What does syntaxes like T1, D2, and p3 mean in music notation?

The notation used in this example is explained in the same textbook http://openmusictheory.com/harmonicFunctions.html and http://openmusictheory.com/harmonicSyntax2.html

In this example T(1 D2p 3):

• T stands for tonic chord
• parentheses mean the whole sequence is what they call a prolongation of the tonic chord
• T1 stands for tonic built on 1st degree of the scale
• D2p stands for passing dominant chord built on the second degree of the scale
• 3 stands for T3, or a tonic chord built on the third degree

You may also notice numbers above the bass notes. This is basso continuo notation (https://en.wikipedia.org/wiki/Basso_continuo)

Guess they refer to:

• Tonic (root)
• Subdominant (4th degree)
• Dominant (5th degree)

T, S and D represent Tonic, Subdominant and Dominant respectively. I = tonic, IV = subdominant, and V = dominant. As in key C, T=I=C. S=IV=F. D=V=G.

The numbers show what the lowest note is - often called its 'inversion'. 1 is root, with the root note (the letter name of the chord) at the bottom. 2 moves everything up, so the third of the chord is at the bottom. 3 moves again, so the 5th is under, and there can be a 3rd inversion, where the 7th of a chord is at the bottom. the order of the other notes isn't reflected in that number.