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Antti H S Laaksonen 2017-02-18 16:22:23 +02:00
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@ -29,7 +29,7 @@ for the number $4$:
A combinatorial problem can often be solved A combinatorial problem can often be solved
using a recursive function. using a recursive function.
In this problem, we can define a function $f(n)$ In this problem, we can define a function $f(n)$
that counts the number of representations for $n$. that gives the number of representations for $n$.
For example, $f(4)=8$ according to the above example. For example, $f(4)=8$ according to the above example.
The values of the function The values of the function
can be recursively calculated as follows: can be recursively calculated as follows:
@ -42,7 +42,7 @@ can be recursively calculated as follows:
The base case is $f(1)=1$, The base case is $f(1)=1$,
because there is only one way to represent the number 1. because there is only one way to represent the number 1.
When $n>1$, we go through all ways to When $n>1$, we go through all ways to
select the last number in the sum. choose the last number in the sum.
For example, in when $n=4$, the sum can end For example, in when $n=4$, the sum can end
with $+1$, $+2$ or $+3$. with $+1$, $+2$ or $+3$.
In addition, we also count the representation In addition, we also count the representation
@ -63,7 +63,7 @@ It turns out that the function also has a closed-form formula
f(n)=2^{n-1}, f(n)=2^{n-1},
\] \]
which is based on the fact that there are $n-1$ which is based on the fact that there are $n-1$
possible positions for +-signs in the sum, possible positions for +-signs in the sum
and we can choose any subset of them. and we can choose any subset of them.
\section{Binomial coefficients} \section{Binomial coefficients}
@ -369,8 +369,8 @@ The following rules precisely define all
valid parenthesis expressions: valid parenthesis expressions:
\begin{itemize} \begin{itemize}
\item The expression \texttt{()} is valid. \item The empty expression is valid.
\item If a expression $A$ is valid, \item If an expression $A$ is valid,
then also the expression then also the expression
\texttt{(}$A$\texttt{)} is valid. \texttt{(}$A$\texttt{)} is valid.
\item If expressions $A$ and $B$ are valid, \item If expressions $A$ and $B$ are valid,
@ -397,7 +397,7 @@ such that both parts are valid
expressions and the first part is as short as possible expressions and the first part is as short as possible
but not empty. but not empty.
For any $i$, the first part contains $i+1$ pairs For any $i$, the first part contains $i+1$ pairs
of parentheses, and the number of expressions of parentheses and the number of expressions
is the product of the following values: is the product of the following values:
\begin{itemize} \begin{itemize}
@ -559,12 +559,12 @@ that corresponds to the area of the region
that belongs to at least one circle. that belongs to at least one circle.
The picture shows that we can calculate The picture shows that we can calculate
the area of $A \cup B$ by first summing the the area of $A \cup B$ by first summing the
areas of $A$ and $B$, and then subtracting areas of $A$ and $B$ and then subtracting
the area of $A \cap B$. the area of $A \cap B$.
The same idea can be applied when the number The same idea can be applied when the number
of sets is larger. of sets is larger.
When there are three sets, the inclusio-exclusion formula is When there are three sets, the inclusion-exclusion formula is
\[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \] \[ |A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |A \cap C| - |B \cap C| + |A \cap B \cap C| \]
and the corresponding picture is and the corresponding picture is
@ -594,7 +594,7 @@ If the intersection contains an odd number of sets,
its size is added to the answer, its size is added to the answer,
and otherwise its size is subtracted from the answer. and otherwise its size is subtracted from the answer.
Note that similar formulas can also be used Note that there are similar formulas
for calculating for calculating
the size of an intersection from the sizes of the size of an intersection from the sizes of
unions. For example, unions. For example,
@ -678,10 +678,10 @@ elements should be changed.
\index{Burnside's lemma} \index{Burnside's lemma}
\key{Burnside's lemma} counts the number of \key{Burnside's lemma} can be used to count
combinations so that the number of combinations so that
only one representative is counted only one representative is counted
for each group of symmetric combinations, for each group of symmetric combinations.
Burnside's lemma states that the number of Burnside's lemma states that the number of
combinations is combinations is
\[\sum_{k=1}^n \frac{c(k)}{n},\] \[\sum_{k=1}^n \frac{c(k)}{n},\]