Is that joist holding up floor joists on the other side? It certainly looks like it comes out further than the wall perpendicular to it on the other side from the stairs.
My take on it is the joist is coming out to the steps because that is where the post is below to support the load. What is at the top of the stairs to the left?
I wouldn't recommend hacking away at the joist based on the pictures you've provided.
The other wall in the background is in line with the top riser in the picture to allow head room for the down staircase,
Here are some more pictures. I'm planning on keeping the front part of the wall and turn it into a newel post. There are no markes on the ceiling that would show that this wall once went all the way up.
I assume that beam is holding up the floor for the landing above. With just the one picture I can't tell what is going on inside the wall, but my bet is the post supporting the end of the beam goes down to the basement floor and is carrying the load of the floor joists for the landing at the top of the stairs. I could be wrong, but I'd rather err on the side of caution than give the go ahead to tear it out based on one picture. JMHO.
My advice may be worth exactly what you paid me for it. I'd need a lot better view than the photos you've provided to commit to anything, and I'd want to have someone look at it in person. If you're going to act based on neal's and my advice based on a couple of photos, good luck.
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