## 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.

1. In the near future I intend to count the number of graphic sequences of lenght m

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https://khairyayman74.wordpress.com/2015/07/14/%D8%B4%D8%B1%D9%83%D8%A9-%D8%BA%D8%B3%D9%8A%D9%84-%D8%A7%D9%84%D9%81%D9%84%D9%84-%D9%88%D8%A7%D9%84%D9%82%D8%B5%D9%88%D8%B1-%D8%A8%D8%A7%D9%84%D8%AF%D9%85%D8%A7%D9%85/
https://khairyayman74.wordpress.com/2015/06/30/%D8%B4%D8%B1%D9%83%D8%A9-%D8%BA%D8%B3%D9%8A%D9%84-%D9%83%D9%86%D8%A8-%D8%A8%D8%A7%D9%84%D8%AF%D9%85%D8%A7%D9%85-0548923301/
https://khairyayman74.wordpress.com/2015/05/30/%D8%B4%D8%B1%D9%83%D8%A9-%D9%86%D9%82%D9%84-%D8%A7%D8%AB%D8%A7%D8%AB-%D8%A8%D8%A7%D9%84%D8%AF%D9%85%D8%A7%D9%85/
https://khairyayman74.wordpress.com/2015/06/14/%D8%BA%D8%B3%D9%8A%D9%84-%D8%A8%D8%B7%D8%B1%D9%85%D9%87-%D8%A8%D8%A7%D9%84%D8%AF%D9%85%D8%A7%D9%85-0548923301/
http://www.abyath.com/2015/07/20/%D8%B4%D8%B1%D9%83%D8%A9-%D9%86%D9%82%D9%84-%D8%A7%D8%AB%D8%A7%D8%AB-%D8%A7%D9%84%D8%AF%D9%85%D8%A7%D9%85-0542065086-2/
http://www.khdmat-sa.com/2015/07/22/%D8%B4%D8%B1%D9%83%D8%A9-%D9%86%D9%82%D9%84-%D8%B9%D9%81%D8%B4-%D8%A8%D8%A7%D9%84%D8%AE%D8%A8%D8%B1-0559328721/
https://khairyayman74.wordpress.com/2015/07/23/%D8%B4%D8%B1%D9%83%D8%A9-%D9%86%D9%82%D9%84-%D8%A7%D8%AB%D8%A7%D8%AB-%D9%81%D9%89-%D8%A7%D9%84%D8%AE%D8%A8%D8%B1-0559328721/
http://www.khdmat-sa.com/2015/07/28/%d8%b4%d8%b1%d9%83%d8%a9-%d9%86%d9%82%d9%84-%d8%a7%d9%84%d8%a7%d8%ab%d8%a7%d8%ab-%d9%81%d9%8a-%d8%a7%d9%84%d9%82%d8%b7%d9%8a%d9%81-0559328721/

15. http://accountingeg.blogspot.com/
http://www.babelsoftco.com
http://www.babelsoftco.com/Mokwlat_Main.html
http://eslamiatview.blogspot.com/
http://www.nakl-eldmam.com/
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http://www.abyath.com/
http://www.abyath.com/2015/06/27/%D8%B4%D8%B1%D9%83%D8%A9-%D8%AA%D9%86%D8%B8%D9%8A%D9%81-%D8%AE%D8%B2%D9%86%D8%A7%D8%AA-%D8%A7%D9%84%D8%AF%D9%85%D8%A7%D9%85-0542065086/
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http://www.tfa2ol.com/
http://www.tfa2ol.com/naklasas.html
http://www.tfa2ol.com/khasil.html
http://www.tfa2ol.com/mokafha.html
http://www.tfa2ol.com/nzafa.html
http://eslamiatview.blogspot.com/

16. http://www.a3lnatk.com/%d9%86%d9%82%d9%84-%d8%a7%d8%ab%d8%a7%d8%ab-%d8%a8%d8%a7%d9%84%d8%af%d9%85%d8%a7%d9%85

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