Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as "interval thinness" and "fellow traveler property". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.
假设 u 和 v 是连通图 G = (V, E) 中的顶点。对于 0 ≤ k ≤ dG (u, v) 的任意整数 k，k 切片 Sk (u, v) 包含最短 uv 路径上的所有顶点 x，且 dG (u, x) = k。这种度量图不变式有不同的研究名称，如 "区间稀疏性 "和 "同路人属性"。精益度等于 0 的图，又称大地图，在图论中也受到特别关注。最近，人们研究了现实生活中复杂网络中精简度的实际计算（穆罕默德等人，COMPLEX NETWORKS'21）。在本文中，我们对两个相关问题进行了更细粒度的复杂性分析，这两个问题分别是：判断图 G 的精简度是否最多为某个小值 ℓ；以及计算特定图类的精简度。我们获得了某些情况下的改进算法，以及在合理假设下的时间复杂度下限。
{"title":"Leanness Computation: Small Values and Special Graph Classes","authors":"David Coudert, Samuel Coulomb, G. Ducoffe","doi":"10.46298/dmtcs.12544","DOIUrl":"https://doi.org/10.46298/dmtcs.12544","url":null,"abstract":"Let u and v be vertices in a connected graph G = (V, E). For any integer k such that 0 ≤ k ≤ dG (u, v), the k-slice Sk (u, v) contains all vertices x on a shortest uv-path such that dG (u, x) = k. The leanness of G is the maximum diameter of a slice. This metric graph invariant has been studied under different names, such as \"interval thinness\" and \"fellow traveler property\". Graphs with leanness equal to 0, a.k.a. geodetic graphs, also have received special attention in Graph Theory. The practical computation of leanness in real-life complex networks has been studied recently (Mohammed et al., COMPLEX NETWORKS'21). In this paper, we give a finer-grained complexity analysis of two related problems, namely: deciding whether the leanness of a graph G is at most some small value ℓ; and computing the leanness on specific graph classes. We obtain improved algorithms in some cases, and time complexity lower bounds under plausible hypotheses.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"109 11","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-07-08","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"141667314","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
The exponential recursive trees model several kinds of networks. At each step of growing of these trees, each node independently attracts a new node with probability p, or fails to do with probability 1 − p. Here, we investigate the number of protected nodes, total path length of protected nodes, and a mean study of the protected node profile of such trees.
指数递归树是几种网络的模型。在这些树的每一步生长过程中，每个节点都会以概率 p 独立吸引一个新节点，或者以概率 1 - p 独立吸引一个新节点。在此，我们将研究受保护节点的数量、受保护节点的总路径长度以及此类树的受保护节点轮廓的平均值。
{"title":"On the protected nodes in exponential recursive trees","authors":"M. Javanian, Rafik Aguech","doi":"10.46298/dmtcs.10524","DOIUrl":"https://doi.org/10.46298/dmtcs.10524","url":null,"abstract":"The exponential recursive trees model several kinds of networks. At each step of growing of these trees, each node independently attracts a new node with probability p, or fails to do with probability 1 − p. Here, we investigate the number of protected nodes, total path length of protected nodes, and a mean study of the protected node profile of such trees.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":" 7","pages":""},"PeriodicalIF":0.0,"publicationDate":"2024-01-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139627132","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We study the enumeration of inversion sequences that avoid the pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of dregular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.
