Chief executives usually have big egos. Ursula Burns, though, is handing bragging rights to the billionaire investor Carl C. Icahn.

Ms. Burns, the Xerox chief executive, says Mr. Icahn was not involved in the board’s decision to split the company in two, which was announced on Friday. But he is getting board seats at one offshoot while Ms. Burns, unusually, has not yet grabbed any role in the new structure.

Mr. Icahn agitated for the split and revealed a stake in Xerox last November. Ms. Burns told investors the previous month that the board was seriously considering strategic options for the company. Xerox suffers from shrinking sales in its photocopying and printing business, known as document technology, and from a steady decline in its stock price since late 2014.

Ms. Burns does have one problem relating to the new plan. Her defining bet in 2009, the same year she took the helm at Xerox, was to buy Affiliated Computer Services for $6.4 billion. This so-called business-process outsourcing operation – managing often paper-based administrative tasks for companies in health care, finance and other sectors – is now being separated again.