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''PBIBD''
 
''PBIBD''
  
A (symmetric) association scheme with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200801.png" /> classes on the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200802.png" /> symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200803.png" /> satisfies:
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A (symmetric) association scheme with $m$ classes on the $v$ symbols $\{1,...,v\}$ satisfies:
  
Two distinct symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200804.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200805.png" /> are termed <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200806.png" />th associates for exactly one <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200807.png" />;
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Two distinct symbols $x$ and $y$ are termed $i$th associates for exactly one $i\in\{1,...,m\}$; each symbol has exactly $n_i$ $i$th associates; and when two distinct symbols $x$ and $y$ are $i$th associates, the number of other symbols that are $j$th associates of $x$ and also $k$th associates of $y$ is $p^i_jk$, independent of the choice of the $i$th associates $x$ and $y$. The matrices $A_0,...,A_m$ of an $m$-class association scheme are defined as $A_0=I$, and for $1\leq \leq m$, $A_i$ is a $(0,1)$-matrix whose entry $(x,y)$ is $1$ exactly when $x$ and $y$ are $i$th associates.
  
each symbol has exactly <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200808.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p1200809.png" />th associates; and
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Let $X$ be a $v$-set with a symmetric $m$-class association scheme defined on it. A partially balanced incomplete block design with $m$ associate classes (or $\mathrm{PBIBD}(m)$) is a [[Block design|block design]] based on $X$ with $b$ sets (the blocks), each of size $k$, and with each symbol appearing in $r$ blocks. Any two symbols that are $i$th associates appear together in $\lambda_i$ blocks of $\mathrm{PBIBD}(m)$. The numbers $v,b,r,k,\lambda_i$ ($1\leq i\leq m$) are the parameters of $\mathrm{PBIBD}(m)$. The notation $\mathrm{PBIBD}(v,k,\lambda_i)$ is also used. $N$ is used for the $v\times b$ $(0,1)$ incidence matrix of $\mathrm{PBIBD}(m)$.
  
when two distinct symbols <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008010.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008011.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008012.png" />th associates, the number of other symbols that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008013.png" />th associates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008014.png" /> and also <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008015.png" />th associates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008016.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008017.png" />, independent of the choice of the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008018.png" />th associates <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008019.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008020.png" />. The matrices <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008021.png" /> of an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008022.png" />-class association scheme are defined as <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008023.png" />, and for <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008024.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008025.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008026.png" />-matrix whose entry <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008027.png" /> is <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008028.png" /> exactly when <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008029.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008030.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008031.png" />th associates.
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Let $A_0,...,A_m$ be the matrices of an association scheme corresponding to a $\mathrm{PBIBD}(m)$. Then $NN^T=rI+\sum_i\lambda_iA_i$ and $JN=kJ$. Conversely, if $N$ is a $(0,1)$-matrix which satisfies these conditions and the $A_i$ are the matrices of an association scheme, then $N$ is the incidence matrix of a $\mathrm{PBIBD}(m)$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008032.png" /> be a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008033.png" />-set with a symmetric <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008034.png" />-class association scheme defined on it. A partially balanced incomplete block design with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008035.png" /> associate classes (or <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008036.png" />) is a [[Block design|block design]] based on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008037.png" /> with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008038.png" /> sets (the blocks), each of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008039.png" />, and with each symbol appearing in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008040.png" /> blocks. Any two symbols that are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008041.png" />th associates appear together in <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008042.png" /> blocks of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008043.png" />. The numbers <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008044.png" /> (<img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008045.png" />) are the parameters of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008046.png" />. The notation <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008047.png" /> is also used. <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008048.png" /> is used for the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008049.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008050.png" /> incidence matrix of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008051.png" />.
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It is easily verified that $vr=bk$, that $\sum^m_{i=1}n_i=v-1$, and that $\sum^m_{i=1}\lambda_in_i=r(k-1)$. A $\mathrm{PBIBD}(1)$ is a balanced incomplete block design (a BIBD; cf. [[Block design|Block design]]); also, a $\mathrm{PBIBD}(2)$ in which $\lambda_1=\lambda_2$ is a BIBD.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008052.png" /> be the matrices of an association scheme corresponding to a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008053.png" />. Then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008054.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008055.png" />. Conversely, if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008056.png" /> is a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008057.png" />-matrix which satisfies these conditions and the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008058.png" /> are the matrices of an association scheme, then <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008059.png" /> is the incidence matrix of a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008060.png" />.
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There are six types of $\mathrm{PBIBD}(2)$s, [[#References|[a3]]], based on the underlying types of association schemes:
 
 
It is easily verified that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008061.png" />, that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008062.png" />, and that <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008063.png" />. A <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008064.png" /> is a balanced incomplete block design (a BIBD; cf. [[Block design|Block design]]); also, a <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008065.png" /> in which <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008066.png" /> is a BIBD.
 
 
 
There are six types of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008067.png" />s, [[#References|[a3]]], based on the underlying types of association schemes:
 
  
 
1) group divisible;
 
1) group divisible;
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6) miscellaneous.
 
6) miscellaneous.
  
