The Halacha discusses three scenarios in which a Jew and a non-Jew establish a partnership in which the business will be open on Shabbat. The ideal arrangement is one in which they sign a contract before entering the partnership, stipulating that all of the profit earned on Shabbat belong to the non-Jew and the Jew will receive the profit earned on a specific other day.
The second scenario is a case in which they did not write a contract and entered the partnership "stam", without stipulations with regard to Shabbat. However, it was tacitly understood that the non-Jew would be the one to work on Shabbat and the Jew would work a different day instead. In such a case, the Rama is lenient and permits splitting even the profit of Shabbat evenly, in the event of severe loss. Maran, however, did not bring this Halacha down, which indicates that he is machmir, and rules that it is forbidden.
The third scenario is also a case where they did not make a contract before establishing the partnership. However, the non-Jew works every day of the week and the Jew only takes off on Shabbat. In such a case the Rama, based on the Ran, is lenient in the event of severe loss. The rationale is that the non-Jew is working on Shabbat on his own initiative. He is not considered the agent of the Jew to work on Shabbat, because he is not requesting a different vacation day in exchange. However, Maran did not bring this case down, which indicates that he prohibits splitting the profit.
If a Jew and non-Jew established a partnership to run a business on Shabbat without a contract, Sepharadim are prohibited to split the profit and Ashkenazim are permitted in the event of severe loss in a case in which:
a. The non-Jew works on Shabbat in exchange for the Jew working a different day
b. The non-Jew works all the days of the week