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Consecutive Patterns in Inversion Sequences 反转序列中的连续模式
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-04-04 DOI: 10.23638/DMTCS-21-2-6
Juan S. Auli, S. Elizalde
{"title":"Consecutive Patterns in Inversion Sequences","authors":"Juan S. Auli, S. Elizalde","doi":"10.23638/DMTCS-21-2-6","DOIUrl":"https://doi.org/10.23638/DMTCS-21-2-6","url":null,"abstract":"An inversion sequence of length $n$ is an integer sequence $e=e_{1}e_{2}dots e_{n}$ such that $0leq e_{i}<i$ for each $i$. Corteel--Martinez--Savage--Weselcouch and Mansour--Shattuck began the study of patterns in inversion sequences, focusing on the enumeration of those that avoid classical patterns of length 3. We initiate an analogous systematic study of consecutive patterns in inversion sequences, namely patterns whose entries are required to occur in adjacent positions. We enumerate inversion sequences that avoid consecutive patterns of length 3, and generalize some results to patterns of arbitrary length. Additionally, we study the notion of Wilf equivalence of consecutive patterns in inversion sequences, as well as generalizations of this notion analogous to those studied for permutation patterns. We classify patterns of length up to 4 according to the corresponding Wilf equivalence relations.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"115 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"124829709","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}
引用次数: 19
The minimal probabilistic and quantum finite automata recognizing uncountably many languages with fixed cutpoints 最小概率和量子有限自动机识别不可数种具有固定截断点的语言
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-04-01 DOI: 10.23638/DMTCS-22-1-13
Aleksejs Naumovs, M. Dimitrijevs, A. Yakaryılmaz
{"title":"The minimal probabilistic and quantum finite automata recognizing uncountably many languages with fixed cutpoints","authors":"Aleksejs Naumovs, M. Dimitrijevs, A. Yakaryılmaz","doi":"10.23638/DMTCS-22-1-13","DOIUrl":"https://doi.org/10.23638/DMTCS-22-1-13","url":null,"abstract":"It is known that 2-state binary and 3-state unary probabilistic finite automata and 2-state unary quantum finite automata recognize uncountably many languages with cutpoints. These results have been obtained by associating each recognized language with a cutpoint and then by using the fact that there are uncountably many cutpoints. In this note, we prove the same results for fixed cutpoints: each recognized language is associated with an automaton (i.e., algorithm), and the proofs use the fact that there are uncountably many automata. For each case, we present a new construction.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"53 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-04-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"131629133","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}
引用次数: 1
The undecidability of joint embedding and joint homomorphism for hereditary graph classes 遗传图类的联合嵌入和联合同态的不可判定性
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-03-28 DOI: 10.23638/DMTCS-21-2-9
S. Braunfeld
{"title":"The undecidability of joint embedding and joint homomorphism for hereditary graph classes","authors":"S. Braunfeld","doi":"10.23638/DMTCS-21-2-9","DOIUrl":"https://doi.org/10.23638/DMTCS-21-2-9","url":null,"abstract":"We prove that the joint embedding property is undecidable for hereditary graph classes, via a reduction from the tiling problem. The proof is then adapted to show the undecidability of the joint homomorphism property as well.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"7 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-28","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"122508759","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}
引用次数: 12
A Characterization of Morphic Words with Polynomial Growth 具有多项式增长的情态词的表征
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-03-24 DOI: 10.23638/DMTCS-22-1-3
Tim Smith
{"title":"A Characterization of Morphic Words with Polynomial Growth","authors":"Tim Smith","doi":"10.23638/DMTCS-22-1-3","DOIUrl":"https://doi.org/10.23638/DMTCS-22-1-3","url":null,"abstract":"A morphic word is obtained by iterating a morphism to generate an infinite word, and then applying a coding. We characterize morphic words with polynomial growth in terms of a new type of infinite word called a $textit{zigzag word}$. A zigzag word is represented by an initial string, followed by a finite list of terms, each of which repeats for each $n geq 1$ in one of three ways: it grows forward [$t(1) t(2) dotsm t(n)]$, backward [$t(n) dotsm t(2) t(1)$], or just occurs once [$t$]. Each term can recursively contain subterms with their own forward and backward repetitions. We show that an infinite word is morphic with growth $Theta(n^k)$ iff it is a zigzag word of depth $k$. As corollaries, we obtain that the morphic words with growth $O(n)$ are exactly the ultimately periodic words, and the morphic words with growth $O(n^2)$ are exactly the multilinear words.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"6 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-03-24","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"133758363","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}
