Why is this variable in equation 2 of the SQAIR paper a random vector of $n$ ones followed by a zero? I've been reading the SQAIR paper lately, and the mathematics involved seems a bit complicated.
Some background, about the paper: SQAIR stands for Sequential Attend, Infer, Repeat - the paper does generative modelling of moving objects. The idea of Attend, Infer, Repeat is to decompose a static scene into constituent objects, where each object is represented by continuous latent variables. The latent variables, $z^{what}$,$z^{where}$ and $z^{pres}$ encode the appearance, position and presence of an object.
Here's a screenshot of the first of many things I'm unable to understand -

Why is $z^{pres,1:n+1}$ a random vector of $n$ ones followed by a zero? Why do we need the zero? How does it help?
Furthermore, an explanation of equation $(2)$ as in the image above, would be great.
P.S. I hope you all find the paper interesting. I'll ask other questions from the paper in separate posts, so as to not crowd one post with too many queries.
From my understanding of the paper, $Z^{pres}$ keeps track of the objects in the scene. For every step of the sequential inference, $z^{pres,i}$ takes either a 0 or a 1. A 1 represents that an object is present and it has to be explained by the remaining latent variables. 0 indicates that all the objects have been explained and inference is complete.