{"title":"New Recurrence Relation for Partitions Into Distinct Parts","authors":"T. Srichan","doi":"10.47443/dml.2022.078","DOIUrl":"https://doi.org/10.47443/dml.2022.078","url":null,"abstract":"Denote by Q n the set of partitions of a positive integer n into distinct parts. For k ∈ N , denote by Q n,k the set of partitions of n into distinct parts whose least part is k + 1 and not equal to n . Let q ( n ) and q ( n, k ) be the number of elements in Q n and Q n,k , respectively. In this paper, several new recurrence relations for partitions into distinct parts are derived from the partition function q ( n, k ) .","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-09-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47923296","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}
{"title":"Bounds on the general eccentric distance sum of graphs","authors":"Yetneberk Kuma Feyissa, T. Vetrík","doi":"10.47443/dml.2022.070","DOIUrl":"https://doi.org/10.47443/dml.2022.070","url":null,"abstract":"Some sharp bounds on the general eccentric distance sum are presented for (i) graphs with given order and chromatic number, (ii) trees with given bipartition, and (iii) bipartite graphs with given order and matching number. Bounds for bipartite graphs hold also if the matching number is replaced by the independence number, vertex cover number or edge cover number.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-30","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"46013079","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}
{"title":"Saturation of Multidimensional 0-1 Matrices","authors":"Shen-Fu Tsai","doi":"10.47443/dml.2022.151","DOIUrl":"https://doi.org/10.47443/dml.2022.151","url":null,"abstract":"A 0-1 matrix $M$ is saturating for a 0-1 matrix $P$ if $M$ does not contain a submatrix that can be turned into $P$ by flipping any number of its $1$-entries to $0$-entries, and changing any $0$-entry to $1$-entry of $M$ introduces a copy of $P$. Matrix $M$ is semisaturating for $P$ if changing any $0$-entry to $1$-entry of $M$ introduces a new copy of $P$, regardless of whether $M$ originally contains $P$ or not. The functions $ex(n;P)$ and $sat(n;P)$ are the maximum and minimum possible number of $1$-entries a $ntimes n$ 0-1 matrix saturating for $P$ can have, respectively. Function $ssat(n;P)$ is the minimum possible number of $1$-entries a $ntimes n$ 0-1 matrix semisaturating for $P$ can have. Function $ex(n;P)$ has been studied for decades, while investigation on $sat(n;P)$ and $ssat(n;P)$ was initiated recently. In this paper, we make nontrivial generalization of results regarding these functions to multidimensional 0-1 matrices. In particular, we find the exact values of $ex(n;P,d)$ and $sat(n;P,d)$ when $P$ is a $d$-dimensional identity matrix. Then we give the necessary and sufficient condition for a multidimensional 0-1 matrix to have bounded semisaturation function.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"44064996","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}
{"title":"Variations of central limit theorems and Stirling numbers of the first kind","authors":"B. Heim, M. Neuhauser","doi":"10.47443/dml.2022.183","DOIUrl":"https://doi.org/10.47443/dml.2022.183","url":null,"abstract":"We construct a new parametrization of double sequences ${A_{n,k}(s)}_{n,k}$ between $A_{n,k}(0)= binom{n-1}{k-1}$ and $A_{n,k}(1)= frac{1}{n!}stirl{n}{k}$, where $stirl{n}{k}$ are the unsigned Stirling numbers of the first kind. For each $s$ we prove a central limit theorem and a local limit theorem. This extends the de,Moivre--Laplace central limit theorem and Goncharov's result, that unsigned Stirling numbers of the first kind are asymptotically normal. Herewith, we provide several applications.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-08-21","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47310983","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}
{"title":"Alphabetic Points and Records in Inversion Sequences","authors":"A. Blecher, A. Knopfmacher, T. Mansour","doi":"10.47443/dml.2022.061","DOIUrl":"https://doi.org/10.47443/dml.2022.061","url":null,"abstract":"An alphabetic point in an inversion sequence is a value j where all the values l to its left satisfy l ≤ j and all the values r to its right satisfy r ≥ j . We study alphabetic points and records in inversion sequences of permutations and obtain formulae for the total numbers of alphabetic points, weak records, and strict records.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-22","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"47865134","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}
