Difference between revisions of "1986 USAMO Problems/Problem 5"

(Solution)
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== Problem ==
 
== Problem ==
By a partition <math>\pi</math> of an integer <math>n\ge 1</math>, we mean here a representation of <math>n</math> as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if <math>n=4</math>, then the partitions <math>\pi</math> are <math>1+1+1+1</math>, <math>1+1+2</math>, <math>1+3, 2+2</math>, and <math>4</math>).
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By a partition <math>\pi</math> of an integer <math>n\ge 1,</math> we mean here a representation of <math>n</math> as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if <math>n=4,</math> then the partitions <math>\pi</math> are <math>1+1+1+1,</math> <math>1+1+2,</math> <math>1+3, 2+2,</math> and <math>4</math>).
  
For any partition <math>\pi</math>, define <math>A(\pi)</math> to be the number of <math>1</math>'s which appear in <math>\pi</math>, and define <math>B(\pi)</math> to be the number of distinct integers which appear in <math>\pi</math>. (E.g., if <math>n=13</math> and <math>\pi</math> is the partition <math>1+1+2+2+2+5</math>, then <math>A(\pi)=2</math> and <math>B(\pi) = 3</math>).
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For any partition <math>\pi,</math> define <math>A(\pi)</math> to be the number of <math>1</math>'s which appear in <math>\pi,</math> and define <math>B(\pi)</math> to be the number of distinct integers which appear in <math>\pi</math> (E.g., if <math>n=13</math> and <math>\pi</math> is the partition <math>1+1+2+2+2+5,</math> then <math>A(\pi)=2</math> and <math>B(\pi) = 3</math>).
  
Prove that, for any fixed <math>n</math>, the sum of <math>A(\pi)</math> over all partitions of <math>\pi</math> of <math>n</math> is equal to the sum of <math>B(\pi)</math> over all partitions of <math>\pi</math> of <math>n</math>.
+
Prove that, for any fixed <math>n,</math> the sum of <math>A(\pi)</math> over all partitions of <math>\pi</math> of <math>n</math> is equal to the sum of <math>B(\pi)</math> over all partitions of <math>\pi</math> of <math>n.</math>
  
 
== Solution ==
 
== Solution ==
Let <math>S(n) = \sum\limits_{\pi} A(\pi)</math> and let <math>T(n) = \sum\limits_{\pi} B(\pi)</math>. We will use generating functions to approach this problem -- specifically, we will show that the generating functions of <math>S(n)</math> and <math>T(n)</math> are equal.
+
Let <math>S(n) = \sum\limits_{\pi} A(\pi)</math> and let <math>T(n) = \sum\limits_{\pi} B(\pi).</math> We will use generating functions to approach this problem -- specifically, we will show that the generating functions of <math>S(n)</math> and <math>T(n)</math> are equal.
  
Let us start by finding the generating function of <math>S(n)</math>. This function counts the total number of 1's in all the partitions of <math>n</math>. Another way to count this is by counting the number of partitions of <math>n</math> that contain <math>x</math> 1's and multiplying this by <math>x</math>, then summing for <math>1\leq x \leq n</math>. However, the number of partitions of <math>n</math> that contain <math>x</math> 1's is the same as the number of partitions of <math>n-x</math> that contain no 1's, so
+
Let us start by finding the generating function of <math>S(n).</math> This function counts the total number of 1's in all the partitions of <math>n.</math> Another way to count this is by counting the number of partitions of <math>n</math> that contain <math>x</math> 1's and multiplying this by <math>x,</math> then summing for <math>1\leq x \leq n.</math> However, the number of partitions of <math>n</math> that contain <math>x</math> 1's is the same as the number of partitions of <math>n-x</math> that contain no 1's, so
  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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Note that there is no <math>(1+x+x^2+x^3+\ldots)</math> term in <math>F(x)</math> because we cannot have any 1's in the partition.
 
