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## What type of mathematical knowledge is needed to understand these mathematical notations?https://arxiv.org/pdf/1603.01121

There are a whole bunch of ways to depict permutations, depending on what you want to use them for. I'll assume you index the set you're permuting with 1,2,\cdots, though some of these notations generalize. Say we want to depict the permutation sending 1 to 4, 2 to 1, 3 to 3, 4 to 5, and 5 to 2. Here are some of the ways we can depict this permutation (if I forgot any, mention them in comments and I'll add them!). 1) Double Line Notation. \begin{pmatrix}1&2&3&4&5\\4&1&3&5&2\end{pmatrix}. Here it's clear that the first line of numbers turns into the second. 2) Line Notation. 41352. This is just the second line of the double-line notation. Due to its brevity, this is the most common version I've seen and heard (it's also easy to say). 3) Cycle Notation. (1452)(3). Within each cycle (parenthesized segment), this means that you move one to the left, wrapping around at the end (so 2 goes to 1). Sometimes the (3) will be omitted since you can figure it out from context. This is most common if you care about the algebraic properties of your permutation, for instance if you want to know what happens when you apply it a bunch of times. 4) Directed Graph Notation. Label 5 vertices and draw corresponding arrows. The result in this case is a square and a loop. This is a nice way to visualize the cyclic structure of the permutation. 5) Matrix Notation. \begin{pmatrix}0&1&0&0&0\\0&0&0&0&1\\0&0&1&0&0\\1&0&0&0&0\\0&0&0&1&0\end{pmatrix}. To see where 4 goes in this notation, you look at the 4th column, and there's a 1 in the 5th row. You can think of this matrix as a permutation on the standard basis vectors (e.g. the column vector (0,0,0,1,0) represents "4"). This is also nice because composing permutations is just matrix multiplication (make sure to get the order right!). 6) Dot Notation. \begin{vmatrix}\hline&&&\bullet&\\\bullet&&&&\\&&\bullet&&\\&&&&\bullet\\&\bullet&&&\\\hline\end{vmatrix}. Think of this as "plotting" the permutation as a function. Notice that this is similar to matrix notation, but upside down relative to it. This is nice if you care about increasing or decreasing subsequences within the permutation.

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So, in conclusion, the math proof on this site is a lie. But not in the way that people think.