Sunday, June 20, 2010

Counting Monotonic Decreasing Sequences

In this blog post I will count the number of monotonic decreasing sequences of length \(m\) and height \(n\).

I recently got interested in graphic sequences. A graphic sequence is an example of a monotonic decreasing non-negative integer sequence i.e. a sequence which never goes up. For example

4,3,3,3,1,1

The next terminology will be used throughout this blog. The height of a sequence is the greatest element which occurs in the sequence. For monotonic decreasing sequences this corresponds with the first element. So the height of the sequence mentioned above is 5.

I set out to count the number of non-negative integer sequences of length <\(m\) and height \(n\). After some considerations I found an elegant manner in which to present the answer.


Consider plotting the above sequence on graph paper. We need a 6 by 5 box to plot the sequence. (Remember, it is a sequence of non-negative integers.) The graph is given below.


When the descenders are added a path is formed from the upper left corner to the lower right corner. (The descenders are in orange.) Because we have plotted a monotonic decreasing sequence the path is formed by making steps due "south" or "east". There are precisely \({m+n-1} \choose {n}\) such paths. (There are \(m+n-1\) choices to make of which \(n\) are "south").

An other relation can be deduced by examining the correspondence between monotonic decreasing sequences and south-east-paths. Let's count the monotonic decreasing sequences of length \(m\) and height at most \(n\). Notice that all those sequences are a tail-subsequence of a monotonic decreasing sequence of length \(m + 1\) and height \(n\). So one can deduce the relation

\[
\sum_{i=0}^{n} {{m+n-1} \choose {n}} = {{m+n} \choose {n}}
\]

On can count the number of monotonic decreasing non-negative integer sequences by considering a nice correspondence between such sequences and south-east-paths on grids.

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