Studying the theory of combinatorial species from this book I came across the notion of the spices of linearly ordered sets with $k$ blocks $\rm{Bal}^k$ as seen in Example 2.12 on the bottom of the page 35 of the mentioned book.
Now evidently we have $$\rm{Bal}^1[\{1,2\}] = \{ \{(1,2)\}, \{(2,1)\}\}.$$
What confuses me is that in the book they say that $$\rm{Bal}^k = (E_{+})^k,$$ where $E_{+}$ is the species of nonempty sets. But then $\rm{Bal}^1 = E_{+}$ and $$E_{+} [\{1,2\}] = \{ \{1,2\}\}$$ which is not equal to $\rm{Bal}^1[\{1,2\}]$ as computed above.
Hence there must be a mistake somewhere in my understanding. Anyone happens to see where?
What you've given for $\text{Bal}^1[\{1,2\}]$ suggests that the ordering occurs within blocks. However, computing $$ E_+^2[\{1,2\}] = \{ (\{1\},\{2\}),(\{2\},\{1\}),(\emptyset,\{1,2\}),(\{1,2\},\emptyset) \} $$ and drawing these structures suggests that, if $E_+^k = \text{Bal}^k$ is to hold, the meaning must be that the ordering occurs between blocks (and therefore is partial on $U$).
If that is the case then $$\text{Bal}^1[\{1,2\}] = \{\{1,2\}\}.$$
Looking at figure 2.5 makes this interpretation seem reasonable, although the more clustering of the points would have left less ambiguity.