我们研究了避免 021 图案和另一个长度为 4 的图案的反转序列的枚举。我们确定了所有可能模式对的生成树，并计算了相应的生成函数。我们引入了不规则生成树的概念，并猜想对于任何避开 021 图案 τ 的生成树 T ({021, τ }) 对于某个整数 d 是不规则的。
{"title":"Inversion sequences avoiding 021 and another pattern of length four","authors":"Toufik Mansour, Gökhan Yıldırım","doi":"10.46298/dmtcs.10444","DOIUrl":"https://doi.org/10.46298/dmtcs.10444","url":null,"abstract":"We study the enumeration of inversion sequences that avoid the pattern 021 and another pattern of length four. We determine the generating trees for all possible pattern pairs and compute the corresponding generating functions. We introduce the concept of dregular generating trees and conjecture that for any 021-avoiding pattern τ , the generating tree T ({021, τ }) is d-regular for some integer d.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"33 6","pages":""},"PeriodicalIF":0.0,"publicationDate":"2023-11-17","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"139263480","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Cláudio Carvalho, J. Costa, Raul Lopes, A. K. Maia, N. Nisse, C. Sales
An s-branching flow f in a network N = (D, u), where u is the capacity function, is a flow thatreaches every vertex in V(D) from s while loosing exactly one unit of flow in each vertex other thans. Bang-Jensen and Bessy [TCS, 2014] showed that, when every arc has capacity n − 1, a network Nadmits k arc-disjoint s-branching flows if and only if its associated digraph D contains k arc-disjoints-branchings. Thus a classical result by Edmonds stating that a digraph contains k arc-disjoints-branchings if and only if the indegree of every set X ⊆ V (D) {s} is at least k also characterizesthe existence of k arc-disjoint s-branching flows in those networks, suggesting that the larger thecapacities are, the closer an s-branching flow is from simply being an s-branching. This observationis further implied by results by Bang-Jensen et al. [DAM, 2016] showing that there is a polynomialalgorithm to find the flows (if they exist) when every arc has capacity n − c, for every fixed c ≥ 1,and that such an algorithm is unlikely to exist for most other choices of the capacities. In this paper,we investigate how a property that is a natural extension of the characterization by Edmonds’ relatesto the existence of k arc-disjoint s-branching flows in networks. Although this property is alwaysnecessary for the existence of the flows, we show that it is not always sufficient and that it is hardto decide if the desired flows exist even if we know beforehand that the network satisfies it. On thepositive side, we show that it guarantees the existence of the desired flows in some particular casesdepending on the choice of the capacity function or on the structure of the underlying graph of D,for example. We remark that, in those positive cases, polynomial time algorithms to find the flowscan be extracted from the constructive proofs.
{"title":"From branchings to flows: a study of an Edmonds' like property to arc-disjoint branching flows","authors":"Cláudio Carvalho, J. Costa, Raul Lopes, A. K. Maia, N. Nisse, C. Sales","doi":"10.46298/dmtcs.9302","DOIUrl":"https://doi.org/10.46298/dmtcs.9302","url":null,"abstract":"An s-branching flow f in a network N = (D, u), where u is the capacity function, is a flow thatreaches every vertex in V(D) from s while loosing exactly one unit of flow in each vertex other thans. Bang-Jensen and Bessy [TCS, 2014] showed that, when every arc has capacity n − 1, a network Nadmits k arc-disjoint s-branching flows if and only if its associated digraph D contains k arc-disjoints-branchings. Thus a classical result by Edmonds stating that a digraph contains k arc-disjoints-branchings if and only if the indegree of every set X ⊆ V (D) {s} is at least k also characterizesthe existence of k arc-disjoint s-branching flows in those networks, suggesting that the larger thecapacities are, the closer an s-branching flow is from simply being an s-branching. This observationis further implied by results by Bang-Jensen et al. [DAM, 2016] showing that there is a polynomialalgorithm to find the flows (if they exist) when every arc has capacity n − c, for every fixed c ≥ 1,and that such an algorithm is unlikely to exist for most other choices of the capacities. In this paper,we investigate how a property that is a natural extension of the characterization by Edmonds’ relatesto the existence of k arc-disjoint s-branching flows in networks. Although this property is alwaysnecessary for the existence of the flows, we show that it is not always sufficient and that it is hardto decide if the desired flows exist even if we know beforehand that the network satisfies it. On thepositive side, we show that it guarantees the existence of the desired flows in some particular casesdepending on the choice of the capacity function or on the structure of the underlying graph of D,for example. We remark that, in those positive cases, polynomial time algorithms to find the flowscan be extracted from the constructive proofs.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"34 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2023-05-02","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115493689","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In the Maker-Breaker domination game played on a graph $G$, Dominator's goal is to select a dominating set and Staller's goal is to claim a closed neighborhood of some vertex. We study the cases when Staller can win the game. If Dominator (resp., Staller) starts the game, then $gamma_{rm SMB}(G)$ (resp., $gamma_{rm SMB}'(G)$) denotes the minimum number of moves Staller needs to win. For every positive integer $k$, trees $T$ with $gamma_{rm SMB}'(T)=k$ are characterized and a general upper bound on $gamma_{rm SMB}'$ is proved. Let $S = S(n_1,dots, n_ell)$ be the subdivided star obtained from the star with $ell$ edges by subdividing its edges $n_1-1, ldots, n_ell-1$ times, respectively. Then $gamma_{rm SMB}'(S)$ is determined in all the cases except when $ellge 4$ and each $n_i$ is even. The simplest formula is obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two smallest such numbers, then $gamma_{rm SMB}'(S(n_1,dots, n_ell))=lceil log_2(n_1+n_2+1)rceil$. For caterpillars, exact formulas for $gamma_{rm SMB}$ and for $gamma_{rm SMB}'$ are established.
