Solid partition

In mathematics, solid partitions are natural generalizations of <a href="/facts/Integer_partition/LtyXWqwH">integer partitions</a> and <a href="/facts/Plane_partition/erlrREf2">plane partitions</a> defined by <a href="/facts/Percy_Alexander_MacMahon/scUegUSn">Percy Alexander MacMahon</a>. A solid partition of 
 
 
 
 n
 
 
 {\displaystyle n}
 
 is a three-dimensional array of non-negative integers 
 
 
 
 
 n
 
 i
 ,
 j
 ,
 k
 
 
 
 
 {\displaystyle n_{i,j,k}}
 
 (with indices 
 
 
 
 i
 ,
 j
 ,
 k
 ≥
 1
 
 
 {\displaystyle i,j,k\geq 1}
 
) such that

∑
          
            i
            ,
            j
            ,
            k
          
        
        
          n
          
            i
            ,
            j
            ,
            k
          
        
        =
        n
      
    
    {\displaystyle \sum _{i,j,k}n_{i,j,k}=n}

and

n
          
            i
            +
            1
            ,
            j
            ,
            k
          
        
        ≤
        
          n
          
            i
            ,
            j
            ,
            k
          
        
        ,
        
        
          n
          
            i
            ,
            j
            +
            1
            ,
            k
          
        
        ≤
        
          n
          
            i
            ,
            j
            ,
            k
          
        
        
        
          and
        
        
        
          n
          
            i
            ,
            j
            ,
            k
            +
            1
          
        
        ≤
        
          n
          
            i
            ,
            j
            ,
            k
          
        
      
    
    {\displaystyle n_{i+1,j,k}\leq n_{i,j,k},\quad n_{i,j+1,k}\leq n_{i,j,k}\quad {\text{and}}\quad n_{i,j,k+1}\leq n_{i,j,k}}
  
 for all 
  
    
      
        i
        ,
        j
        
           and 
        
        k
        .
      
    
    {\displaystyle i,j{\text{ and }}k.}

Let 
 
 
 
 
 p
 
 3
 
 
 (
 n
 )
 
 
 {\displaystyle p_{3}(n)}
 
 denote the number of solid partitions of 
 
 
 
 n
 
 
 {\displaystyle n}
 
. As the definition of solid partitions involves three-dimensional arrays of numbers, they are also called three-dimensional partitions in notation where plane partitions are two-dimensional partitions and partitions are one-dimensional partitions. Solid partitions and their higher-dimensional generalizations are discussed in the book by <a href="/facts/George_Andrews_(mathematician)/x58GmSuR">Andrews</a>.

Solid partition open-in-new

Solid partition