Quick question for proof on unimodal sequence formula in Enumerative Combinatorics

Question

I am looking at page $238$ of Stanley's Enumerative Combinatorics where he says that $\#V_n = \#D_n - \#V_n^1$ because every element in $V_n^1$ appears twice as a value of $\Gamma_1$. Can someone clarify why this is true?

Answer 1

I would have thought that this issue is related to the possibility that two elements of $D_n$

$$\left[\begin{array} \, & c& a_1 & a_2 & \ldots \\ \, & b_1 & b_2 & \ldots\end{array}\right]$$ and

$$\left[\begin{array} \, & a_1 & a_2 & \ldots \\ \, &c & b_1 & b_2 & \ldots\end{array}\right]$$

could each correspond to the same element of $V_n$

$$\left[\begin{array} \, & a_1 & a_2 & \ldots \\ c & \, & \, & \, \\ \, & b_1 & b_2 & \ldots\end{array}\right]$$

providing that $c \ge a_1$ and $c \ge b_1$.