in Fig.4a.

 On the other hand, for the number N together with two arbitrary integers $n$ ($\leq $N), and $j^{\prime }$, there is an identity

$\frac{N(N-1)}{2}+j^{\prime }N=\frac{n(n+2j^{\prime }-1)}{%
2}+\frac{(N-n)(N+n+2j^{\prime }-1)}{2}\qquad \qquad \qquad (16)$

Let the left hand side be equal to $L$, and the two terms at the right be equal to $l_{in}$  and $l_{out}$. Then this identity is associated with a division of $L$. When $j^{\prime }=0$ , the left hand side of (16) is equal to 78 . It is easy to see that the pair N$_{o}=n-1$ and $l_{in}$ meet the requirement of eq.(9b), while the pair N$_{out}$=$N-n$ and $l_{out}$ meet the requirement of eq.(9a).  Thus, eq.(16) implies that all the (N$%
_{o}+1)_{c}$-N$_{out}$ structures with N$_{o}=0$ to 12 are the CFS of the $%
L=78$ states. Therefore the wave function of the (78)$_{1}$can get access to all the  symmetric configurations$^{12,13}$, and thus is nodeless (except a pair of electrons overlap with each other). Accordingly, the energy of this state is lower.

For the (79)$_{1}$ state, there is also only one BF {0-11,13} belonging to the LLL. Thus this state is also liquid-like as shown in Fig.4b.  However, on the contrary with the (78)$_{1}$ , all the (N$_{in})$-N$_{out}$ structures 