$3$部图的最大边数，该图的顶点集由三个部分组成，每个部分都有$n$个顶点，并且没有同$C_3$或$C_4$同构的子图。我们构造了没有3环和4环的稠密平衡3部图，并证明了ex $(n, n, n, {C_{3}, C_{4}})ge (frac{6sqrt{2}-8}{(sqrt{2}-1)^{3/2}}+o(1))n^{3/2}$。
{"title":"Density of Balanced 3-Partite Graphs without 3-Cycles or 4-Cycles","authors":"Zequn Lv, Mei Lu, Chunqiu Fang","doi":"10.37236/10958","DOIUrl":"https://doi.org/10.37236/10958","url":null,"abstract":"Let $C_k$ be a cycle of order $k$, where $kge 3$. Let ex$(n, n, n, {C_{3}, C_{4}})$ be the maximum number of edges in a balanced $3$-partite graph whose vertex set consists of three parts, each has $n$ vertices that has no subgraph isomorphic to $C_3$ or $C_4$. We construct dense balanced 3-partite graphs without 3-cycles or 4-cycles and show that ex$(n, n, n, {C_{3}, C_{4}})ge (frac{6sqrt{2}-8}{(sqrt{2}-1)^{3/2}}+o(1))n^{3/2}$.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"16 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82731889","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}
Given a graph $G$ and an odd prime $p$, for a mapping $f: E(G) to {mathbb Z}_psetminus{0}$ and a ${mathbb Z}_p$-boundary $b$ of $G$, an orientation $tau$ is called an $(f,b;p)$-orientation if the net out $f$-flow is the same as $b(v)$ in ${mathbb Z}_p$ at each vertex $vin V(G)$ under orientation $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations and closely related to Tutte's nowhere zero 3-flow conjecture. They proved that $(6p^2 - 14p + 8)$-edge-connected graphs have all possible $(f,b;p)$-orientations. In this paper, the framework of such orientations is extended to signed graph through additive bases. We also study the $(f,b;p)$-orientation problem for some (signed) graphs families including complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.
给定一个图$G$和一个奇数素数$p$，对于映射$f: E(G) 到$G$的{mathbb Z}_psetminus{0}$和$G$的${mathbb Z}_p$-边界$b$，如果净流出$f$流与${mathbb Z}_p$中的$b(v)$在v (G)$中的每个顶点$v $在取向$D$下的$b(f,b;p)$-取向$tau$称为$(f,b;p)$-取向。这个概念是由Esperet et al.(2018)引入的，它推广了mod $p$-取向，与Tutte的nowhere zero 3-flow猜想密切相关。他们证明了$(6p^2 - 14p + 8)$-边连通图具有所有可能的$(f,b;p)$-方向。本文通过加性基将这种定向的框架扩展到签名图。我们还研究了一些(有符号)图族的$(f,b;p)$取向问题，这些图族包括完全图、弦图、序列-平行图和二部图，表明了更低的边连通界仍然保证了这些图族的这种取向的存在。
{"title":"Weighted Modulo Orientations of Graphs and Signed Graphs","authors":"Jianbing Liu, Miaomiao Han, H. Lai","doi":"10.37236/10740","DOIUrl":"https://doi.org/10.37236/10740","url":null,"abstract":"Given a graph $G$ and an odd prime $p$, for a mapping $f: E(G) to {mathbb Z}_psetminus{0}$ and a ${mathbb Z}_p$-boundary $b$ of $G$, an orientation $tau$ is called an $(f,b;p)$-orientation if the net out $f$-flow is the same as $b(v)$ in ${mathbb Z}_p$ at each vertex $vin V(G)$ under orientation $D$. This concept was introduced by Esperet et al. (2018), generalizing mod $p$-orientations and closely related to Tutte's nowhere zero 3-flow conjecture. They proved that $(6p^2 - 14p + 8)$-edge-connected graphs have all possible $(f,b;p)$-orientations. In this paper, the framework of such orientations is extended to signed graph through additive bases. We also study the $(f,b;p)$-orientation problem for some (signed) graphs families including complete graphs, chordal graphs, series-parallel graphs and bipartite graphs, indicating that much lower edge-connectivity bound still guarantees the existence of such orientations for those graph families.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"36 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-16","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"87094159","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}
Michael Anastos, David Fabian, Alp Müyesser, T. Szab'o
We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3in mathbb{N}$ with $a_1+a_2+a_3leq n-2$, we show there exists a matching $M$ of $G$ with $|Mcap M_i|=a_i$ for each $iin {1,2,3}$. The bound $n-2$ in the theorem is best possible in general.We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.
