## 4 Representation and Characterization: Scenarios and States
### S7 Scenario  
Scenario S7 can be broken down into 13 subsets of states (or sub-scenarios). These subsets are obtained by considering characterization. By characterization, we mean the different ways in which the n-tuple of the seven CSFs can be obtained, as in Table 2. \begin{table}
\begin{tabular}{l l} \hline \hline
**Scenario** & **Scenario Characterization** \\ \hline S7 & All 7 CSFs are variable \\ \hline S6 & 6 out of the 7 CSFs are variable, and 1 is fixed \\ \hline S5 & 5 out of the 7 CSFs are variable, and 2 are fixed \\ \hline S4 & 4 out of the 7 CSFs are variable, and 3 are fixed \\ \hline S3 & 3 out of the 7 CSFs are variable, and 4 are fixed \\ \hline S2 & 2 out of the 7 CSFs are variable, and 5 are fixed \\ \hline S1 & 1 out of the 7 CSFs is variable, and 6 are fixed \\ \hline S0 & 1 out of the 7 CSFs is variable, and 6 are fixed \\ \hline \hline \end{tabular}
\end{table} 
Table 1: Scenarios and their characterizations. \begin{table}
\begin{tabular}{l l l l l l} \hline \hline
**Sub-Scenario** & **Characterization** & **N.Sub-Scenarios** & **N.States** & **E** & **S** \\ \hline S7\_1 & 7 & 5 & 5 \(\times\) (7!/7!) = 5 & [7; 35] & 0 \\ \hline S7\_2 & 601 & 20 & 20 \(\times\) (7!/6!) = 140 & [8; 34] & [1; 4] \\ \hline S7\_3 & 502 & 20 & 20 \(\times\) (7!/5!2!) = 420 & [9; 33] & [2; 8] \\ \hline S7\_4 & 50101 & 30 & 30 \(\times\) (7!/5!) = 1260 & [10; 32] & [2; 7] \\ \hline S7\_5 & 403 & 20 & 20 \(\times\) (7!/4!3!) = 700 & [10; 32] & [3; 12] \\ \hline S7\_6 & 40201 & 60 & 60 \(\times\) (7!/4!2!) = 6300 & [11; 31] & [3; 11] \\ \hline S7\_7 & 4010101 & 20 & 20 \(\times\) (7!/4!) = 4200 & [13; 29] & [4; 9] \\ \hline \hline \end{tabular}
\end{table} 
Table 2: S7 sub-scenarios, where the symbol \(\lozenge\) indicates the composition of the micro-state string as described above. The symbol \(\lozenge\) indicates the composition of the micro-state string and so for example 6\(\lozenge\)1 means that six CSFs are equal, and one is different; 5\(\lozenge\)2 means that five CSFs are equal, and two are different but equal to each other; and 4\(\lozenge\)2\(\lozenge\)1 means that four CSFs have one value, two have another one, and one has yet another one, and so on  
The total number of sub-scenarios is 330. This value is calculated using the well-known combinatorial expression for calculating combinations with repetition of n = 5 distinct elements taken k = 7 at a time, which is  
\[C_{n,k}=\binom{n+k-1}{k}=\frac{(n+k-1)!}{k!(n-1)!} \tag{15}\]  
which in this case becomes  
\[C_{5,7}=\binom{5+7-1}{7}=\frac{11!}{7!4!}=330. \tag{16}\]  
The sum of the sub-scenarios is indeed 330; however, it contains extra information because the scenario under consideration, with seven CSFs, has been broken down into sub-scenarios of various types, such as seven identical CSFs, six identical, and one different, etc. This breakdown was necessary to calculate the number of states belonging to each sub-space, or using thermodynamic language, to calculate the number of microscopic states corresponding to the same macroscopic state. It is given using the product of the number of sub-scenarios and the number of permutations with repetition of the specific characterization. In other words, to calculate the states, the number of components of the sub-scenario has been multiplied using the following relation  
\[P_{k,k_{1},k_{2},\ldots,k_{r}}=\frac{k!}{k_{1}!k_{2}!\ldots k_{r}!} \tag{17}\]  
with k being the length of the sequence of elements, and k\({}_{1}\), k\({}_{2}\),..., k\({}_{r}\) representing the number of times that the first, second,..., r-th element repeats, respectively. The sum of the states reported in the previous table yields a partition of the set of states, which is 5\({}^{7}\) = 78,125. Figure 2 shows the table of sub-scenarios in the ES plane. In gray, it displays the 13 levels of entropy within the allowable range of 0-12 and the 29 levels of energy within the allowable range of 7-35. Additionally, for brevity, the boxes indicate the sub-scenarios S7_# with only the last number #.