{"title":"Maker-Breaker domination game on trees when Staller wins","authors":"Csilla Bujt'as, Pakanun Dokyeesun, Sandi Klavvzar","doi":"10.46298/dmtcs.10515","DOIUrl":"https://doi.org/10.46298/dmtcs.10515","url":null,"abstract":"In the Maker-Breaker domination game played on a graph $G$, Dominator's goal\u0000is to select a dominating set and Staller's goal is to claim a closed\u0000neighborhood of some vertex. We study the cases when Staller can win the game.\u0000If Dominator (resp., Staller) starts the game, then $gamma_{rm SMB}(G)$\u0000(resp., $gamma_{rm SMB}'(G)$) denotes the minimum number of moves Staller\u0000needs to win. For every positive integer $k$, trees $T$ with $gamma_{rm\u0000SMB}'(T)=k$ are characterized and a general upper bound on $gamma_{rm SMB}'$\u0000is proved. Let $S = S(n_1,dots, n_ell)$ be the subdivided star obtained from\u0000the star with $ell$ edges by subdividing its edges $n_1-1, ldots, n_ell-1$\u0000times, respectively. Then $gamma_{rm SMB}'(S)$ is determined in all the cases\u0000except when $ellge 4$ and each $n_i$ is even. The simplest formula is\u0000obtained when there are at least two odd $n_i$s. If $n_1$ and $n_2$ are the two\u0000smallest such numbers, then $gamma_{rm SMB}'(S(n_1,dots, n_ell))=lceil\u0000log_2(n_1+n_2+1)rceil$. For caterpillars, exact formulas for $gamma_{rm\u0000SMB}$ and for $gamma_{rm SMB}'$ are established.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"30 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-12-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122391365","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
An edge-colored graph is emph{rainbow }if no two edges of the graph have the same color. An edge-colored graph $G^c$ is called emph{properly colored} if every two adjacent edges of $G^c$ receive distinct colors in $G^c$. A emph{strongly edge-colored} graph is a proper edge-colored graph such that every path of length $3$ is rainbow. We call an edge-colored graph $G^c$ emph{rainbow vertex pair-pancyclic} if any two vertices in $G^c$ are contained in a rainbow cycle of length $ell$ for each $ell$ with $3 leq ell leq n$. In this paper, we show that every strongly edge-colored graph $G^c$ of order $n$ with minimum degree $delta geq frac{2n}{3}+1$ is rainbow vertex pair-pancyclicity.
