C major and modes

I’m not sure I understand the question if I’m honest.

When you say “omitted the flats” are you talking about on my article here?

https://www.guitar-chords.org.uk/modes/modesexplained-part1.html

I haven’t used flats because I’ve come in at a different perspective using series modes, i.e., modes that share the same notes, all having the same parent key. Rather than explain, I’ll just list the differences.

Series modes belonging to C major, all share the same notes.

C Ionian: C D E F G A B
D Dorian: D E F G A B C
E Phrygian: E F G A B C D
F Lydian: F G A B C D E
G Mixolydian: G A B C D E F
A Aeolian: A B C D E F G
B Locrian: B C D E F G A

Parallel Modes, all share the same root note.

C Ionian: C D E F G A B
C Dorian: C D Eb F G A Bb
C Phrygian: C Db Eb F G Ab Bb
C Lydian: C D E F# G A B
C Mixolydian: C D E F G A Bb
C Aeolian: C D Eb F G Ab Bb
C Locrian: C Db Eb F Gb Ab Bb

The formulas don’t change

C Ionian: 1 2 3 4 5 6 7
C Dorian: 1 2 b3 4 5 6 b7
C Phrygian: 1 b2 b3 4 5 b6 b7
C Lydian: 1 2 3 #4 5 6 7
C Mixolydian: 1 2 3 4 5 6 b7
C Aeolian: 1 2 b3 4 5 b6 b7
C Locrian: 1 b2 b3 4 b5 b6 b7