Denote the collection of partitions of $n$ by $\mathcal{P}(n)$, with the property that for $\lambda = (\lambda_1,\cdots,\lambda_s)\in\mathcal{P}(n)$ we have $$\lambda_1\geq \cdots \geq \lambda_s \geq 1,\qquad \lambda_1+\cdots +\lambda_s =n$$ Then we define a partial order on $\mathcal{P}(n)$, denoted $\geq $, by taking for $\lambda,\mu \in \mathcal{P}(n)$ where $\lambda$ is as above, and $\mu = (\mu_1,\cdots,\mu_t)$: $$\lambda \geq \mu \iff \sum_{i=1}^k \lambda_i \geq \sum_{i=1}^k \mu_i \quad \text{for} \,\, 1\leq k \leq \text{min}(s,t).$$
Are $(5,2,1,1,1,1)$ and $(3,3,3,2)$ incomparable? We have $\text{min}(s,t)= 4$ and $n=11$:
\begin{array}{c|c} k& \text{sum up to k for }\lambda \text{ and }\mu\\ k=1 & 5> 3\\ k=2& 7>6\\ k=3& 8<9\\ k=4& 9<11\\ \end{array}
It seems so. Am I understanding this dominance ordering correctly? I think then I have this top one, dominant over both of the below. But the below are incomparable.
$$\begin{array}{|c|c|c|c|c|} \hline \blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare\\\hline \blacksquare&\blacksquare&\blacksquare&\\\hline \blacksquare&\blacksquare\\\hline \blacksquare\\\hline \\\hline \\\hline \end{array} $$ Is dominant over both of these: $$\begin{array}{|c|c|c|c|c|} \hline \blacksquare&\blacksquare&\blacksquare&\blacksquare&\blacksquare\\\hline \blacksquare&\blacksquare&&\\\hline \blacksquare\\\hline \blacksquare\\\hline \blacksquare\\\hline \blacksquare\\\hline \end{array} \qquad\begin{array}{|c|c|c|c|c|} \hline \blacksquare&\blacksquare&\blacksquare&\color{white}{\blacksquare}&\color{white}{\blacksquare}\\\hline \blacksquare&\blacksquare&\blacksquare&\\\hline \blacksquare&\blacksquare&\blacksquare\\\hline \blacksquare&\blacksquare\\\hline \\\hline \\\hline \end{array}$$