Laurence has provided a nice answer, and he's pointed out the crucial fact that the E section is in the key of
FMaj rather than
CMaj. I would add that the chords you've included in your original posting seem slightly incorrect to me. Here's how I would name the chords if I were doing the transcription:
| FMaj FMaj/E | Dmin Dmin/C | Gmin Gmin/F | C7/E C7 |
Perhaps there is a misconception that every bass note requires a different chord. I don't think that's true: the bass will often "walk a bass line" and change bass notes while the chord stays the same. And in a stride style (as we have here), the bass will often contain scalar passing tones that transition to the next chord. In measures 1 and 2, we hear a new bass note on beat 3. But those bass notes are simply scalar passing tones that move the song from the prior chord to the next chord. The piano never plays a
C/E chord in measure 1, and it never plays a
C7 chord in measure 2. The only chord we hear the piano play in measure 1 is
FMaj, and the only chord we hear the piano play in measure 2 is
Dmin. Additionally, in the second half of measure 2, the melody surrounds an
A), and the
A is a chord tone of
So using a Roman Numeral analysis, I would analyze this progression as:
I-vi-ii-V7 in the key of
FMaj, which is an extremely common progression.
As a side note, there is a deceptive cadence of sorts that occurs in the next measure (measure 5) of the E section. The V7 chord doesn't resolve to the I chord as we might expect, and instead, it moves to a III7 chord, which then resolves to the vi chord.
For reference, I'm looking at your sheet music and listening to this version of the song.