I assume $\lambda$ is a partition with parts $\lambda_1 \geq \lambda_2 \geq \dots$, while the parts of the conjugate partition $\lambda'$ are $\lambda'_1 \geq \lambda'_2 \geq \dots$.

But not so with exclamation marks, since 5! = 120 but 5!! = 15.

Your claim for type A is not true. Take $\underline{y} = s_1 s_2 s_3 s_1 s_4 s_2 s_1$, a reduced expression for $y = 43251$ and $\underline{w} = s_1 s_2 s_3 s_4 s_2 s_1$, a reduced expression for $w = 42351$.

A comment about the trick itself: You can vary where the card identifying the suit appears as follows. After the assistant gives the mystery card to the mark, they add the ranks of the remaining 4 cards modulo 4 and use that number to determine the position of the suit card, which the magician can recover with the same computation. The order of the other three cards is determined by ignoring the suit card. This small addition to the trick makes it much harder for the mark to crack.