This problem is asking for an arrangement of integers between $1$ and $mn$ such that there is no increasing subsequence of length $m+1$ and no decreasing subsequence of length $n+1$. A well-known solution to this problem is based on the Robinson-Schensted-Knuth (RSK) correspondence and the construction of a standard Young tableau. Here’s a construction for such a sequence:

Solution Approach:

One common method to find such a sequence is to use a permutation that avoids long increasing or decreasing subsequences. The Ferrer diagram and the RSK algorithm ensure that there are bounds on the lengths of the increasing and decreasing subsequences.

We can use a two-line standard representation approach, which helps control both the increasing and decreasing subsequences. Specifically, the reverse row-reading word of an $m \times n$ tableau provides such a sequence.

Construction:

An arrangement that satisfies the conditions for this problem can be obtained using the Lexicographic order of the integers organized into an $m \times n$ grid.

For simplicity, we can take the following example construction when $m = 3$ and $n = 3$ (though the method works for any $m$ and $n$):

1. List the integers between $1$ and $mn = 9$. $1, 2, 3, 4, 5, 6, 7, 8, 9$

2. Arrange these integers in a grid as