No smoking permitted in this area/on behalf of

I came across a sign "No smoking permitted in this area", which to me is incorrect. I understand there are many ways to convey the same message such as "Smoking not permitted in this area" or "NO smoking area" etc.

In my estimation the said message is incorrect as "No" and "permitted" are used together. Am I correct?

Secondly, the phrase "on behalf of" means "as the agent of, on the part of," as in the guardian signed the contract on behalf of the minor child.

However, I have read the phrase being used “on behalf of my wife and I”. How can I serve as an agent of myself?

Also note that because a person acting on somebody's behalf is acting for another, on behalf of myself is illogical and I think it should be avoided.

Kindly suggest what the expression should be in this kind of a situation.