Well, I seem to be a voice in the wilderness on this one, but I would certainly assume that my children are invited to a wedding that I am invited to, unless explicitly told otherwise.
I’ve never encountered accidental ambiguity on this, either. Maybe about 5 of the many weddings I have been invited to since becoming a parent have been of the “no children” variety, were explicitly stated as such in the invitation, and in all but one case was also followed by a direct email from the friend in question on the side apologizing for the exclusion and describing it as being based on cost (but no doubt also serving as a polite reminder of their wishes in this regard).
To me, a wedding is the joining of two families, and families have children – that’s what defines a family, after all (your family wouldn’t be your family if you hadn’t grown up in it!). Excluding my own family’s children from attending my wedding would have been like, well, having a housewarming party in a hotel conference room instead of my actual house.
Now I understand that this is not everybody’s attitude, but I would go so far as to say it is the default/prevailing one in most American sub-cultures. Weddings have children the way that McDonald’s hamburgers have pickles: most people like 'em that way, but you have the right to ask to not have them and deserve to have your request honored, but even so be forewarned that unless you take extra steps (like trying to watch them make it) you might well get a couple anyway.
So my advice would be to hold firm on the no children policy, and if it’s not too late, send that secondary email/letter! Dis-inviting people’s kids is one thing; having the parents feel like they were somehow singled out, or that some kind of double standard was applied, is quite another. One time I attended a friend’s wedding after leaving my kids with my parents, and was annoyed (my wife even more so) to find quite that there were 5 or 6 kids there, that my kids had even met before and would have liked to have played with.
This was “that one case” where the friend in question had NOT sent the separate follow-up messages to those invitees he knew had children to reinforce the message, and it turned out that the parents of these kids (close family of the bride’s) hadn’t really read the invitation and just brought the whole family along. Basically their reaction had been “oh look, it’s the invitation to <X>'s wedding, let’s mark the date and venue/directions”, and the idea that a wedding might not include children never occured to them.
BTW, in response to all the replies about “bratty children” “ruining” weddings, I have not yet been to a wedding I would consider having been disrupted by a child. There may have been a few cases of crying infants/small children whisked away by a parent, and one case of a 2-year-old piping up “I don’t like this story, I want a different one” during the priest’s sermon at a Catholic wedding, which is still remembered with great amusement by the groom and bride, while nobody remembers anything about the homily.