So, a family member ran into an editor of a mid-sized house that only takes agented submissions, and they were discussing my work, and then the editor gave him his business card and direct email and said, "Tell her to write me."

As luck would have it, I just finished a book that I think would fit with their house, which the family member did not even know about! Do I write, mention the connection and business card, and then ask for permission to pitch my novel, or do I write, mention the connection, and include the pitch in the letter?

On one hand, if this is an opportunity to get their eyes on my work, then I feel like asking for permission to include the pitch is just giving them an extra chance to say "No thanks" before they even know what my novel is about, but on the other hand, if they were just being nice when they said, "Tell her to write to me," wouldn't it be presumptuous to go ahead and include a pitch without being invited to do so?