I was studying an algorithm on $k$-combinations of $n$-bit strings and realised that, my brute-force approach would spend lots of time on "structurally equivalent" bitstring combinations, that could be transformed into another bitstring combination for which a solution had already been computed by permuting bit-positions.
For readability I'll denote $n$-bit strings as elements of $2^{\{0,\dots,n-1\}}$, e.g. 1011 $= \{0,1,3\}$
Then, for $n=3$ and $k=2$, the only "unique" combinations are the following: $$ \{\emptyset, \{0\}\}\\ \{\emptyset, \{0, 1\}\}\\ \{\emptyset, \{0, 1, 2\}\}\\ \{\{0\}, \{1\}\}\\ \{\{0\}, \{0, 1\}\}\\ \{\{0\}, \{1, 2\}\}\\ \{\{0\}, \{0, 1, 2\}\}\\ \{\{0, 1\}, \{0, 2\}\}\\ \{\{0, 1\}, \{0, 1, 2\}\}\\ $$ $\{\{0\}, \{0, 2\}\}$ is not "unique", as it can be mapped to $\{\{0\}, \{0, 1\}\}$ by permuting $1$ and $2$.
Since it seems to me that others should have had similar problems before:
Is there a name for this subset of combinations?
P.S.: My best guess at a formal definition would be a set $X$, such that forall $A \in {2^n}^k$ ($k$-combinations of the powerset of $\{0,\dots,n-1\}$) $$ A \in X \Leftrightarrow A = \min\limits_\sigma(\sigma(A)) $$ i.e., $A$ is in $X$, iff it is the minimal element in the set of all pictures over all permutations (lifting permutations to sets of sets and taking the minimum over the lexicographical ordering).
You can characterize your pairs by giving the following:
coupled with the requirement that the first set have no more elements than the second. Because you run out of distinct elements you must have $t \ge f+s-n$. The first set consists of the lowest $f$ numbers, $0$ through $f-1$. The second set consists of the lowest $t$ numbers and then the numbers from $f$ through $f+s-1-t$. If $k$ gets larger you can extend the number of elements list easily but the number of overlaps grows as $2^k-k-1$. I don't know of a name for it.