Partition the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008068.png" />-set <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008069.png" /> into <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008070.png" /> groups each of size <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008071.png" />. In a group-divisible association scheme the first associates are the symbols in the same group and the second associates are all the other symbols. The eigenvalues of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008072.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008073.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008074.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008075.png" />, with multiplicities <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008076.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008077.png" />, and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008078.png" />, respectively. A group-divisible partially balanced incomplete block design is singular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008079.png" />; semi-regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008080.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008081.png" />; and regular if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008082.png" /> and <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008083.png" />.
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Partition the $v$-set $X$ into $m$ groups each of size $n$. In a group-divisible association scheme the first associates are the symbols in the same group and the second associates are all the other symbols. The eigenvalues of $NN^T$ are $rk$, $r-\lambda_1$ and $rk-v\lambda_2$, with multiplicities $1$, $m(n-1)$, and $m-1$, respectively. A group-divisible partially balanced incomplete block design is singular if $r=\lambda_1$; semi-regular if $r>\lambda_1$, $rk-v\lambda_2=0$; and regular if $r>\lambda_1$ and $rk>v\lambda_2$.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008084.png" />, <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008085.png" />, and arrange the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008086.png" /> elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008087.png" /> in a symmetrical <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008088.png" />-array with the diagonal entries blank. In the triangular association scheme, the first associates of a symbol are those in the same row or column of the array; all other symbols are second associates. The duals of triangular <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008089.png" />s are the residual designs of symmetric BIBDs with <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008090.png" />. Triangular schemes and generalized triangular schemes are also known as Johnson schemes.
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Let $v=n(n-1)/2$, $n\geq 5$, and arrange the $v$ elements of $X$ in a symmetrical $(n\times n)$-array with the diagonal entries blank. In the triangular association scheme, the first associates of a symbol are those in the same row or column of the array; all other symbols are second associates. The duals of triangular $\mathrm{PBIBD}(2)$s are the residual designs of symmetric BIBDs with $\lambda=2$. Triangular schemes and generalized triangular schemes are also known as Johnson schemes.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008091.png" /> and arrange the <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008092.png" /> symbols in an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008093.png" /> array. Superimpose on this array a set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008094.png" /> mutually orthogonal Latin squares (see [[#References|[a1]]] and also [[Latin square|Latin square]]) of order <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008095.png" />. Let the first associates of any symbol be those in the same row or column of the array or be associated with the same symbols in one of the Latin squares. This is an <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008097.png" />-type association scheme. If <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008098.png" />, then the scheme is group divisible; if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p12008099.png" />, then all the symbols are first associates of each other.
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Let $v=n^2$ and arrange the $v$ symbols in an $n\times n$ array. Superimpose on this array a set of $i-2$ mutually orthogonal Latin squares (see [[#References|[a1]]] and also [[Latin square|Latin square]]) of order $n$. Let the first associates of any symbol be those in the same row or column of the array or be associated with the same symbols in one of the Latin squares. This is an $L_i$-type association scheme. If $i=n$, then the scheme is group divisible; if $i=n+1$, then all the symbols are first associates of each other.
  
Let <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080100.png" />. A non-group divisible association scheme defined on <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080101.png" /> is cyclic if <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080102.png" /> and if the set of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080103.png" /> differences of distinct elements of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080104.png" /> has each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080105.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080106.png" /> times and each element of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080107.png" /> <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080108.png" /> times. The first associates of <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080109.png" /> are <img align="absmiddle" border="0" src="https://www.encyclopediaofmath.org/legacyimages/p/p120/p120080/p120080110.png" />.
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Let $\{1,...,v-1\}=D\cup E$. A non-group divisible association scheme defined on $Z_v$ is cyclic if $D=-D$ and if the set of $n_1(n_1-1)$ differences of distinct elements of $D$ has each element of $D$ $p^1_11$ times and each element of $E$ $p^2_11$ times. The first associates of $i$ are $i+D$.
  
 
In a partial-geometry-type association scheme, two symbols are first associates if they are incident with a line of the geometry and second associates if they are not incident with a line of the geometry.
 
In a partial-geometry-type association scheme, two symbols are first associates if they are incident with a line of the geometry and second associates if they are not incident with a line of the geometry.

Latest revision as of 06:10, 23 December 2020

PBIBD

A (symmetric) association scheme with $m$ classes on the $v$ symbols $\{1,...,v\}$ satisfies:

Two distinct symbols $x$ and $y$ are termed $i$th associates for exactly one $i\in\{1,...,m\}$; each symbol has exactly $n_i$ $i$th associates; and when two distinct symbols $x$ and $y$ are $i$th associates, the number of other symbols that are $j$th associates of $x$ and also $k$th associates of $y$ is $p^i_jk$, independent of the choice of the $i$th associates $x$ and $y$. The matrices $A_0,...,A_m$ of an $m$-class association scheme are defined as $A_0=I$, and for $1\leq \leq m$, $A_i$ is a $(0,1)$-matrix whose entry $(x,y)$ is $1$ exactly when $x$ and $y$ are $i$th associates.

