Ryser–Woodallλ$\lambda$设计猜想

IF 0.5 4区数学 Q3 MATHEMATICS

Journal of Combinatorial Designs Pub Date : 2023-02-26 DOI:10.1002/jcd.21878

Navin M. Singhi, Mohan S. Shrikhande, Rajendra M. Pawale

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We denote the block size <math>\n <semantics>\n <mrow>\n <mo>∣</mo>\n \n <msub>\n <mi>B</mi>\n \n <mi>j</mi>\n </msub>\n \n <mo>∣</mo>\n </mrow>\n <annotation> $| {B}_{j}| $</annotation>\n </semantics></math> by <math>\n <semantics>\n <mrow>\n <msub>\n <mi>k</mi>\n \n <mi>j</mi>\n </msub>\n </mrow>\n <annotation> ${k}_{j}$</annotation>\n </semantics></math> and by <math>\n <semantics>\n <mrow>\n <msubsup>\n <mi>k</mi>\n \n <mi>j</mi>\n \n <mo>′</mo>\n </msubsup>\n </mrow>\n <annotation> ${k}_{j}^{^{\\prime} }$</annotation>\n </semantics></math> (respectively, <math>\n <semantics>\n <mrow>\n <msubsup>\n <mi>k</mi>\n \n <mi>j</mi>\n \n <mo>*</mo>\n </msubsup>\n </mrow>\n <annotation> ${k}_{j}^{* }$</annotation>\n </semantics></math>) the number of points with replication number <math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mn>1</mn>\n </msub>\n </mrow>\n <annotation> ${r}_{1}$</annotation>\n </semantics></math> (respectively, <math>\n <semantics>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mn>2</mn>\n </msub>\n </mrow>\n <annotation> ${r}_{2}$</annotation>\n </semantics></math>) in block <math>\n <semantics>\n <mrow>\n <msub>\n <mi>B</mi>\n \n <mi>j</mi>\n </msub>\n </mrow>\n <annotation> ${B}_{j}$</annotation>\n </semantics></math> of <math>\n <semantics>\n <mrow>\n <mi>D</mi>\n </mrow>\n <annotation> $D$</annotation>\n </semantics></math>. Take <math>\n <semantics>\n <mrow>\n <mi>g</mi>\n \n <mo>=</mo>\n \n <mtext>gcd</mtext>\n \n <mfenced>\n <mrow>\n <mfrac>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>−</mo>\n \n <msub>\n <mi>r</mi>\n \n <mn>2</mn>\n </msub>\n </mrow>\n \n <mrow>\n <mtext>gcd</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>r</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>−</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <msub>\n <mi>r</mi>\n \n <mn>2</mn>\n </msub>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mfrac>\n \n <mo>,</mo>\n \n <mi>λ</mi>\n </mrow>\n </mfenced>\n \n <mo>,</mo>\n \n <mi>λ</mi>\n \n <mo>=</mo>\n \n <msub>\n <mi>λ</mi>\n \n <mn>1</mn>\n </msub>\n \n <mi>g</mi>\n \n <mo>,</mo>\n \n <mfrac>\n <mrow>\n <msub>\n <mi>r</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>−</mo>\n \n <msub>\n <mi>r</mi>\n \n <mn>2</mn>\n </msub>\n </mrow>\n \n <mrow>\n <mtext>gcd</mtext>\n \n <mrow>\n <mo>(</mo>\n \n <mrow>\n <msub>\n <mi>r</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>−</mo>\n \n <mn>1</mn>\n \n <mo>,</mo>\n \n <msub>\n <mi>r</mi>\n \n <mn>2</mn>\n </msub>\n \n <mo>−</mo>\n \n <mn>1</mn>\n </mrow>\n \n <mo>)</mo>\n </mrow>\n </mrow>\n </mfrac>\n \n <mo>=</mo>\n \n <mi>g</mi>\n \n <mi>m</mi>\n </mrow>\n <annotation> $g=\\text{gcd}\\left(\\frac{{r}_{1}-{r}_{2}}{\\text{gcd}({r}_{1}-1,{r}_{2}-1)},\\lambda \\right),\\lambda ={\\lambda }_{1}g,\\frac{{r}_{1}-{r}_{2}}{\\text{gcd}({r}_{1}-1,{r}_{2}-1)}=gm$</annotation>\n </semantics></math>, for positive integers <math>\n <semantics>\n <mrow>\n <msub>\n <mi>λ</mi>\n \n <mn>1</mn>\n </msub>\n \n <mo>,</mo>\n \n <mi>m</mi>\n </mrow>\n <annotation> ${\\lambda }_{1},m$</annotation>\n </semantics></math> and let <math>\n <semantics>\n <mrow>\n <msub>\n <mi>g</mi>\n \n <mn>1</mn>\n </msub>\n </mrow>\n <annotation> ${g}_{1}$</annotation>\n </semantics></math> be the