I have used the solution of integer partitioning using dynamic programming explained in this post and in this article.
Following is the resultant matrix when $N$ is equal to $6$: $$\begin{bmatrix} 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 2 & 2 & 2 & 2 & 2 \\ 0 & 1 & 2 & 3 & 3 & 3 & 3 \\ 0 & 1 & 3 & 4 & 5 & 5 & 5 \\ 0 & 1 & 3 & 5 & 6 & 7 & 7 \\ 0 & 1 & 4 & 7 & 9 & 10 & 11 \end{bmatrix}$$
I have tried to obtain the respective partitions of the integer by traversing the matrix again. But it does not add up to the sum required. How do we extract the sequences of the partitioning from the matrix above?