<s>[INST] Finding a set of representatives in a certain array made from a multiset Let $M$ be an $n \times n$ array whose entries are from the multiset consisting of a single one, three twos, five threes,..., and $2n-1$ $n$'s, with the additional condition that, for $1\leq k\leq n$, no row or column have more than $k$ $k$'s.
Is it the case that one can always find $r_1, r_2,\dots,r_n$, all different, such that row $r_1$ contains a $1$, row $r_2$ contains a $2,\dots,$ and row $r_n$ contains an $n$, or one can find $c_1, c_2,\dots,c_n$, all different, such that column $c_1$ contains a $1$, column $c_2$ contains a $2,\dots,$ column cn contains an $n$?
 [/INST] It turns out that it is not the case that one can always find a set of distinct representatives for matrices as described in the question.  Here is an example of a 9x9 matrix where no set can be chosen:
$$\begin{pmatrix}x & x & x & 6 & 5 & 5 & 5 & 5 & 5\\x & x & x & 5 & 6 & 6 & 6 & 6 & 5\\x & x & x & 6 & 6 & 6 & 6 & 6 & 6\\x & x & x & 7 & 7 & 7 & 7 & 7 & 7\\x & x & x & 7 & 7 & 7 & 7 & 7 & 7\\x & x & x & 7 & 8 & 8 & 8 & 8 & 8\\9 & 8 & 8 & 8 & 8 & 8 & 8 & 8 & 8\\8 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 8\\9 & 9 & 9 & 9 & 9 & 9 & 9 & 9 & 9\end{pmatrix}$$
The $x$'s may be any appropriate arrangement of the proper number of $1$'s, $2$'s, $3$'s, $4$'s, and $5$'s.
The bottom three rows contain only two different numbers: $8$ and $9$. So its not possible to choose three of them containing three different numbers.
Similarly, the rightmost six columns contain only five different numbers, so its not possible to choose six of them containing six different numbers.
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