largest divisor of <math>\n <semantics>\n <mrow>\n <msub>\n <mi>λ</mi>\n \n <mn>1</mn>\n </msub>\n </mrow>\n <annotation> ${\\lambda }_{1}$</annotation>\n </semantics></math> such that if <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n </mrow>\n <annotation> $p$</annotation>\n </semantics></math> is a prime dividing <math>\n <semantics>\n <mrow>\n <msub>\n <mi>g</mi>\n \n <mn>1</mn>\n </msub>\n </mrow>\n <annotation> ${g}_{1}$</annotation>\n </semantics></math>, then <math>\n <semantics>\n <mrow>\n <mi>p</mi>\n </mrow>\n <annotation> $p$</annotation>\n </semantics></math> divides <math>\n <semantics>\n <mrow>\n <mi>g</mi>\n </mrow>\n <annotation> $g$</annotation>\n </semantics></math>. We obtain the following results:\n\n </p>","PeriodicalId":15389,"journal":{"name":"Journal of Combinatorial Designs","volume":"31 5","pages":"267-276"},"PeriodicalIF":0.5000,"publicationDate":"2023-02-26","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"","citationCount":"0","resultStr":"{\"title\":\"Towards the Ryser–Woodall \\n \\n \\n λ\\n \\n $\\\\lambda $\\n -design conjecture\",\"authors\":\"Navin M. Singhi, Mohan S. Shrikhande, Rajendra M. Pawale\",\"doi\":\"10.1002/jcd.21878\",\"DOIUrl\":null,\"url\":null,\"abstract\":\"<p>Let <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation> ${r}_{1}$</annotation>\\n </semantics></math> and <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>2</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>></mo>\\n \\n <msub>\\n <mi>r</mi>\\n \\n <mn>2</mn>\\n </msub>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n <annotation> ${r}_{2},({r}_{1}\\\\gt {r}_{2})$</annotation>\\n </semantics></math> be the two replication numbers of a <math>\\n <semantics>\\n <mrow>\\n <mi>λ</mi>\\n </mrow>\\n <annotation> $\\\\lambda $</annotation>\\n </semantics></math>-design <math>\\n <semantics>\\n <mrow>\\n <mi>D</mi>\\n </mrow>\\n <annotation> $D$</annotation>\\n </semantics></math>. We denote the block size <math>\\n <semantics>\\n <mrow>\\n <mo>∣</mo>\\n \\n <msub>\\n <mi>B</mi>\\n \\n <mi>j</mi>\\n </msub>\\n \\n <mo>∣</mo>\\n </mrow>\\n <annotation> $| {B}_{j}| $</annotation>\\n </semantics></math> by <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>k</mi>\\n \\n <mi>j</mi>\\n </msub>\\n </mrow>\\n <annotation> ${k}_{j}$</annotation>\\n </semantics></math> and by <math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mi>k</mi>\\n \\n <mi>j</mi>\\n \\n <mo>′</mo>\\n </msubsup>\\n </mrow>\\n <annotation> ${k}_{j}^{^{\\\\prime} }$</annotation>\\n </semantics></math> (respectively, <math>\\n <semantics>\\n <mrow>\\n <msubsup>\\n <mi>k</mi>\\n \\n <mi>j</mi>\\n \\n <mo>*</mo>\\n </msubsup>\\n </mrow>\\n <annotation> ${k}_{j}^{* }$</annotation>\\n </semantics></math>) the number of points with replication number <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation> ${r}_{1}$</annotation>\\n </semantics></math> (respectively, <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>2</mn>\\n </msub>\\n </mrow>\\n <annotation> ${r}_{2}$</annotation>\\n </semantics></math>) in block <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>B</mi>\\n \\n <mi>j</mi>\\n </msub>\\n </mrow>\\n <annotation> ${B}_{j}$</annotation>\\n </semantics></math> of <math>\\n <semantics>\\n <mrow>\\n <mi>D</mi>\\n </mrow>\\n <annotation> $D$</annotation>\\n </semantics></math>. Take <math>\\n <semantics>\\n <mrow>\\n <mi>g</mi>\\n \\n <mo>=</mo>\\n \\n <mtext>gcd</mtext>\\n \\n <mfenced>\\n <mrow>\\n <mfrac>\\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <msub>\\n <mi>r</mi>\\n \\n <mn>2</mn>\\n </msub>\\n </mrow>\\n \\n <mrow>\\n <mtext>gcd</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>r</mi>\\n \\n <mn>2</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mfrac>\\n \\n <mo>,</mo>\\n \\n <mi>λ</mi>\\n </mrow>\\n </mfenced>\\n \\n <mo>,</mo>\\n \\n <mi>λ</mi>\\n \\n <mo>=</mo>\\n \\n <msub>\\n <mi>λ</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mi>g</mi>\\n \\n <mo>,</mo>\\n \\n <mfrac>\\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <msub>\\n <mi>r</mi>\\n \\n <mn>2</mn>\\n </msub>\\n </mrow>\\n \\n <mrow>\\n <mtext>gcd</mtext>\\n \\n <mrow>\\n <mo>(</mo>\\n \\n <mrow>\\n <msub>\\n <mi>r</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n \\n <mo>,</mo>\\n \\n <msub>\\n <mi>r</mi>\\n \\n <mn>2</mn>\\n </msub>\\n \\n <mo>−</mo>\\n \\n <mn>1</mn>\\n </mrow>\\n \\n <mo>)</mo>\\n </mrow>\\n </mrow>\\n </mfrac>\\n \\n <mo>=</mo>\\n \\n <mi>g</mi>\\n \\n <mi>m</mi>\\n </mrow>\\n <annotation> $g=\\\\text{gcd}\\\\left(\\\\frac{{r}_{1}-{r}_{2}}{\\\\text{gcd}({r}_{1}-1,{r}_{2}-1)},\\\\lambda \\\\right),\\\\lambda ={\\\\lambda }_{1}g,\\\\frac{{r}_{1}-{r}_{2}}{\\\\text{gcd}({r}_{1}-1,{r}_{2}-1)}=gm$</annotation>\\n </semantics></math>, for positive integers <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>λ</mi>\\n \\n <mn>1</mn>\\n </msub>\\n \\n <mo>,</mo>\\n \\n <mi>m</mi>\\n </mrow>\\n <annotation> ${\\\\lambda }_{1},m$</annotation>\\n </semantics></math> and let <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>g</mi>\\n \\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation> ${g}_{1}$</annotation>\\n </semantics></math> be the largest divisor of <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>λ</mi>\\n \\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation> ${\\\\lambda }_{1}$</annotation>\\n </semantics></math> such that if <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n <annotation> $p$</annotation>\\n </semantics></math> is a prime dividing <math>\\n <semantics>\\n <mrow>\\n <msub>\\n <mi>g</mi>\\n \\n <mn>1</mn>\\n </msub>\\n </mrow>\\n <annotation> ${g}_{1}$</annotation>\\n </semantics></math>, then <math>\\n <semantics>\\n <mrow>\\n <mi>p</mi>\\n </mrow>\\n <annotation> $p$</annotation>\\n </semantics></math> divides <math>\\n <semantics>\\n <mrow>\\n <mi>g</mi>\\n </mrow>\\n <annotation> $g$</annotation>\\n </semantics></math>. We obtain the following results:\\n\\n </p>\",\"PeriodicalId\":15389,\"journal\":{\"name\":\"Journal of Combinatorial Designs\",\"volume\":\"31 5\",\"pages\":\"267-276\"},\"PeriodicalIF\":0.5000,\"publicationDate\":\"2023-02-26\",\"publicationTypes\":\"Journal Article\",\"fieldsOfStudy\":null,\"isOpenAccess\":false,\"openAccessPdf\":\"\",\"citationCount\":\"0\",\"resultStr\":null,\"platform\":\"Semanticscholar\",\"paperid\":null,\"PeriodicalName\":\"Journal of Combinatorial Designs\",\"FirstCategoryId\":\"100\",\"ListUrlMain\":\"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21878\",\"RegionNum\":4,\"RegionCategory\":\"数学\",\"ArticlePicture\":[],\"TitleCN\":null,\"AbstractTextCN\":null,\"PMCID\":null,\"EPubDate\":\"\",\"PubModel\":\"\",\"JCR\":\"Q3\",\"JCRName\":\"MATHEMATICS\",\"Score\":null,\"Total\":0}","platform":"Semanticscholar","paperid":null,"PeriodicalName":"Journal of Combinatorial Designs","FirstCategoryId":"100","ListUrlMain":"https://onlinelibrary.wiley.com/doi/10.1002/jcd.21878","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}

