Triangle of partition numbers

<p>In the <a href="/facts/Number_theory/zhoa7zrc">number theory</a> of <a href="/facts/Integer_partition/LtyXWqwH">integer partitions</a>, the numbers 
  
    
      
        
          p
          
            k
          
        
        (
        n
        )
      
    
    {\displaystyle p_{k}(n)}
  
 denote both the number of partitions of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 into exactly 
  
    
      
        k
      
    
    {\displaystyle k}
  
 parts (that is, sums of 
  
    
      
        k
      
    
    {\displaystyle k}
  
 positive integers that add to 
  
    
      
        n
      
    
    {\displaystyle n}
  
), and the number of partitions of 
  
    
      
        n
      
    
    {\displaystyle n}
  
 into parts of maximum size exactly 
  
    
      
        k
      
    
    {\displaystyle k}
  
. These two types of partition are in <a href="/facts/Bijection/ayC7YE3z">bijection</a> with each other, by a diagonal reflection of their <a href="/facts/Young_diagram/e5lI4bck">Young diagrams</a>. Their numbers can be arranged into a triangle, the triangle of partition numbers, in which the 
  
    
      
        n
      
    
    {\displaystyle n}
  
th row gives the partition numbers 
  
    
      
        
          p
          
            1
          
        
        (
        n
        )
        ,
        
          p
          
            2
          
        
        (
        n
        )
        ,
        …
        ,
        
          p
          
            n
          
        
        (
        n
        )
      
    
    {\displaystyle p_{1}(n),p_{2}(n),\dots ,p_{n}(n)}
  
:
</p>
<table><tbody><tr><th> <i>k</i><i>n</i> </th><th>1</th><th>2</th><th>3</th><th>4</th><th>5</th><th>6</th><th>7</th><th>8</th><th>9</th></tr><tr><th>1</th><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>2</th><td>1</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>3</th><td>1</td><td>1</td><td>1</td><td></td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>4</th><td>1</td><td>2</td><td>1</td><td>1</td><td></td><td></td><td></td><td></td><td></td></tr><tr><th>5</th><td>1</td><td>2</td><td>2</td><td>1</td><td>1</td><td></td><td></td><td></td><td></td></tr><tr><th>6</th><td>1</td><td>3</td><td>3</td><td>2</td><td>1</td><td>1</td><td></td><td></td><td></td></tr><tr><th>7</th><td>1</td><td>3</td><td>4</td><td>3</td><td>2</td><td>1</td><td>1</td><td></td><td></td></tr><tr><th>8</th><td>1</td><td>4</td><td>5</td><td>5</td><td>3</td><td>2</td><td>1</td><td>1</td><td></td></tr><tr><th>9</th><td>1</td><td>4</td><td>7</td><td>6</td><td>5</td><td>3</td><td>2</td><td>1</td><td>1</td></tr></tbody></table>

Triangle of partition numbers open-in-new

Triangle of partition numbers