Let $X$ be a $v$-set with a symmetric $m$-class association scheme defined on it. A partially balanced incomplete block design with $m$ associate classes (or $\mathrm{PBIBD}(m)$) is a block design based on $X$ with $b$ sets (the blocks), each of size $k$, and with each symbol appearing in $r$ blocks. Any two symbols that are $i$th associates appear together in $\lambda_i$ blocks of $\mathrm{PBIBD}(m)$. The numbers $v,b,r,k,\lambda_i$ ($1\leq i\leq m$) are the parameters of $\mathrm{PBIBD}(m)$. The notation $\mathrm{PBIBD}(v,k,\lambda_i)$ is also used. $N$ is used for the $v\times b$ $(0,1)$ incidence matrix of $\mathrm{PBIBD}(m)$.

Let $A_0,...,A_m$ be the matrices of an association scheme corresponding to a $\mathrm{PBIBD}(m)$. Then $NN^T=rI+\sum_i\lambda_iA_i$ and $JN=kJ$. Conversely, if $N$ is a $(0,1)$-matrix which satisfies these conditions and the $A_i$ are the matrices of an association scheme, then $N$ is the incidence matrix of a $\mathrm{PBIBD}(m)$.

It is easily verified that $vr=bk$, that $\sum^m_{i=1}n_i=v-1$, and that $\sum^m_{i=1}\lambda_in_i=r(k-1)$. A $\mathrm{PBIBD}(1)$ is a balanced incomplete block design (a BIBD; cf. Block design); also, a $\mathrm{PBIBD}(2)$ in which $\lambda_1=\lambda_2$ is a BIBD.

There are six types of $\mathrm{PBIBD}(2)$s, [a3], based on the underlying types of association schemes:

1) group divisible;

2) triangular;

3) Latin-square-type;

4) cyclic;

5) partial-geometry-type; and

6) miscellaneous.

Partition the $v$-set $X$ into $m$ groups each of size $n$. In a group-divisible association scheme the first associates are the symbols in the same group and the second associates are all the other symbols. The eigenvalues of $NN^T$ are $rk$, $r-\lambda_1$ and $rk-v\lambda_2$, with multiplicities $1$, $m(n-1)$, and $m-1$, respectively. A group-divisible partially balanced incomplete block design is singular if $r=\lambda_1$; semi-regular if $r>\lambda_1$, $rk-v\lambda_2=0$; and regular if $r>\lambda_1$ and $rk>v\lambda_2$.

Let $v=n(n-1)/2$, $n\geq 5$, and arrange the $v$ elements of $X$ in a symmetrical $(n\times n)$-array with the diagonal entries blank. In the triangular association scheme, the first associates of a symbol are those in the same row or column of the array; all other symbols are second associates. The duals of triangular $\mathrm{PBIBD}(2)$s are the residual designs of symmetric BIBDs with $\lambda=2$. Triangular schemes and generalized triangular schemes are also known as Johnson schemes.

Let $v=n^2$ and arrange the $v$ symbols in an $n\times n$ array. Superimpose on this array a set of $i-2$ mutually orthogonal Latin squares (see [a1] and also Latin square) of order $n$. Let the first associates of any symbol be those in the same row or column of the array or be associated with the same symbols in one of the Latin squares. This is an $L_i$-type association scheme. If $i=n$, then the scheme is group divisible; if $i=n+1$, then all the symbols are first associates of each other.

Let $\{1,...,v-1\}=D\cup E$. A non-group divisible association scheme defined on $Z_v$ is cyclic if $D=-D$ and if the set of $n_1(n_1-1)$ differences of distinct elements of $D$ has each element of $D$ $p^1_11$ times and each element of $E$ $p^2_11$ times. The first associates of $i$ are $i+D$.

In a partial-geometry-type association scheme, two symbols are first associates if they are incident with a line of the geometry and second associates if they are not incident with a line of the geometry.

See [a2], [a4], [a5] for further information.

References

[a1] R.J.R. Abel, A.E. Brouwer, C.J. Colbourn, J.H. Dinitz, "Mutually orthogonal latin squares" C.J. Colbourn (ed.) J.H. Dinitz (ed.) , CRC Handbook of Combinatorial Designs , CRC (1996) pp. 111–141
[a2] R.A. Bailey, "Partially balanced designs" N.L. Johnson (ed.) S. Kotz (ed.) C. Read (ed.) , Encycl. Stat. Sci. , 6 , Wiley (1985) pp. 593–610
[a3] W.H. Clatworthy, "Tables of two-associate-class partially balanced designs" , Applied Math. Ser. , 63 , Nat. Bureau of Standards (US) (1973)
[a4] D. Raghavarao, "Constructions and combinatorial problems in design of experiments" , Wiley (1971)
[a5] D.J. Street, A.P. Street, "Partially balanced incomplete block designs" C.J. Colbourn (ed.) J.H. Dinitz (ed.) , CRC Handbook of Combinatorial Designs , CRC (1996) pp. 419–423
How to Cite This Entry:
Partially balanced incomplete block design. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Partially_balanced_incomplete_block_design&oldid=17069
This article was adapted from an original article by C.J. Colbourn (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article