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\begin{document}
\centerline{Math 372 lecture for Wednesday, Week 2}
\bigskip

\medskip

{\bf Definition.} A \emph{poset} (\emph{partially ordered set}) is a finite
set~$P$ and a relation~$\leq$ defined for some (not necessarily all) elements
of~$P$ satisfying:
\begin{enumerate}
  \item $x\leq x$ for all~$x\in P$ (reflexivity)
  \item $x\leq y$ and~$y\leq x$ imply~$x=y$ (antisymmetry)
  \item $x\leq y$ and~$y\leq z$ imply~$x\leq z$ (transitivity).
\end{enumerate}
Given~$\leq$, we use the usual conventions for the symbols~$\geq$, $<$,
$>$, $\lneq$,~$\not\leq$ etc.  If neither~$x\leq y$ nor~$y\leq x$, we say~$x$
and~$y$ are {\em incomparable}.
\medskip

{\bf Example.} Let~$X$ be a finite set, and let~$P=2^X$, the collection of all
subsets of~$X$.  For~$U,V\in P$ define~$U\leq V$ if~$U\subseteq V$ in~$X$.
Then~$(P,\leq)$ is a poset called a {\em boolean poset} and denoted~$B_X$.
If~$X=[n]:=\left\{ 1,\dots,n \right\}$, then we write~$B_n$ for~$B_X$.

If~$P$ is a poset and~$x,y\in P$, we say~$y$ {\em covers}~$x$ and write~$x\lessdot y$
if~$x<y$ and there does not exist~$z\in P$ such that~$x<z<y$.  The {\em Hasse
diagram} for a poset is a graph in the plane whose vertices are the elements of
$P$, the element $x$ is drawn below~$y$ if~$x<y$, and there is an edge
between~$x$ and~$y$ if~$y$ covers~$x$.  Here is the Hasse diagram for~$B_4$:
\begin{center}
  \begin{tikzpicture}[scale=1.5]
    \node at (0,0) (0) {$\emptyset$};
    \node at (-1.5,1) (1) {$1$};
    \node at (-0.5,1) (2) {$2$};
    \node at (0.5,1) (3) {$3$};
    \node at (1.5,1) (4) {$4$};
    \node at (-3,2) (12) {$12$};
    \node at (-1.8,2) (13) {$13$};
    \node at (-0.7,2) (14) {$14$};
    \node at (0.7,2) (23) {$23$};
    \node at (1.8,2) (24) {$24$};
    \node at (3,2) (34) {$34$};
    \node at (-1.5,3) (123) {$123$};
    \node at (-0.5,3) (124) {$124$};
    \node at (0.5,3) (134) {$134$};
    \node at (1.5,3) (234) {$234$};
    \node at (0,4) (1234) {$1234$};
    \draw (0)--(1);
    \draw (0)--(2);
    \draw (0)--(3);
    \draw (0)--(4);
    \draw (12)--(1);
    \draw (12)--(2);
    \draw (13)--(1);
    \draw (13)--(3);
    \draw (14)--(1);
    \draw (14)--(4);
    \draw (23)--(2);
    \draw (23)--(3);
    \draw (24)--(2);
    \draw (24)--(4);
    \draw (34)--(3);
    \draw (34)--(4);
    \draw (123)--(12);
    \draw (123)--(13);
    \draw (123)--(23);
    \draw (124)--(12);
    \draw (124)--(14);
    \draw (124)--(24);
    \draw (134)--(13);
    \draw (134)--(14);
    \draw (134)--(34);
    \draw (234)--(23);
    \draw (234)--(24);
    \draw (234)--(34);
    \draw (1234)--(123);
    \draw (1234)--(124);
    \draw (1234)--(134);
    \draw (1234)--(234);
    \node at (3.4,1.9) {.};
  \end{tikzpicture}
\end{center}
\medskip

{\bf Example.} There are five posets with three elements:
\begin{center}
  \begin{tikzpicture}
    \node[ball] at (0,0) {};
    \node[ball] at (0.4,0) {};
    \node[ball] at (0.8,0) {};
    \begin{scope}[xshift=2cm]
      \node[ball] at (0,0) (1) {};
      \node[ball] at (0.4,0) (2) {};
      \node[ball] at (0,0.4) (3) {};
      \draw (1)--(3);
    \end{scope}
    \begin{scope}[xshift=4cm]
      \node[ball] at (0,0) (1) {};
      \node[ball] at (0.4,0) (2) {};
      \node[ball] at (0.2,0.4) (3) {};
      \draw (1)--(3);
      \draw (2)--(3);
    \end{scope}
    \begin{scope}[xshift=6cm]
      \node[ball] at (0,0.4) (1) {};
      \node[ball] at (0.4,0.4) (2) {};
      \node[ball] at (0.2,0) (3) {};
      \draw (1)--(3);
      \draw (2)--(3);
    \end{scope}
    \begin{scope}[xshift=8cm]
      \node[ball] at (0,0) (1) {};
      \node[ball] at (0.0,0.4) (2) {};
      \node[ball] at (0.0,0.8) (3) {};
      \draw (1)--(2)--(3);
    \end{scope}
  \end{tikzpicture}
\end{center}
\medskip

\textbf{Question.} What is the size of the largest collection of
incomparable elements in~$B_n$?  
\medskip

So we are looking for the largest collection of
subsets of~$[n]$ such that if~$U$ and~$V$ are in the collection, then
neither~$U\subseteq V$ nor~$V\subseteq U$.  It turns out that for~$B_4$, the
answer is
\[
  \left\{ 12,13,14,23,24,34 \right\}.
\]
\medskip

