 This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right)  .
<meaning-changed>  This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left({2}}}\right)  .
 This provides an alternative proof of the previously known result, as well as new structural insights into these diagonal flip paths\left(n^{\frac{3{2}}}\right) .