{"title":"Splitting Matchings and the Ryser-Brualdi-Stein Conjecture for Multisets","authors":"Michael Anastos, David Fabian, Alp Müyesser, T. Szab'o","doi":"10.37236/11714","DOIUrl":"https://doi.org/10.37236/11714","url":null,"abstract":"We study multigraphs whose edge-sets are the union of three perfect matchings, $M_1$, $M_2$, and $M_3$. Given such a graph $G$ and any $a_1,a_2,a_3in mathbb{N}$ with $a_1+a_2+a_3leq n-2$, we show there exists a matching $M$ of $G$ with $|Mcap M_i|=a_i$ for each $iin {1,2,3}$. The bound $n-2$ in the theorem is best possible in general.We conjecture however that if $G$ is bipartite, the same result holds with $n-2$ replaced by $n-1$. We give a construction that shows such a result would be tight. We also make a conjecture generalising the Ryser-Brualdi-Stein conjecture with colour multiplicities.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"458 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75102745","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}
Arnbjörg Soffía Árnadóttir, Waltraud Lederle, R. G. Möller
We study groups acting vertex-transitively and non-discretely on connected, cubic graphs (regular graphs of degree 3). Using ideas from Tutte's fundamental papers in 1947 and 1959, it is shown that if the action is edge-transitive, then the graph has to be a tree. When the action is not edge-transitive Tutte's ideas are still useful and can, amongst other things, be used to fully classify the possible two-ended graphs. Results about cubic graphs are then applied to Willis' scale function from the theory of totally disconnected, locally compact groups. Some of the results in this paper have most likely been known to experts but most of them are not stated explicitly with proofs in the literature.
{"title":"On Infinite, Cubic, Vertex-Transitive Graphs with Applications to Totally Disconnected, Locally Compact Groups","authors":"Arnbjörg Soffía Árnadóttir, Waltraud Lederle, R. G. Möller","doi":"10.37236/10709","DOIUrl":"https://doi.org/10.37236/10709","url":null,"abstract":"We study groups acting vertex-transitively and non-discretely on connected, cubic graphs (regular graphs of degree 3). Using ideas from Tutte's fundamental papers in 1947 and 1959, it is shown that if the action is edge-transitive, then the graph has to be a tree. When the action is not edge-transitive Tutte's ideas are still useful and can, amongst other things, be used to fully classify the possible two-ended graphs. Results about cubic graphs are then applied to Willis' scale function from the theory of totally disconnected, locally compact groups. Some of the results in this paper have most likely been known to experts but most of them are not stated explicitly with proofs in the literature.","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"140 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"82103296","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}
In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime $ell$ for the function $pbig(n,m,Nbig)$, which enumerates the partitions of $n$ into at most $m$ parts with no part larger than $N$. We also treat the function $pbig(n,m,(a,b]big)$, which bounds the largest part above and below, and obtain similar infinite families of congruences. For $m leq 4$ and $ell = 3$, simple combinatorial statistics called "cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons. For $m leq 4$ and $ell = 3$, simple combinatorial statistics called ``cranks" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons.
{"title":"Congruences for Consecutive Coefficients of Gaussian Polynomials with Crank Statistics","authors":"Dennis Eichhorn, Lydia Engle, Brandt Kronholm","doi":"10.37236/10493","DOIUrl":"https://doi.org/10.37236/10493","url":null,"abstract":"\u0000 \u0000 \u0000In this paper, we establish infinite families of congruences in consecutive arithmetic progressions modulo any odd prime $ell$ for the function $pbig(n,m,Nbig)$, which enumerates the partitions of $n$ into at most $m$ parts with no part larger than $N$. We also treat the function $pbig(n,m,(a,b]big)$, which bounds the largest part above and below, and obtain similar infinite families of congruences. \u0000For $m leq 4$ and $ell = 3$, simple combinatorial statistics called \"cranks\" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons. \u0000 \u0000For $m leq 4$ and $ell = 3$, simple combinatorial statistics called ``cranks\" witness these congruences. We prove this analytically for $m=4$, and then both analytically and combinatorially for $m = 3$. Our combinatorial proof relies upon explicit dissections of convex lattice polygons. \u0000 \u0000 \u0000","PeriodicalId":11515,"journal":{"name":"Electronic Journal of Combinatorics","volume":"8 1","pages":""},"PeriodicalIF":0.7,"publicationDate":"2022-12-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"75679116","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}
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