Note that there is no <math>(1+x+x^2+x^3+\ldots)</math> term in <math>F(x)</math> because we cannot have any 1's in the partition.
  
Let <math>c_m</math> be the coefficient of <math>x^m</math> in the expansion of <math>F(x)</math>, so we can rewrite it as <math>F(x)=c_0+c_1x+c_2x^2+c_3x^3+\ldots</math>. We wish to compute <math>S(n)=1\cdot c_{n-1}+2\cdot c_{n-2}+\ldots+n\cdot c_0</math>.
+
Let <math>c_m</math> be the coefficient of <math>x^m</math> in the expansion of <math>F(x),</math> so we can rewrite it as <math>F(x)=c_0+c_1x+c_2x^2+c_3x^3+\ldots.</math> We wish to compute <math>S(n)=1\cdot c_{n-1}+2\cdot c_{n-2}+\ldots+n\cdot c_0.</math>  
  
Consider the power series <math>G(x)=(x+2x^2+3x^3+4x^4+\ldots)F(x)</math>.
+
Consider the power series <math>G(x)=(x+2x^2+3x^3+4x^4+\ldots)F(x).</math>
  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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\end{align*}</cmath>
 
\end{align*}</cmath>
  
If we expand the first line, we see that the coefficient of <math>x^n</math> in <math>G(x)</math> for any <math>n</math> is <math>1\cdot c_{n-1}+2\cdot c_{n-2}+\ldots+n\cdot c_0</math>, which is exactly <math>S(n)</math>! So by definition, <math>G(x)=\frac{x}{1-x} \prod\limits_{i=1}^{\infty}\frac{1}{1-x^i}</math> is the generating function of <math>S(n)</math>.
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If we expand the first line, we see that the coefficient of <math>x^n</math> in <math>G(x)</math> for any <math>n</math> is <math>1\cdot c_{n-1}+2\cdot c_{n-2}+\ldots+n\cdot c_0,</math> which is exactly <math>S(n)</math>! So by definition, <math>G(x)=\frac{x}{1-x} \prod\limits_{i=1}^{\infty}\frac{1}{1-x^i}</math> is the generating function of <math>S(n).</math>
  
Now let's find the generating function of <math>T(n)</math>. Notice that counting the number of distinct elements in each partition and summing over all partitions is equivalent to counting how many partitions of <math>n</math> contain i and then summing for all <math>1\leq i \leq n</math>.
+
Now let's find the generating function of <math>T(n).</math> Notice that counting the number of distinct elements in each partition and summing over all partitions is equivalent to counting how many partitions of <math>n</math> contain i and then summing for all <math>1\leq i \leq n.</math>  
  
 
So,  
 
So,  
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\end{align*}</cmath>
 
\end{align*}</cmath>
  
However, the number of partitions of n that contain a <math>i</math> is the same as the total number of partitions of <math>n-i</math>, so  
+
However, the number of partitions of n that contain a <math>i</math> is the same as the total number of partitions of <math>n-i,</math> so  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
 
T(n) &= 1\cdot (\text{\# of partitions of n-1})\\ &+ 2\cdot (\text{\# of partitions of n-2})\\ &\ldots \\ &+ n\cdot (\text{\# of partitions of 0})
 
T(n) &= 1\cdot (\text{\# of partitions of n-1})\\ &+ 2\cdot (\text{\# of partitions of n-2})\\ &\ldots \\ &+ n\cdot (\text{\# of partitions of 0})
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\end{align*}</cmath>
 
\end{align*}</cmath>
  
Let's write the expansion of <math>P(x)</math> as <math>P(x)=d_0+d_1x+d_2x^2+d_3x^3+\ldots</math>, so we wish to find <math>T(n)=d_0+d_1+d_2+\ldots+d_{n-1}</math>.
+
Let's write the expansion of <math>P(x)</math> as <math>P(x)=d_0+d_1x+d_2x^2+d_3x^3+\ldots,</math> so we wish to find <math>T(n)=d_0+d_1+d_2+\ldots+d_{n-1}.</math>
  
