Counting Total Spin for N Two-Level Systems (TLS)

[Haris]

TL;DR

I am having considerable trouble understanding the addition of spins. The context in which I am studying this is the Tavis-Cummings Model.
Statement:
"Assume that r and m mean total spin and projection of spin along z, respectively. For N-TLS the total spin (r) can assume N+1 to 1/2 or 0 spin depending on N being even or odd. For a fixed r the value of m varies from +j to -j in integer steps. R is the operator whose eigen-values are r. The basis choice is |r,m>."

Now then, if I intend to make a matrix pretaining to single transitions of the composite system, i align the states with fixed r. For fixed r I have 2m+1 states. When r=N/2 my states are N+1 as simple substitution verifies. However, when r=N/2 -1 the number of states are (N-1)^2. It gets weirder for N/2 -2

Questions:
1) Why is r =N/2 -1 valid as individual spin is half not 1.
2) The counting on main diagonals are pretty confusing. For each irrep of SU(2) there's 2m+1 states and that's fine. But the multiplicity of each state is entirely vague to me.

I have linked the original paper and the figure 1 is where the counting is shown.

 

[Haris]

Correction: its 2r+1 instead of 2m+1