This is a problem from my professor, who claimed that it's open:

Combinatorial problem.

Fill $1,2,...,mn$ into a rectangle of size $m\times n$, such that for every number other than $mn$, there is a larger number which is in the same row or column.

Prove there are $\frac{(mn)!m!n!}{(m+n-1)!}$ ways to fill.

It is said that the problem is from a high school student.

I would like to know:

Is the problem really open?

If it's open, are there any references? Is it from a high school student, as claimed?

There's no need to answer the problem to any extent.

  • 1
    $\begingroup$ I have rolled the title back to the wording of the OP, which is consistent with the wording of the question being quoted by the OP, rather than the pernickety change of title made by the edit-happy user @user64494 $\endgroup$ – Yemon Choi Jun 29 '19 at 13:56
  • $\begingroup$ @Yemon Choi: As far as I know it, the rectangle is a geometric term. I have never before seen rows and columns of a rectangle. Also the use of the notation $<$ in a title does not make a good impression. $\endgroup$ – user64494 Jun 29 '19 at 17:45
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    $\begingroup$ @user64494 for example, "Latin rectangles" have rows and columns, and they are matrices/boards/tableaux filled with prescribed numbers, just as in this question. The word "matrix" is more algebraic-flavoured, as for me. $\endgroup$ – Fedor Petrov Jun 30 '19 at 7:05

Here it is.

Number of collinear ways to fill a grid

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