Look at the deed. If you are named as "X and Y as tenants in common", you own 50% and his heirs own the other 50%. If you are named as "X and Y as joint tenants with rights of survivorship", the property passed to you on your partner's death. If it says nothing either way regarding "tenants in common vs. joint", I believe "default" is tenants in common.
Your partner's will may also have some bearing on the question. As to the joint bank account, since either party could freely draw on it, it's entirely yours. I assume the gay marriage law arrived in New York too late to help your situation and you are not married (where you would spilt assets with his heirs automatically).
You should consult an estates practice lawyer to help you. Try to determine if your partner had a will or blood relatives and who they all are.
Atty. Lebowitz gives you the correct analysis on the title. That it was your "home" has no bearing on the situation. A joint tenancy could've/should've been set up, even prior to the recognition of same sex marriage, to address this if the intent was for you to inherit it all. Now, perhaps your partner's will leaves his estate to you, and if so, the his 50% of the tenancy in common, if that's what exists, would go to you, so no worries. Now, if it is a tenancy in common and the other 50% is in fact now inherited by his heirs, you face the possible situation of them wanting their money sooner rather than later. They can force the issue thru an expensive partition action, forcing a sale and then dividing the proceeds, or you could come to an agreement to buy out their 50% interest. I agree that the joint account is yours. Also check if your partner left any life insurance or other non-probate assets like 401Ks and IRAs. If you are the sole named beneficiary of those, they're yours too.
To questioners from West Virginia & New York: Although I am licensed to practice in your state, I practice on a day-to-day basis in Massachusetts. I answer questions in your state in areas of the law in which I practice, and in which I feel comfortable trying to offer you assistance based on my knowledge of specific statutes in your state and/or general principles applicable in all states. It is always best, however, to work with attorneys and court personnel in your own area to deal with specific problems and factual situations.
The nature of your relationship, is not usually relevant, if a deed by its terms provides a means for disposition of the property.
If you were both owners you and the estate of the deceased owner are the successor owners. If the decedent provided for disposition of his share to you by a last will and testament, the estate's executor will at some point during the probate period convey full title to you.
You asked this question several times today and along the way altered the facts you presented.
Hire an attorney, as no attorney answering here is your attorney.
The answer provided to you is in the nature of general information. The general proposition being that you should try to avoid a bad outcome if you can.