Calculating Sequences with Specified Longest Increasing Subsequence Length

Calculating the number of sequences with a specified longest increasing subsequence (LIS) length involves a combinatorial understanding of the problem and the application of specific formulas and techniques. Understanding how to approach this problem is crucial for anyone dealing with sequences and subsequences in mathematical and computational contexts.

Step 1: Understanding the Problem

A sequence of length n has a longest increasing subsequence of length k if there exists a subsequence of k elements that is increasing, and no subsequence of length greater than k exists. This problem can be quite complex and requires a combinatorial approach to find the number of such sequences.

Step 2: Utilizing the Hook-Length Formula

One effective method to count such sequences is through the use of the hook-length formula, a powerful tool in combinatorial theory, particularly involving Young tableaux.

Partitions

Consider the partition of n into k parts. Each part represents a number of elements that can form an increasing subsequence.

Young Tableaux

The number of ways to fill a Young tableau of shape corresponding to a partition can be calculated using the hook-length formula. The formula for the number of standard Young tableaux of shape lambda is given by:

[ f^{lambda} frac{n!}{prod_{(i,j) in lambda} hook_{ij}} ]

where hook_{ij} is the hook-length of the cell (i,j) in the Young diagram of the partition.

Step 3: Counting Sequences

To count the sequences with a specific LIS length k:

Choose k positions for the increasing subsequence: You can choose k positions from n total positions. Fill the chosen positions: The values in these positions must be increasing. Fill the remaining n-k positions: The values in these positions can be filled with any integers that do not form an increasing sequence with the chosen k.

Step 4: Using Combinatorial Techniques

You may also use generating functions or advanced combinatorial techniques to derive the exact count depending on the constraints on the numbers in the sequences. For example, if the elements must be distinct, this can be handled using binomial coefficients.

Example

For instance, if you want to calculate the number of sequences of length n with an LIS of length k:

Choose k elements from n if the elements are distinct, this is simply [ binom{n}{k} ]. Arrange the k elements in increasing order, which can be done in only one way. Arrange the remaining n-k elements such that they do not increase the length of the LIS beyond k.

Conclusion

The calculation can get complex depending on the constraints of the sequences you are considering, such as distinct integers, bounded integers, etc. For specific cases, you might need to adapt the approach or use computational tools to enumerate valid sequences. If you need more specific details or further context on specific cases, please provide additional information!