For years, Jim Coulter and David Bonderman, the billionaire co-founders of private equity titan TPG, flirted with the idea of selling shares in their own firm to the public. No longer. Coulter and Jon Winkelried, the former Goldman Sachs Group Inc. president who took over for Bonderman as co-chief executive three years ago, have decided to remain an old-school partnership. TPG has instead explored other private financing arrangements, including a possible sale of a stake in the company, according to people with knowledge of the deliberations.