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{"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}
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Abstract
Let
r
1
${r}_{1}$
and
r
2
,
(
r
1
>
r
2
)
${r}_{2},({r}_{1}\gt {r}_{2})$
be the two replication numbers of a
λ
$\lambda $
-design
D
$D$
. We denote the block size
∣
B
j
∣
$| {B}_{j}| $
by
k
j
${k}_{j}$
and by
k
j
′
${k}_{j}^{^{\prime} }$
(respectively,
k
j
*
${k}_{j}^{* }$
) the number of points with replication number
r
1
${r}_{1}$
(respectively,
r
2
${r}_{2}$
) in block
B
j
${B}_{j}$
of
D
$D$
. Take
g
=
gcd
r
1
−
r
2
gcd
(
r
1
−
1
,
r
2
−
1
)
,
λ
,
λ
=
λ
1
g
,
r
1
−
r
2
gcd
(
r
1
−
1
,
r
2
−
1
)
=
g
m
$g=\text{gcd}\left(\frac{{r}_{1}-{r}_{2}}{\text{gcd}({r}_{1}-1,{r}_{2}-1)},\lambda \right),\lambda ={\lambda }_{1}g,\frac{{r}_{1}-{r}_{2}}{\text{gcd}({r}_{1}-1,{r}_{2}-1)}=gm$
, for positive integers
λ
1
,
m
${\lambda }_{1},m$
and let
g
1
${g}_{1}$
be the largest divisor of
λ
1
${\lambda }_{1}$
such that if
p
$p$
is a prime dividing
g
1
${g}_{1}$
, then
p
$p$
divides
g
$g$
. We obtain the following results:
Ryser–Woodallλ$\lambda$设计猜想
设r 1${r}_{1} $和r2，（r1>；r2）${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} $$D$D$。取g=gcd r 1−r 2 gcd（r 1−1.r 2−1），λ，λ= 设r 1${r}_{1} $和r2，（r1>；r2）${r}_{2} ({r}_{1} \gt{r}_{2} ）$是λ$\lambda$设计D$D$的两个复制数。我们表示块大小ŞBjŞ$|{B}_{j} |$by kj${k}_{j} $和by k j′${k}_{j} ^｛^｛\prime｝｝$（分别为kj*${k}_{j} ^｛*｝$）复制编号为r 1的点数${r}_{1} $（分别，r 2${r}_{2} $）${B}_{j} $$D$D$。
本文章由计算机程序翻译，如有差异，请以英文原文为准。
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