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I've come across a function from the set of integer partitions to the natural numbers which I don't recognise but which probably ought to be familiar; it arises in the homogeneous Garnir relations for …

The answer is yes, and it follows from the results in my paper "A generalisation of core partitions", J. Combin. Theory 127. In that paper I define a class of partitions called $[p:q]$-cores. One ch …

For $j\geqslant0$ let $c_j$ denote the $2$-core partition $(j,j-1,\dots,1)$. Your conditions on partitions of $2n$ can be re-phrased as asking for $2$-restricted partitions of $2$-weight $n$ and $2$- …

The answer to Question 1 is no. Here's a construction. First we construct a simple graph $G_n$ on $n$ vertices for each $n\geqslant3$ which has every degree from $1$ to $n-1$ inclusive occurring, wi …

I think the following is a simple combinatorial argument which constructs the most dominant semistandard $\lambda$-tableau of content $\mu$ whenever $\lambda\trianglerighteq\mu$. (n.b. I haven't foll …

I can't give you your desired "most general" theorem, but I can say a little about this. In (b), the condition "shape(A) is lexicographically larger than shape(B)" is much stronger than it needs to be …

The equivalence of Kleshchev's algorithm and Mullineux's algorithm was proved by Ford and Kleshchev, but the result they prove is slightly weaker than you want. The result you're asking for is Coroll …