{"title":"Rainbow vertex pair-pancyclicity of strongly edge-colored graphs","authors":"Peixue Zhao, Fei Huang","doi":"10.46298/dmtcs.10142","DOIUrl":"https://doi.org/10.46298/dmtcs.10142","url":null,"abstract":"An edge-colored graph is emph{rainbow }if no two edges of the graph have the\u0000same color. An edge-colored graph $G^c$ is called emph{properly colored} if\u0000every two adjacent edges of $G^c$ receive distinct colors in $G^c$. A\u0000emph{strongly edge-colored} graph is a proper edge-colored graph such that\u0000every path of length $3$ is rainbow. We call an edge-colored graph $G^c$\u0000emph{rainbow vertex pair-pancyclic} if any two vertices in $G^c$ are contained\u0000in a rainbow cycle of length $ell$ for each $ell$ with $3 leq ell leq n$.\u0000In this paper, we show that every strongly edge-colored graph $G^c$ of order\u0000$n$ with minimum degree $delta geq frac{2n}{3}+1$ is rainbow vertex\u0000pair-pancyclicity.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"36 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-10-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117208936","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
This paper considers the following three Roman domination graph invariants on Kneser graphs: Roman domination, total Roman domination, and signed Roman domination. For Kneser graph $K_{n,k}$, we present exact values for Roman domination number $gamma_{R}(K_{n,k})$ and total Roman domination number $gamma_{tR}(K_{n,k})$ proving that for $ngeqslant k(k+1)$, $gamma_{R}(K_{n,k}) =gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman domination number $gamma_{sR}(K_{n,k})$, the new lower and upper bounds for $K_{n,2}$ are provided: we prove that for $ngeqslant 12$, the lower bound is equal to 2, while the upper bound depends on the parity of $n$ and is equal to 3 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller dimensions, exact values are found by applying exact methods from literature.
{"title":"Several Roman domination graph invariants on Kneser graphs","authors":"Tatjana Zec, Milana Grbi'c","doi":"10.46298/dmtcs.10506","DOIUrl":"https://doi.org/10.46298/dmtcs.10506","url":null,"abstract":"This paper considers the following three Roman domination graph invariants on\u0000Kneser graphs:\u0000 Roman domination, total Roman domination, and signed Roman domination.\u0000 For Kneser graph $K_{n,k}$, we present exact values for Roman domination\u0000number $gamma_{R}(K_{n,k})$ and total Roman domination number\u0000$gamma_{tR}(K_{n,k})$ proving that for $ngeqslant k(k+1)$,\u0000$gamma_{R}(K_{n,k}) =gamma_{tR}(K_{n,k}) = 2(k+1)$. For signed Roman\u0000domination number $gamma_{sR}(K_{n,k})$, the new lower and upper bounds for\u0000$K_{n,2}$ are provided: we prove that for $ngeqslant 12$, the lower bound is\u0000equal to 2, while the upper bound depends on the parity of $n$ and is equal to\u00003 if $n$ is odd, and equal to $5$ if $n$ is even. For graphs of smaller\u0000dimensions, exact values are found by applying exact methods from literature.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"129 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2022-04-12","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"115220642","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
In a generalized Tur'an problem, two graphs $H$ and $F$ are given and the question is the maximum number of copies of $H$ in an $F$-free graph of order $n$. In this paper, we study the number of double stars $S_{k,l}$ in triangle-free graphs. We also study an opposite version of this question: what is the maximum number edges/triangles in graphs with double star type restrictions, which leads us to study two questions related to the extremal number of triangles or edges in graphs with degree-sum constraints over adjacent or non-adjacent vertices.
{"title":"Extremal problems of double stars","authors":"Ervin GyHori, Runze Wang, Spencer Woolfson","doi":"10.46298/dmtcs.8499","DOIUrl":"https://doi.org/10.46298/dmtcs.8499","url":null,"abstract":"In a generalized Tur'an problem, two graphs $H$ and $F$ are given and the\u0000question is the maximum number of copies of $H$ in an $F$-free graph of order\u0000$n$. In this paper, we study the number of double stars $S_{k,l}$ in\u0000triangle-free graphs. We also study an opposite version of this question: what\u0000is the maximum number edges/triangles in graphs with double star type\u0000restrictions, which leads us to study two questions related to the extremal\u0000number of triangles or edges in graphs with degree-sum constraints over\u0000adjacent or non-adjacent vertices.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"31 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-09-03","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124267339","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
We obtain an explicit formula for the variance of the number of $k$-peaks in a uniformly random permutation. This is then used to obtain an asymptotic formula for the variance of the length of longest $k$-alternating subsequence in random permutations. Also a central limit is proved for the latter statistic.
