inversion?

  1. Under no circumstances children are allowed in the bar.
  2. Under no circumstances are children allowed in the bar.
    Are these sentences OK?
    Is #2 sentence written in inversion?

Sentence 2 is the grammatically correct version.
Sentence 1 is incorrect.
The alternative would be:
Children are not allowed in the bar under any circumstances.
(‘Children are allowed in the bar under no circumstances’ sounds odd in this context - the negative would be used.)

Beeesneees,

  1. Under no circumstances children are allowed in the bar.
    How do you say that it is wrong?
    “children are allowed in the bar” is correct sentence.
    “Under no circumstances” is added as a connecting phrase to the sentence.
    Please explain to me in what way it is wrong?

Please could you explain me what is an inversion?

‘are’ is modified by the negative ‘under no circumstances’. It isn’t simply a connecting phrase.
As I have already stated, the correct form is:
Under no circumstances are children allowed in the bar.
‘Not under any circumstances are children allowed in the bar’ is also possible.

However, in these forms, which are not negative, ‘children are’ is correct:
Under certain circumstances, children are allowed in the bar.
Under some circumstances, children are allowed in the bar.
Under special circumstances, children are allowed in the bar.
Under most circumstances, children are allowed in the bar.

oald8.oxfordlearnersdictionaries … /inversion

Can I not use comma as below:
“Under no circumstances, children are allowed in the bar.”
When positive form is correct, how is negative form wrong?
Please enlighten me.

No, use of the comma there would not improve matters.
I have explained it. It’s simply the form the pattern takes.