Consider the power series <math>H(x)=(x+x^2+x^3+x^4+\ldots)P(x)</math>.
+
Consider the power series <math>H(x)=(x+x^2+x^3+x^4+\ldots)P(x).</math>
  
 
<cmath>\begin{align*}
 
<cmath>\begin{align*}
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\end{align*}</cmath>
 
\end{align*}</cmath>
  
If we expand the first line, we see that the coefficient of <math>x^n</math> in <math>H(x)</math> for any <math>n</math> is <math>d_{n-1}+d_{n-2}+\ldots+d_0</math>, which is precisely <math>T(n)</math>. This means <math>H(x)</math> is the generating function of <math>T(n)</math>.
+
If we expand the first line, we see that the coefficient of <math>x^n</math> in <math>H(x)</math> for any <math>n</math> is <math>d_{n-1}+d_{n-2}+\ldots+d_0,</math> which is precisely <math>T(n).</math> This means <math>H(x)</math> is the generating function of <math>T(n).</math>
  
 
Thus, the generating functions of <math>S(n)</math> and <math>T(n)</math> are the same, so <math>S(n)=T(n)</math> for all <math>n</math> and we are done.
 
Thus, the generating functions of <math>S(n)</math> and <math>T(n)</math> are the same, so <math>S(n)=T(n)</math> for all <math>n</math> and we are done.

Revision as of 13:54, 30 August 2018

Problem

By a partition $\pi$ of an integer $n\ge 1,$ we mean here a representation of $n$ as a sum of one or more positive integers where the summands must be put in nondecreasing order. (E.g., if $n=4,$ then the partitions $\pi$ are $1+1+1+1,$ $1+1+2,$ $1+3, 2+2,$ and $4$).

For any partition $\pi,$ define $A(\pi)$ to be the number of $1$'s which appear in $\pi,$ and define $B(\pi)$ to be the number of distinct integers which appear in $\pi$ (E.g., if $n=13$ and $\pi$ is the partition $1+1+2+2+2+5,$ then $A(\pi)=2$ and $B(\pi) = 3$).

Prove that, for any fixed $n,$ the sum of $A(\pi)$ over all partitions of $\pi$ of $n$ is equal to the sum of $B(\pi)$ over all partitions of $\pi$ of $n.$

Solution

Let $S(n) = \sum\limits_{\pi} A(\pi)$ and let $T(n) = \sum\limits_{\pi} B(\pi).$ We will use generating functions to approach this problem -- specifically, we will show that the generating functions of $S(n)$ and $T(n)$ are equal.

Let us start by finding the generating function of $S(n).$ This function counts the total number of 1's in all the partitions of $n.$ Another way to count this is by counting the number of partitions of $n$ that contain $x$ 1's and multiplying this by $x,$ then summing for $1\leq x \leq n.$ However, the number of partitions of $n$ that contain $x$ 1's is the same as the number of partitions of $n-x$ that contain no 1's, so

\begin{align*} S(n) &= 1\cdot (\text{\# of partitions of n-1 with no 1's})\\ &+ 2\cdot (\text{\# of partitions of n-2 with no 1's})\\ &\cdots \\ &+ n\cdot (\text{\# of partitions of 0 with no 1's}) \end{align*}

The number of partitions of $m$ with no 1's is the coefficient of $x^m$ in

\begin{align*} F(x) &= (1+x^2+x^4+\ldots)(1+x^3+x^6+\ldots)(1+x^4+x^8+\ldots)\ldots \\ &= \prod\limits_{i=2}^{\infty}\frac{1}{1-x^i} \end{align*}

Note that there is no $(1+x+x^2+x^3+\ldots)$ term in $F(x)$ because we cannot have any 1's in the partition.

