As per the Indian Partnership Act, in case of death of ONE partner in the Partner ship Firm having TWO partner the partnership firm unless written agreement that legal heirs or the nominee or legal representative of the deceased partner can put his/her lag in the shoe of the of deceased partner and business of the firm can be continued with the presumption that the partnership was never dissolved on the death of that partner, incase of death of ONE partner out of TWO of the partnership Firm having only TWO partner the Partnership Firm will be DESOLVED.
In my view a Partnership Firm is a Created by execution Partnership DEED which is again an agreement or contract between or among the partners the partner ship DEED will not have any effect if ONE partner cease to be in existence or expires because partner ship DEED is contract and in Contract there should be minimum TWO parties one party can not enter into contract like self can not enter into contract with self. So in Two partners partnership if ONE partner is expire it will be deemed that there is no contract unless otherwise specifically provided in the partner ship DEED.
So, in my view the old partnership should be dissolved and a new partner ship should be made with new partner. Name may be kept the same but all registration for business the change of the composition so business should be done wherever applicable. Permanent Account Number will have to be surrendered and NEW PAN will have to be taken.