Hello :')

Recently I've solving this: link, and I've read solution here.

When the solution says, we can infer the following recurrences:

A[n][i][j] = D[n-1][i][j-1] + D[n-1][i+1][j-1] +…+ D[n-1][n-1][j-1]
D[n][i][j] = A[n-1][1][j-1] + A[n-1][2][j-1] +…+ A[n-1][i-1][j-1],

Is this only valid for i<j? Because the solution says, that when renumbering, we remove i and relabel everything greater than i. But if i>j, then the recurrence will not be D[n-1][smth][j-1], but rather D[n-1][smth][j].

But if both A and D are valid for only i<j, then do the formulas really work? Because in the sum for A[n][i][j], we will add some D[n-1][i'][j'] such that i'>j'. Then the recurrence wouldn't really work.

Can anyone provide some explanation or clarification on this?

Thanks,

-minimario