ILP representation of the number of maximal 1 sequences in a row

Question

I am currently using an ILP to model events which occur on some input sequence from $1...n$. These events modify the input sequence in order to obtain a desired sequence. Each event can happen on some consecutive range $(i,j)$ and no two events can overlap. Therefore, I can represent events by some matrix $a=\{0, 1\}^{n\times n}$ where $a_{i,j} = 1$ iff there is an event that begins at $i$ and ends at $j$ ($i<=j$).

Unfortunately, this results in $O(n^2)$ variables and in some cases $O(n^3)$ or greater number of constraints. It would be ideal if instead I could represent these events as a row $a'=\{0,1\}^n$, where $a'_i = 1 \iff \exists a_{p,q}\quad \text{s.t.}\quad p\leq i\leq q \land a_{p,q} = 1$
i.e. a position $a'_i=1$ iff there is an event that modifies position $i$.

Since I am optimizing over the number of events, at the moment I am minimizing $\sum a_{p,q}$, since it represents the number of events. If I ditch $a$ and try to only use $a'$, I am left with some row, for example $a'=[1,1,0,...,1,0,1]$.

Question 1: I want to represent the total number of events given $a'$. Currently, I am thinking $\sum a_i(1-a_{i+1})$ to count the number of "ends" of events. Obviously I can deal with the edge case as well. Unfortunately, this results in a product of binary variables. I know that I can represent this with a helper variable, and just sum over that, but I was wondering if there was any more efficient ways.

Question 2: This question is more important, since my whole model revolves around this. An event will only affect the input elements after the end of the event. Think of events as duplications. If some range $(i,j)$ is duplicated, all elements after $j$ would shift to the right by $j-i+1$ slots. Following this idea of events being duplications, consider the input below:

$a,b,c,d,e$ and the desired sequence $a,b,a,b,c,d,e,d,e$. We can achieve this by duplicating $a,b$ and $d,e$ i.e. $a'=[1,1,0,1,1]$. Notice how positions $0$ and $1$ did not shift at all, positions $2,3,4,5,6$ were originally $0,1,2,3,4$, and positions $7$ and $8$ were originally $3$ and $4$.

Essentially, for every position $i$ in the resulting sequence, I would like to be able to determine which position $j$ it came from in the original sequence. Consider position $6$ in the resulting sequence, we can clearly see it came from position $4$. We notice that this shift of $2$. Similarly, position $5$ came from position $3$. A pattern unfolds such that the shift of position $i$ is always equal to the $\sum_{j<k} a'_j$ where $k$ is the largest index such that $k\leq i \land a_k=0$. This is the same as saying that we are taking the sum of all values of $a'$ up to the most recent $0$ before or at $i$ in $a'$.

How can I model such shifting in an ILP? I.e. given $a’$, how can I create a variable $h$ in the ILP such that $h_{r, l}=1$ iff $\sum_{j<k} a'_j= l$ where $k$ is the largest index such that $k\leq r \land a_k=0$.