In preparation for answering this question in general, we review some vocabulary from the
text (Chapter 4). Let~$P$ be a poset.  Then a string of elements of~$P$ of the
form
\[
  x_0\leq x_1\leq\dots\leq x_n
\]
is called a {\em chain} in~$P$.  A chain of the form
\[
  x_0<x_1<\dots< x_n
\]
has {\em length}~$n$. A chain is {\em maximal} if it is not contained in a
larger chain.  If every maximal chain has length~$n$, then we say~$P$ is {\em
graded of rank~$n$}.  A chain is {\em saturated} if it has the form
\[
  x_0\lessdot x_1\lessdot\dots\lessdot x_i
\]
for some~$i$.
If~$P$ is graded of rank~$n$ and~$x\in P$, then the {\em rank} of~$x$
is~$j$, written~$\rho(x)=j$ if~$j$ is the length of the largest saturated chain
in~$P$ having top element~$x$.  The~$i$-th {\em level} of~$P$ is
\[
  P_i=\left\{ x\in P:\rho(x)=i \right\}.
\]
Let~$p_i:=|P_i|$.  Then the rank generating function for the graded
poset~$P$ is
\[
  F(P,q)=\sum_{i=0}^np_iq^i.
\]
We say~$P$ is {\em rank-symmetric} if~$p_i=p_{n-i}$ for~$0\leq i\leq n$, and
{\em rank unimodal} if 
\[
  p_0\leq p_1\leq\dots\leq p_j\geq\dots\geq p_{n-1}\geq p_n
\]
for some~$j$.
\medskip

{\bf Example.} The boolean poset~$B_3$ is graded of rank~$3$ and has~$6$ maximal
chains.  The levels are:
\begin{center}
  \begin{tikzpicture}
    \node at (0,0) (0) {$\emptyset$};
    \node at (-1,1) (1) {$1$};
    \node at (0,1) (2) {$2$};
    \node at (1,1) (3) {$3$};
    \node at (-1,2) (12) {$12$};
    \node at (0,2) (13) {$13$};
    \node at (1,2) (23) {$23$};
    \node at (0,3) (123) {$123$};
    \draw (0)--(1);
    \draw (0)--(2);
    \draw (0)--(3);
    \draw (12)--(1);
    \draw (12)--(2);
    \draw (13)--(1);
    \draw (13)--(3);
    \draw (23)--(2);
    \draw (23)--(3);
    \draw (123)--(12);
    \draw (123)--(13);
    \draw (123)--(23);
    \node at (-3,0) {level $0$};
    \node at (-3,1) {level $1$};
    \node at (-3,2) {level $2$};
    \node at (-3,3) {level $3$};
    \node at (2,1.5) {.};
  \end{tikzpicture}
\end{center}
The sizes of the levels are~$p_0=1$, $p_1=3$, $p_2=3$, and $p_3=1$.  This
sequence, $1,3,3,1$ is symmetric and unimodal.  The rank generating function is
\[
  F(B_3,q)=1+3q+3q^2+q^3.
\]
In general,~$B_n$ is rank-symmetric and unimodal with~$p_i=\binom{n}{i}$
(since~$P_i$ is the number of subsets of~$[n]$ of cardinality~$i$) and
rank generating function
\[
  F(B_n,q)=\sum_{i=0}^{n}\binom{n}{i}q^i=(1+q)^n.
\]
\medskip

An {\em antichain} in any poset~$P$ is a subset of~$P$ in which no distinct pair of
elements is comparable.  For instance, in~$B_3$, the following sets are
antichains:
\[
  \left\{ 2 \right\},\quad \left\{ 1,3 \right\},\quad\left\{ 12,34
  \right\},\quad\left\{ 1,23,24 \right\}.
\]
The following is then a rewording of our question:
\bigskip

\textbf{Question.} What is the size of the largest antichain in~$B_n$?  
\bigskip

{\bf Definition.} A graded poset~$P$ of rank~$n$ has the {\em Sperner property}
if the size of the largest antichain in~$P$ is the maximal cardinality of a level of~$P$:
\[
  \max\left\{ |A|:A\text{ an antichain of~$P$} \right\}
  =
  \max\left\{ |P_i|:i=0,\dots,n \right\}.
\]
Note that~$P_i$ is always an antichain.
\medskip

{\bf Non-example.} The following is the Hasse diagram for a graded poset of rank~$2$ that does not
have the Sperner property:
\begin{center}
  \begin{tikzpicture}
    \node[ball,label={below:$a$}] at (0,0) (a) {};
    \node[ball,label={below:$b$}] at (1,0) (b) {};
    \node[ball,label={below:$c$}] at (2,0) (c) {};
    \node[ball,label={above:$d$}] at (0,1) (d) {};
    \node[ball,label={above:$e$}] at (1,1) (e) {};
    \node[ball,label={above:$f$}] at (2,1) (f) {};
    \draw (a)--(d);
    \draw (b)--(d);
    \draw (c)--(d);
    \draw (c)--(e);
    \draw (c)--(f);
  \end{tikzpicture}
\end{center}
There is one largest antichain, $\left\{ a,b,e,f \right\}$, and it has size~$4$,
whereas~$|P_0|=|P_1|=3$.
\medskip

{\bf Exercise.} Draw the Hasse diagrams of all posets of size~$3$ having the
Sperner property. 
\end{document}


