single child parents had purchased a vacation home together 50/50.mother passes no will for my mother how and what should legally happen

In New York, if a married person dies without a will, the surviving spouse is entitled to the first $50,000 in the estate plus half of the remainder, and the other half is divided equally between their children.  The ownership of the house depends upon how the deed was drafted. If the deed refers to your parents as "husband and wife" or "tenants by the entirety", your mother's share of the house automatically passed to your father when she died. His intestate share ($50K plus 50% of the estate) may be made up in part or completely by the value of your mother's share of the house. However, if the deed says your parents owned the property as "tenants in common" (unusual for a married couple unless they purchase the property late in life, and have children of other marriages)- or if the deed does not state they were husband and wife or tenants by the entirety, your mother's share of the house belongs to you- IF there is enough value in the rest of the estate to satisfy the statutory requirement that your father gets the first $50,000 plus half the estate. For example, if your parents were tenants in common in the house and your mother left a million dollar estate which included the value of the house, and the house was worth $300,000, your father would get $550,000 ($50K + 500K), and you would get your mother's share of the house (worth $150K) and the residue of the estate, $300,000. You would then become a tenant in common (50% co-owner) with your father.  If you wanted the value of the house immediately, your dad would need to buy you out, or you would have to take him to court and seek a partition and judicial sale. Otherwise, since you are the only child, you will own the whole house when your dad passes away.