{"title":"The Variance and the Asymptotic Distribution of the Length of Longest $k$-alternating Subsequences","authors":"Altar cCicceksiz, Yunus Emre Demirci, Umit Icslak","doi":"10.46298/dmtcs.10296","DOIUrl":"https://doi.org/10.46298/dmtcs.10296","url":null,"abstract":"We obtain an explicit formula for the variance of the number of $k$-peaks in\u0000a uniformly random permutation. This is then used to obtain an asymptotic\u0000formula for the variance of the length of longest $k$-alternating subsequence\u0000in random permutations. Also a central limit is proved for the latter\u0000statistic.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"142 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-07-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124493807","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
Let $Bigllanglematrix{ncr k}Bigrrangle$, $Bigllanglematrix{B_ncr k}Bigrrangle$, and $Bigllanglematrix{D_ncr k}Bigrrangle$ be the Eulerian numbers in the types A, B, and D, respectively -- that is, the number of permutations of n elements with $k$ descents, the number of signed permutations (of $n$ elements) with $k$ type B descents, the number of even signed permutations (of $n$ elements) with $k$ type D descents. Let $S_n(t) = sum_{k = 0}^{n-1} Bigllanglematrix{ncr k}Bigrrangle t^k$, $B_n(t) = sum_{k = 0}^n Bigllanglematrix{B_ncr k}Bigrrangle t^k$, and $D_n(t) = sum_{k = 0}^n Bigllanglematrix{D_ncr k}Bigrrangle t^k$. We give bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^n tS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) - n2^{n-1}tS_{n-1}(t).$$ These bijective proofs rely on a representation of signed permutations as paths. Using this representation we also establish a bijective correspondence between even signed permutations and pairs $(w, E)$ with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$, which we use to obtain bijective proofs of enumerative results for threshold graphs.
{"title":"Bijective proofs for Eulerian numbers of types B and D","authors":"L. Santocanale","doi":"10.46298/dmtcs.7413","DOIUrl":"https://doi.org/10.46298/dmtcs.7413","url":null,"abstract":"Let $Bigllanglematrix{ncr k}Bigrrangle$, $Bigllanglematrix{B_ncr\u0000k}Bigrrangle$, and $Bigllanglematrix{D_ncr k}Bigrrangle$ be the\u0000Eulerian numbers in the types A, B, and D, respectively -- that is, the number\u0000of permutations of n elements with $k$ descents, the number of signed\u0000permutations (of $n$ elements) with $k$ type B descents, the number of even\u0000signed permutations (of $n$ elements) with $k$ type D descents. Let $S_n(t) =\u0000sum_{k = 0}^{n-1} Bigllanglematrix{ncr k}Bigrrangle t^k$, $B_n(t) =\u0000sum_{k = 0}^n Bigllanglematrix{B_ncr k}Bigrrangle t^k$, and $D_n(t) =\u0000sum_{k = 0}^n Bigllanglematrix{D_ncr k}Bigrrangle t^k$. We give\u0000bijective proofs of the identity $$B_n(t^2) = (1 + t)^{n+1}S_n(t) - 2^n\u0000tS_n(t^2)$$ and of Stembridge's identity $$D_n(t) = B_n(t) -\u0000n2^{n-1}tS_{n-1}(t).$$ These bijective proofs rely on a representation of\u0000signed permutations as paths. Using this representation we also establish a\u0000bijective correspondence between even signed permutations and pairs $(w, E)$\u0000with $([n], E)$ a threshold graph and $w$ a degree ordering of $([n], E)$,\u0000which we use to obtain bijective proofs of enumerative results for threshold\u0000graphs.","PeriodicalId":412397,"journal":{"name":"Discrete Mathematics & Theoretical Computer Science","volume":"57 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2021-04-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133511708","PeriodicalName":null,"FirstCategoryId":null,"ListUrlMain":null,"RegionNum":0,"RegionCategory":"","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":"","EPubDate":null,"PubModel":null,"JCR":null,"JCRName":null,"Score":null,"Total":0}
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