What The Threshold Remembers

One standard recurrence for longest increasing subsequence walks the array asking whether each element can be appended after the last one already chosen. Writing $i$ for the position of that last chosen element and $j$ for the candidate:

$\text{LISbigger}(i,j) = \begin{cases}0 & j > n \\text{LISbigger}(i, j+1) & A[i] \ge A[j] \\max\big(\text{LISbigger}(i, j+1),\ 1 + \text{LISbigger}(j, j+1)\big) & \text{otherwise}\end{cases}$

A student memoizes on $j$ alone, reasoning that $i$ is just where the recursion came from. What does dropping $i$ destroy?