Moving washer/install for drain plumbing

I am moving our washer/dryer from kitchen into an old walk-in closet in the hallway. It backs up to the guest bathroom, so hot/cold water and 1 1/2" drain are right there. There is also a 4" vertical stack about 40" away, but hard to get to.

Can I drop my standpipe just above or just below the existing horizontal sink drain, install P-trap and "Y" into the horizontal drain? Would there be any vent problems with the 4" being that close?

I'm trying to install a picture so you can see what I'm looking at.
thanks for the help.
John

Washing machine must drain into a 2" line, and by most codes cannot share a branch line with another fixture.

My existing washer is 1 1/2" so I thought 1 1/2" here would be okay. It may also be posible to use 2", bring it down below sink drain, then run horizontally through 3 studs into 4" vertical. Would I need a separate vent for that?

Thanks. I did go ahead and run 2" over to the 3" (not 4" like I first stated) vertical stack. Took a little planning, but came out well and hopefully all within code. My standpipe is 30", p-trap within 12" of floor, and a vertical run of 44" to vertical stack, so don't need extra venting.

Please let me know if anyone sees a problem.
Thanks.
John