Consider two sets of real numbers $\{a,b,c\}, \{d,e,f\}$ with some linear ordering. I am interested in trying to understand the behaviour of the sum of the middle values of each linear ordering
- It is known that if $a<b<c$ and $d<e<f$, then $b+e$ stays in the middle in the new ordering $a+d<b+e<c+f$
- It is also known that if $a<b<c$ and $d>e>f$, then 3 possibilities can happen:
a. The two orderings are complementary, thus $k=a+d=c+f$. Then the max and min of the parent orderings are evened out, leaving behind $b+e$ which can be max or min or equality depending on whether $b+e<k,b+e>k,b+e=k$
b. The magnitudes involved in the ordering $a<b<c$ is not large enough to complement $d<e<f$ thus $a+d<b+e<c+f$ results
c. The magnitudes involved in the ordering $d>e>f$ is not larger enough to complement $a<b<c$ thus $a+d>b+e>c+f$ results
But what about the general case where the two orderings are formed by all permutations of symbols $d,e,f,<,>$ and $a,b,c,<,>$ respectively, such as $a < b < c$ and $d < f < e$, what condition is needed to ensure $b+e$ is a maximum?