I don't think you can have permanent partners but I can show you how to set someone as your partner. While in play:
1. Click custom game.
2. Change the variant to "Doubles (Bughouse)"
3. Underneath the time control option, you will see your username and avatar and beside it an icon of a person and a question mark. Click on the icon.
4. You can invite someone to be your partner by typing in their username or clicking on them in your friend's list.
5. After choosing a partner, you will be taken back to the custom game menu. However, the play button will be replaced with a "partner" button. Click "Partner".
6. Once your partner accepts you can send out a challenge to play a game.
i dont see any buttons to pic a partner in bughouse, which can be frustrating because you can be paired with someone who resigns every game... yet one time a person requested me to be a partner, but i dont see a button anywhere for how he did it... where do you select a permanent partner, so matchs go faster and better and no instant resigns or disconnects....