引用次数: 0

摘要

设r 1${r}_{1} $和r2，（r1&gt；r2）${r}_{2} ({r}_{1} \gt{r}_{2} ）$是λ$\lambda$设计D$D$的两个复制数。我们表示块大小ŞBjŞ$|{B}_{j} |$by kj${k}_{j} $和by k j′${k}_{j} ^｛^｛\prime｝｝$（分别为kj*${k}_{j} ^｛*｝$）复制编号为r 1的点数${r}_{1} $（分别，r 2${r}_{2} $）${B}_{j} $$D$D$。取g=gcd r 1−r 2 gcd（r 1−1.r 2−1），λ，λ= 设r 1${r}_{1} $和r2，（r1&gt；r2）${r}_{2} ({r}_{1} \gt{r}_{2} ）$是λ$\lambda$设计D$D$的两个复制数。我们表示块大小ŞBjŞ$|{B}_{j} |$by kj${k}_{j} $和by k j′${k}_{j} ^｛^｛\prime｝｝$（分别为kj*${k}_{j} ^｛*｝$）复制编号为r 1的点数${r}_{1} $（分别，r 2${r}_{2} $）${B}_{j} $$D$D$。

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Towards the Ryser–Woodall λ $\lambda $ -design conjecture

Let $r_{1}$ and $r_{2}, (r_{1} > r_{2})$ be the two replication numbers of a $λ$ -design $D$ . We denote the block size $∣ B_{j} ∣$ by $k_{j}$ and by $k_{j}^{'}$ (respectively, $k_{j}^{*}$ ) the number of points with replication number $r_{1}$ (respectively, $r_{2}$ ) in block $B_{j}$ of $D$ . Take $g = gcd (\frac{r_{1} - r_{2}}{gcd (r_{1} - 1, r_{2} - 1)}, λ), λ = λ_{1} g, \frac{r_{1} - r_{2}}{gcd (r_{1} - 1, r_{2} - 1)} = g m$ , for positive integers $λ_{1}, m$ and let $g_{1}$ be the largest divisor of $λ_{1}$ such that if $p$ is a prime dividing $g_{1}$ , then $p$ divides $g$ . We obtain the following results:

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来源期刊

Journal of Combinatorial Designs 数学-数学

CiteScore

1.60

自引率

14.30%

发文量

审稿时长

>12 weeks

期刊介绍： The Journal of Combinatorial Designs is an international journal devoted to the timely publication of the most influential papers in the area of combinatorial design theory. All topics in design theory, and in which design theory has important applications, are covered, including: block designs, t-designs, pairwise balanced designs and group divisible designs Latin squares, quasigroups, and related algebras computational methods in design theory construction methods applications in computer science, experimental design theory, and coding theory graph decompositions, factorizations, and design-theoretic techniques in graph theory and extremal combinatorics finite geometry and its relation with design theory. algebraic aspects of design theory. Researchers and scientists can depend on the Journal of Combinatorial Designs for the most recent developments in this rapidly growing field, and to provide a forum for both theoretical research and applications. All papers appearing in the Journal of Combinatorial Designs are carefully peer refereed.