Let $c_m$ be the coefficient of $x^m$ in the expansion of $F(x),$ so we can rewrite it as $F(x)=c_0+c_1x+c_2x^2+c_3x^3+\ldots.$ We wish to compute $S(n)=1\cdot c_{n-1}+2\cdot c_{n-2}+\ldots+n\cdot c_0.$

Consider the power series $G(x)=(x+2x^2+3x^3+4x^4+\ldots)F(x).$

\begin{align*} G(x) &= (x+2x^2+3x^3+4x^4+\ldots)(c_0+c_1x+c_2x^2+c_3x^3+\ldots)\\ &= x(1+2x+3x^2+4x^3+\ldots) \prod\limits_{i=2}^{\infty}\frac{1}{1-x^i}\\ &= x\cdot\frac{1}{(1-x)^2} \prod\limits_{i=2}^{\infty}\frac{1}{1-x^i}\\ &= \frac{x}{1-x} \prod\limits_{i=1}^{\infty}\frac{1}{1-x^i} \end{align*}

If we expand the first line, we see that the coefficient of $x^n$ in $G(x)$ for any $n$ is $1\cdot c_{n-1}+2\cdot c_{n-2}+\ldots+n\cdot c_0,$ which is exactly $S(n)$! So by definition, $G(x)=\frac{x}{1-x} \prod\limits_{i=1}^{\infty}\frac{1}{1-x^i}$ is the generating function of $S(n).$

Now let's find the generating function of $T(n).$ Notice that counting the number of distinct elements in each partition and summing over all partitions is equivalent to counting how many partitions of $n$ contain i and then summing for all $1\leq i \leq n.$

So,

\begin{align*} T(n) &= 1\cdot (\text{\# of partitions of n that contain a 1})\\ &+ 2\cdot (\text{\# of partitions of n that contain a 2})\\ &\cdots \\ &+ n\cdot (\text{\# of partitions of n that contain a n}) \end{align*}

However, the number of partitions of n that contain a $i$ is the same as the total number of partitions of $n-i,$ so \begin{align*} T(n) &= 1\cdot (\text{\# of partitions of n-1})\\ &+ 2\cdot (\text{\# of partitions of n-2})\\ &\ldots \\ &+ n\cdot (\text{\# of partitions of 0}) \end{align*}

The generating function for the number of partitions is \begin{align*} P(x) &= (1+x+x^2+\ldots)(1+x^2+x^4+\ldots)(1+x^3+x^6+\ldots)\cdots \\ &= \prod\limits_{i=1}^{\infty} \frac{1}{1-x^i}. \end{align*}

Let's write the expansion of $P(x)$ as $P(x)=d_0+d_1x+d_2x^2+d_3x^3+\ldots,$ so we wish to find $T(n)=d_0+d_1+d_2+\ldots+d_{n-1}.$

Consider the power series $H(x)=(x+x^2+x^3+x^4+\ldots)P(x).$

\begin{align*} H(x) &= (x+x^2+x^3+x^4+\ldots)(d_0+d_1x+d_2x^2+d_3x^3+\ldots) \\ &= x(1+x+x^2+x^3+\ldots) \prod\limits_{i=1}^{\infty} \frac{1}{1-x^i}\\ &= x\cdot \frac{1}{1-x}\prod\limits_{i=1}^{\infty} \frac{1}{1-x^i}\\ &= \frac{x}{1-x} \prod\limits_{i=1}^{\infty} \frac{1}{1-x^i} \end{align*}

If we expand the first line, we see that the coefficient of $x^n$ in $H(x)$ for any $n$ is $d_{n-1}+d_{n-2}+\ldots+d_0,$ which is precisely $T(n).$ This means $H(x)$ is the generating function of $T(n).$

Thus, the generating functions of $S(n)$ and $T(n)$ are the same, so $S(n)=T(n)$ for all $n$ and we are done.

~Peggy

See Also

1986 USAMO (ProblemsResources)
Preceded by
Problem 4
Followed by
Last Question
1 2 3 4 5
All USAMO Problems and Solutions

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