In Symmetric Functions and Hall Polynomials by Manin, Manin claims that for a skew diagram $\theta = \lambda - \mu$ to be a horizontal $m$-strip, "the sequences $\lambda$ and $\mu$ are interlaced, in the sense that $\lambda_1 \geq \mu_1 \geq \lambda_2 \geq \mu_2 \geq \dots$" (top paragraph of page 5). However, isn't a necessary condition for a horizontal $m$-strip that that for each $\theta_i = \lambda_i - \mu_i = 0$, for $i > 1$? Additionally, shouldn't $\theta_1 = \lambda_1 - \mu_1 > 0$ so the "interlaced" inequality would look more like $\lambda_1 > \mu_1 \geq \lambda_2 = \mu_2 \geq \lambda_3 = \mu_3 \dots$?
I also have another question in regard to how the author addresses skew diagrams. In this skew diagram below, skew diagram, when he says the first column of the skew diagram, is that the first column of the whole diagram, or is that the first column of dark squares? Basically, what I'm asking is, does $\theta_1' = 1$ or 4?
Also, I would really appreciate it if anyone had any suggestions for other resources or problem sets that could be helpful for self-studying this book!
You must mean Macdonald's Symmetric Functions and Hall Polynomials.
Let me address your second question first: In the diagram of $\theta = (5,4,4,1) - (4,3,2)$, the skew partition consists of the shaded squares. From the definition $\theta_i = \lambda_i - \mu_i$, we have $\theta_1 = 5 - 4 = 1$, $\theta_2 = 4 - 3 = 1$, $\theta_3 = 4 - 2 = 2$, and $\theta_4 = 1-0 = 1$.
The terminology "horizontal strip" is a little confusing. As he says on p5, the notion is equivalent to having at most one shaded box in each column. For, instance, $(10,9,6,4,1) - (9,7,4,1)$ is a horizontal 9-strip, shown below. The name "horizontal" makes some sense as each connected component is part of a row.
The image is from some online notes by Andres Buch. You might also look at Fulton's book Young Tableaux and Sagan's The Symmetric Group.