{"title":"A Note on the Locally Irregular Edge Colorings of Cacti","authors":"J. Sedlar, Riste vSkrekovski","doi":"10.47443/dml.2022.069","DOIUrl":"https://doi.org/10.47443/dml.2022.069","url":null,"abstract":"A graph is locally irregular if the degrees of the end-vertices of every edge are distinct. An edge coloring of a graph G is locally irregular if every color induces a locally irregular subgraph of G. A colorable graph G is any graph which admits a locally irregular edge coloring. The locally irregular chromatic index X'irr(G) of a colorable graph G is the smallest number of colors required by a locally irregular edge coloring of G. The Local Irregularity Conjecture claims that all colorable graphs require at most 3 colors for a locally irregular edge coloring. Recently, it has been observed that the conjecture does not hold for the bow-tie graph B, since B is colorable and requires at least 4 colors for a locally irregular edge coloring. Since B is a cactus graph and all non-colorable graphs are also cacti, this seems to be a relevant class of graphs for the Local Irregularity Conjecture. In this paper we establish that X'irr(G)<= 4 for all colorable cactus graphs.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-07-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42390599","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}
{"title":"Families of specialized Euler sums","authors":"A. Sofo","doi":"10.47443/dml.2022.064","DOIUrl":"https://doi.org/10.47443/dml.2022.064","url":null,"abstract":"Abstract A family of Euler sums is investigated that adds a new important class to the vast literature of existing knowledge of representation of Euler sums in terms of well-known special functions such as the Riemann zeta and Dirichet beta functions. Some examples are given to highlight the obtained theorems.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-07","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"42650780","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}
{"title":"Involutions containing exactly r pairs of intersecting arcs","authors":"T. Mansour","doi":"10.47443/dml.2022.046","DOIUrl":"https://doi.org/10.47443/dml.2022.046","url":null,"abstract":"Abstract The generating function Fr(x) that counts the involutions on n letters containing exactly r pairs of intersecting arcs in their graphical representation is studied. More precisely, an algorithm that computes the generating function Fr(x) for any given r ≥ 0 is presented. To derive the result for a given r, the algorithm performs certain routine checks on involutions of length 2r + 2 without fixed points. The algorithm is implemented in Maple and yields explicit formulas for 0 ≤ r ≤ 4.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-06-04","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"45458764","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}
{"title":"A Curious Family of Convex Benzenoids and Their Altans","authors":"N. Bašić, P. Fowler","doi":"10.47443/dml.2021.s218","DOIUrl":"https://doi.org/10.47443/dml.2021.s218","url":null,"abstract":"The altan graph of G , a ( G, H ) , is constructed from graph G by choosing an attachment set H from the vertices of G and attaching vertices of H to alternate vertices of a new perimeter cycle of length 2 | H | . When G is a polycyclic plane graph with maximum degree 3 , the natural choice for the attachment set is to take all perimeter degree- 2 vertices in the order encountered in a walk around the perimeter. The construction has implications for the electronic structure and chemistry of carbon nanostructures with molecular graph a ( G, H ) , as kernel eigenvectors of the altan correspond to non-bonding π molecular orbitals of the corresponding unsaturated hydrocarbon. Benzenoids form an important subclass of carbon nanostructures. A convex benzenoid has a boundary on which all vertices of degree 3 have exactly two neighbours of degree 2 . The nullity of a graph is the dimension of the kernel of its adjacency matrix. The possible values for the excess nullity of a ( G, H ) over that of G are 2 , 1 , or 0 . Moreover, altans of benzenoids have nullity at least 1 . Examples of benzenoids where the excess nullity is 2 were found recently. It has been conjectured that the excess nullity when G is a convex benzenoid is at most 1 . Here, we exhibit an infinite family of convex benzenoids with 3 -fold dihedral symmetry (point group D 3h ) where nullity increases from 2 to 3 under altanisation. This family accounts for all known examples with the excess nullity of 1 where the parent graph is a singular convex benzenoid.","PeriodicalId":36023,"journal":{"name":"Discrete Mathematics Letters","volume":null,"pages":null},"PeriodicalIF":0.8,"publicationDate":"2022-05-19","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":null,"resultStr":null,"platform":"Semanticscholar","paperid":"48575180","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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