This question has its genesis in a group assignment: $k$ students are to be given oral exams. Each student will be asked one distinct question from $n$ questions given to them earlier, no two students will be asked the same question. In order to save time, the group pools their answers to share before the exams. To their surprise, on the exam date, no person was asked a question that they worked on. What is the probability that this would happen? We'll assume the examiners had no knowledge of how they worked, and that every permutation of questions asked was equally likely.

Assuming that no people work together on a question, we can regard the distribution of questions among students as a partition of $n$ into $k$ parts. Then we are counting the number of ways to write $1,2,\ldots,k$ into the Young diagram with the shape of the partition, such that each number $i$ does not appear in row $i$. By including the numbers $k+1,\ldots,n$, which can be freely added to the remaining squares of the diagram in $(n-k)!$ ways, we can assume that there are $n$ students, some of whom may not contribute to any question. For a partition $\lambda$, call the number of ways to fill in the diagram $f(\lambda)$.

If every student solves one question, then $f([1^n])$ is the number of the derangements of $n$. If some student does not solve a question, then they can be placed anywhere, which gives a recurrence relation:

$$ f([a_1,a_2,\ldots,a_k]) = \sum_{i=1}^k a_if([a_1,\ldots,a_i-1,\ldots,a_k]) $$

Using this, I found the numbers $f([m,1^{n-m}])$ are given by $m!\cdot(n-m)\cdot (\text{an OEIS sequence})$ where the sequences for $m=2,\ldots,6$ are: A000153, A000261, A001909, A001910, A176732 and there are similar entries for $m=7,8,9,10$.

I could not find any formula or reference related to other simple families of partitions, or the problem in general.

Do these derangement numbers have a name?