In number theory and combinatorics, a partition of a positive integer n, also called an integer partition, is a way of writing n as a sum of positive integers.

Two sums that differ only in the order of their summands are considered the same partition. For example, 4 can be partitioned in five distinct ways:

1
2
3
4
5

4
3 + 1
2 + 2
2 + 1 + 1
1 + 1 + 1 + 1

The order-dependent composition 1 + 3 is the same partition as 3 + 1, while the two distinct compositions 1 + 2 + 1 and 1 + 1 + 2 represent the same partition 2 + 1 + 1.

Young diagrams associated to the partitions of the positive integers 1 through 8. They are arranged so that images under the reflection about the main diagonal of the square are conjugate partitions.

Integer Partition

References

Implementations

  • 1

    # TODO: Implement in Python

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    function partition(A, n, p) {
    
    if (p == 0) console.log(`[${A.split('').join(', ')}]`);
    
    else {
        if (n > 1) partition(A, n - 1, p);
        if (n <= p) partition(n + A, n, p - n);
    }
}

function integerPartition(n) {

  // cycle through each cell of matrix
  for (let i = 1; i <= n; i++) {
    for (let j = 1; j <= n; j++) {
      if (i > j)  {
        
        // set cell to cell above it
        D[i][j] = D[i - 1][j];
        
      }
      else {
        
        // grab above cell and add it to previous cell
        const above = D[i - 1][j];
        const left = D[i][j - i];
        D[i][j] = above + left;
        
      }
    }
  }
  return D[n][n];
}

console.log(`Partitioning: ${integer}`);

partition(A, integer, integer);
const part = integerPartition(integer);

console.log(part);