引用次数: 1
Taking-and-merging games as rewrite games 将合并游戏作为重写游戏
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-02-19 DOI: 10.23638/DMTCS-22-4-5
Éric Duchêne, Victor Marsault, Aline Parreau, M. Rigo
{"title":"Taking-and-merging games as rewrite games","authors":"Éric Duchêne, Victor Marsault, Aline Parreau, M. Rigo","doi":"10.23638/DMTCS-22-4-5","DOIUrl":"https://doi.org/10.23638/DMTCS-22-4-5","url":null,"abstract":"This work is a contribution to the study of rewrite games. Positions are finite words, and the possible moves are defined by a finite number of local rewriting rules. We introduce and investigate taking-and-merging games, that is, where each rule is of the form a^k->epsilon. \u0000We give sufficient conditions for a game to be such that the losing positions (resp. the positions with a given Grundy value) form a regular language or a context-free language. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games. \u0000Finally we show that more general rewrite games quickly lead to undecidable problems. Namely, it is undecidable whether there exists a winning position in a given regular language, even if we restrict to games where each move strictly reduces the length of the current position. We formulate several related open questions in parallel with the famous conjecture of Guy about the periodicity of the Grundy function of octal games.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"44 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"117303336","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}
引用次数: 2
New Results on Directed Edge Dominating Set 关于有向边支配集的新结果
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-02-13 DOI: 10.46298/dmtcs.5378
R. Belmonte, T. Hanaka, I. Katsikarelis, Eun Jung Kim, M. Lampis
{"title":"New Results on Directed Edge Dominating Set","authors":"R. Belmonte, T. Hanaka, I. Katsikarelis, Eun Jung Kim, M. Lampis","doi":"10.46298/dmtcs.5378","DOIUrl":"https://doi.org/10.46298/dmtcs.5378","url":null,"abstract":"We study a family of generalizations of Edge Dominating Set on directed\u0000graphs called Directed $(p,q)$-Edge Dominating Set. In this problem an arc\u0000$(u,v)$ is said to dominate itself, as well as all arcs which are at distance\u0000at most $q$ from $v$, or at distance at most $p$ to $u$.\u0000 First, we give significantly improved FPT algorithms for the two most\u0000important cases of the problem, $(0,1)$-dEDS and $(1,1)$-dEDS (that correspond\u0000to versions of Dominating Set on line graphs), as well as polynomial kernels.\u0000We also improve the best-known approximation for these cases from logarithmic\u0000to constant. In addition, we show that $(p,q)$-dEDS is FPT parameterized by\u0000$p+q+tw$, but W-hard parameterized by $tw$ (even if the size of the optimal is\u0000added as a second parameter), where $tw$ is the treewidth of the underlying\u0000graph of the input.\u0000 We then go on to focus on the complexity of the problem on tournaments. Here,\u0000we provide a complete classification for every possible fixed value of $p,q$,\u0000which shows that the problem exhibits a surprising behavior, including cases\u0000which are in P; cases which are solvable in quasi-polynomial time but not in P;\u0000and a single case $(p=q=1)$ which is NP-hard (under randomized reductions) and\u0000cannot be solved in sub-exponential time, under standard assumptions.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"45 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-13","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"116925760","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}
引用次数: 6
On the number of pancake stacks requiring four flips to be sorted 关于需要四次翻转才能排序的煎饼堆的数量
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-02-11 DOI: 10.23638/DMTCS-21-2-5
Saúl A. Blanco, Charles Buehrle, Akshay Patidar
{"title":"On the number of pancake stacks requiring four flips to be sorted","authors":"Saúl A. Blanco, Charles Buehrle, Akshay Patidar","doi":"10.23638/DMTCS-21-2-5","DOIUrl":"https://doi.org/10.23638/DMTCS-21-2-5","url":null,"abstract":"Using existing classification results for the 7- and 8-cycles in the pancake graph, we determine the number of permutations that require 4 pancake flips (prefix reversals) to be sorted. A similar characterization of the 8-cycles in the burnt pancake graph, due to the authors, is used to derive a formula for the number of signed permutations requiring 4 (burnt) pancake flips to be sorted. We furthermore provide an analogous characterization of the 9-cycles in the burnt pancake graph. Finally we present numerical evidence that polynomial formulae exist giving the number of signed permutations that require $k$ flips to be sorted, with $5leq kleq9$.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"16 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-11","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121843320","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}
