There are two regimes that work independently and together to allocate shares of a decedent's intestate estate. The per capita that counts the number of people who survive the decedent. The per stirpes regime counts people who stand in the shoes of the decedent and who take by representation. Which designation you select can have wide implications on who and how much your beneficiaries will ultimately inherit.
In the per stirpes regime, the decedent's estate is divided into as many shares as there are surviving children or deceased children who left descendants and then to subdivide the decedent's estate from there. This regime views the family vertically so that property is shared equally at soon as a member of a decedent's family is located but descendants in differerent generations may receive widely differering shares depending on how many children their parents had. Some people view this regime as unfair because a more remote descendant may receive a larger share than a closer descendant.
In the per capita regime, all descendants of who survive the decedent receive an equal share, regardless of how remote the descendant is from the decedent. This regime works well when all descendants survive the decedent and are in the same generation. However, in this regime, a surviving child could theoretically receive the same share of the decednt's estate as a grandchild, merely because a decedent's child predeceased and left a child. Many people also view this outcome as unfair because of the presumption that the decedent would have wanted to leave a greater share of the decedent's estate to a child rather than to a grandchild.
To resolve the shortcomings of each regime, many states use a hybrid regime called per capita with representation. In this regime, older "empty" generations where there are no surviving descendants are skipped and the decedent's estate is divided into shares at the first generation leaving survivors. Each descendant at that first generation receives a share, the remaining shares from predeceased descendants if any, are combined, and then these shares are divided equally among the descendants of the following generation. Many people view this generational or horizontal approach as more equitable to all family members, based on how remote they are to the decedent.
The default rule in New York for all Wills and other testamentary instruments executed after September 1, 1992, absent the Will maker designating otherwise, is per capita with representation.