10. Let m and n be positive integers. Exhibit an arrangement of the integersbetween 1 and mn which has no increasing subsequence of length m + 1,and no decreasing subsequence of length n + 1

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

Let m and n be positive integers. Exhibit an arrangement of the integers between 1 and mn which has no increasing subsequence of length m + 1, and no decreasing subsequence of length n + 1.

Explore the combinatorics problem of arranging integers between 1 and mn with no increasing subsequence of length m + 1 and no decreasing subsequence of length n + 1. This solution involves concepts from permutation theory, the Robinson-Schensted-Knuth correspondence, and Ferrer diagrams, making it suitable for advanced high school and undergraduate students.

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