Reversed Letter Word Formations and Graph Theory

Robleh Wais 11/5/2023

Take the following words:

                    stop

                    star

Reverse them and you get:

                          pots

                          rats

Of course, there are many such words (I believe) in English when reversed yield another English word.  But how many are there? What is the percentage of the total lexicon that exhibits this property? Is this percentage growing?  How does it compare to other languages?  I don't have the answer to the latter question for sure but would like to one day research this aspect of human languages.  One definite answer comes to mind immediately, reversed letter word formation does NOT increase the size of the English vocabulary.  The reason is obvious, but it does highlight an odd ability to make a directed graphical map of the language.  If that is the case, we can ask does this directed graphical mapping lead to further mappings?  Let me clarify this idea. We can rearrange the reversed words, i.e., make an anagram, then reverse it again, to map to another English word.  The word could be the same as the original word.  For example, take the word stop above, it can be anagrammed to be tops So, its directed graph, would be the map below:

stop

pots

tops

The graph would be a triangle with 3 nodes and 3 edges

What is needed is to enumerate the subset of words with this quality, and then a second test of reversibility to locate new words after they are anagrammed.  And yes, of course, further operations of this type to see if this directed graph can be extended.  This is a simple straightforward process any high-level programming language like Python could perform given an input dataset.  What is not clear is where in graph theory mathematics this would lead.  We might discover that a rich network of reversed/anagrammed/reversed words can be discovered to connect English words.

Well, this is exactly what I've begun to do.  Included in this article is an Excel sheet with a VBA macro that reverses common English words. I have made a sample of approximately 50,000 words, most of which are 3 to 5 letters long, and there are some amazing results.  Above is an image of the directed graph using reversed words. A black and white image of lines and dots Description automatically generated

Here is a much better image after some processing and mapping of directed word pairs. Maybe not as esthetically attractive, but more informative.  Look closely at the image below, you can see that matching pairs of reversed words are shown in purple.  They represent 389 words. That is, words that reverse to form other English words, and the remaining grey nodes with edges are non-English word mappings.

A green spiral on a white background Description automatically generated

I have uploaded the source data for anyone interested in reviewing it.  There is a VBA Excel sheet that was used to generate the source data.  The Gephi program was used to create the images above.

Below is the link for downloading.

Reversed English Words

Permutations, Graph Theory, and Reversed Words

Well, this idea becomes even more fascinating if we consider some combination versus permutations of these words. First, let's set the stage with a little background on combinatoric formulations.

   Permutations of sets of objects can be calculated using the formula, = n! / (n-r), where N = set of objects and r = a subset sample of N. The arrangement of subsets is important. A permutation is a distinct set pairing operation, which differs from a combination by simple omission of the variable n in the formula. This omission significantly reduces the number of ordered pairs as will become obvious from exemplification.

   Combinations of sets of objects can be calculated using the formula, = n! /n! (n-r), where n = set of objects and r = a subset sample of n. The bold n indicates the factor difference in the formulation.

So, let's calculate P(4,2) and C(4,2).

P(4,2) = 4!/(4!-2!) =12 and C(4,2) = 4!/2!(4-2)! = 6

There are more permutations of a given number pair set than combinations of the same number pair set. This is the key to our next step. What is that step? It is crying out to be heard! How many more reversed words would be generated if we allowed the reversed words to be rearranged based on their permutation structure? The number sets would grow immensely if we increased r in permutations. But, since we don't know if these subsets would generate English words, it is the randomicity that would determine the number of reversed English words that would be generated. Another important point, the set will explode, I have an example to illustrate this, but the remapped set would not. So, as an example take the word reward. Calculate P(6,4) and the result is 360. Here is the list to prove it.

This is an abbreviation of the total list, and yes you guessed it there are a few English words in there, I have marked them in bold case. The subset of permutated English words of REWARD is small. This is for a 6-letter word but think of  7, or 8-letter words. If we take the new word formed and reverse it again, we can find perhaps more of these reversed word pairs. This is the mind-bending part. Right before us is the possibility to find new connections within the linguistic structure of this language (and others by association), through combinatorics, and graph theory. I suspect the lack of diacritical characters in English increases the probability of finding reversed word pairs. However, I can't offer a probabilistic proof of this guess.  I could be quite wrong on this point. Another reason this is becoming so interesting to me.

More to Come.....Where there is Permutation there is Probability....now it gets even more astounding.....