引用次数: 4
Consecutive patterns in restricted permutations and involutions 有限排列和对合中的连续模式
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-02-06 DOI: 10.23638/DMTCS-21-3-21
M. Barnabei, F. Bonetti, N. Castronuovo, M. Silimbani
{"title":"Consecutive patterns in restricted permutations and involutions","authors":"M. Barnabei, F. Bonetti, N. Castronuovo, M. Silimbani","doi":"10.23638/DMTCS-21-3-21","DOIUrl":"https://doi.org/10.23638/DMTCS-21-3-21","url":null,"abstract":"It is well-known that the set $mathbf I_n$ of involutions of the symmetric\u0000group $mathbf S_n$ corresponds bijectively - by the Foata map $F$ - to the set\u0000of $n$-permutations that avoid the two vincular patterns $underline{123},$\u0000$underline{132}.$ We consider a bijection $Gamma$ from the set $mathbf S_n$\u0000to the set of histoires de Laguerre, namely, bicolored Motzkin paths with\u0000labelled steps, and study its properties when restricted to $mathbf\u0000S_n(1underline{23},1underline{32}).$ In particular, we show that the set\u0000$mathbf S_n(underline{123},{132})$ of permutations that avoids the\u0000consecutive pattern $underline{123}$ and the classical pattern $132$\u0000corresponds via $Gamma$ to the set of Motzkin paths, while its image under $F$\u0000is the set of restricted involutions $mathbf I_n(3412).$ We exploit these\u0000results to determine the joint distribution of the statistics des and inv over\u0000 $mathbf S_n(underline{123},{132})$ and over $mathbf I_n(3412).$\u0000 Moreover, we determine the distribution in these two sets of every\u0000consecutive pattern of length three. To this aim, we use a modified version of\u0000the well-known Goulden-Jacson cluster method.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"27 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-06","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"126853036","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}
引用次数: 0
Statistics on Linear Chord Diagrams 线性弦图统计
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-02-01 DOI: 10.23638/DMTCS-21-2-11
N. Cameron, K. Killpatrick
{"title":"Statistics on Linear Chord Diagrams","authors":"N. Cameron, K. Killpatrick","doi":"10.23638/DMTCS-21-2-11","DOIUrl":"https://doi.org/10.23638/DMTCS-21-2-11","url":null,"abstract":"Linear chord diagrams are partitions of $left[2nright]$ into $n$ blocks of size two called chords. We refer to a block of the form ${i,i+1}$ as a short chord. In this paper, we study the distribution of the number of short chords on the set of linear chord diagrams, as a generalization of the Narayana distribution obtained when restricted to the set of noncrossing linear chord diagrams. We provide a combinatorial proof that this distribution is unimodal and has an expected value of one. We also study the number of pairs $(i,i+1)$ where $i$ is the minimal element of a chord and $i+1$ is the maximal element of a chord. We show that the distribution of this statistic on linear chord diagrams corresponds to the second-order Eulerian triangle and is log-concave.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"33 6","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"121000571","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}
引用次数: 8
(Open) packing number of some graph products (开)某些图形产品的包装编号
Discret. Math. Theor. Comput. Sci. Pub Date : 2019-01-21 DOI: 10.23638/DMTCS-22-4-1
D. Mojdeh, Iztok Peterin, B. Samadi, I. Yero
{"title":"(Open) packing number of some graph products","authors":"D. Mojdeh, Iztok Peterin, B. Samadi, I. Yero","doi":"10.23638/DMTCS-22-4-1","DOIUrl":"https://doi.org/10.23638/DMTCS-22-4-1","url":null,"abstract":"The packing number of a graph $G$ is the maximum number of closed neighborhoods of vertices in $G$ with pairwise empty intersections. Similarly, the open packing number of $G$ is the maximum number of open neighborhoods in $G$ with pairwise empty intersections. We consider the packing and open packing numbers on graph products. In particular we give a complete solution with respect to some properties of factors in the case of lexicographic and rooted products. For Cartesian, strong and direct products, we present several lower and upper bounds on these parameters.","PeriodicalId":110830,"journal":{"name":"Discret. Math. Theor. Comput. Sci.","volume":"116 1","pages":"0"},"PeriodicalIF":0.0,"publicationDate":"2019-01-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"123451315","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}
引用次数: 1
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