Question 6
Karl starts with cards $1,2,\ldots,n$ in any order. A pair $(a,b)$ is swapped if $a\gt b$ and card $a$ is left of card $b$. He moves card 1 to its opposite position: if it had $i$ cards to its left, it now has $i$ cards to its right. He does the same for cards $2,3,\ldots,n$, each exactly once in that order.
For $n=4$, the source example is
$$3142\longrightarrow3412\longrightarrow2341\longrightarrow2431\longrightarrow2341,$$
with three swapped pairs initially and finally. Prove that the final order always has the same number of swapped pairs as the initial order.
Proposed by Maria Monks Gillespie.
For $n=4$, the source example is
$$3142\longrightarrow3412\longrightarrow2341\longrightarrow2431\longrightarrow2341,$$
with three swapped pairs initially and finally. Prove that the final order always has the same number of swapped pairs as the initial order.
Proposed by Maria Monks Gillespie.