The Cascading-E notation (E^ for short) is a further extension and generalization of Hyper-E notation introduced by Sbiis Saibian in January 22, 2013. The limit of its growth rate is comparable to $$f_{\varepsilon_0}(n)$$ in FGH, making it about as powerful as nested arrays in Bird's Array Notation.

E^ is part of the general Extensible-E System, which includes all future extensions of the notation, such as Extended Cascading-E Notation.

## Terminology

Separators of the form $$\#^{X}*\#^{X}\cdots\#^{X}*\#^{X}$$ are known as a hyper-product of cascaders and each $$\#^{X}$$ is a single cascader. X can be any positive integer or itself can be a hyper-product of cascaders. This recursive definition defines all possible separators in Cascading-E. Consequently, even separators in the form $$\#^{n}$$ where n is a positive integer are considered a hyper-product of cascaders. Sometimes in basic numbers, like Godgahlah, we can say that En#^#n = En####...####n with n #'s..

### Key Band

The Key Band is a cascader of the form $$\#^{n}$$ (n is a positive integer), situated at the right most position of the hyper-product.

The following search-algorithm finds and defines the Key Band:

1. Begin at ground level (level 0), and proceed to step 2.
2. For the current level, find the last cascader of the current hyper-product, and proceed to step 3.
3. If this cascader is in the form $$\#^{n}$$, then it is the Key Band by definition, otherwise proceed to step 4.
4. Go up to the next exponent level of the last cascader and go back to step 2.

## Definition

Let Ea1&a2& ... &an be any expression in E^, where a1 through an are n positive integers, and all & are hyper-products (which may or may not be distinct.) Each individual & may be chosen from the set of legal separators.

Below are the 5 formal rules of E^. Let $$\&_k$$ be the kth hyper-product and $$L(\&_k)$$ be the last cascader of the kth hyper-product.

• Rule 1. With no hyperions, we have $$E(a)b = a^b$$.
• Rule 2. If $$L(\&_{n-1}) \neq \#^n$$ (the last cascader is not of the form $$\#^{n}$$):
$$E(a)b@X\#^{(X\#^{n})}@c = E(a)b@X\#^{(X\#^{n-1})^{^b}}@b$$ (@ indicates the unchanged remainder of the expression.)
• Rule 3. If the last argument is 1, it can be removed: $$E@a\#^{n}1 = E@a$$
• Rule 4. $$L(\&_{n-1}) = \#^n$$ and $$\&_k \neq \#$$:
$$E@aX\#^{n}b = E@aX\#^{n-1}aX\#^{n}(b-1)$$.
• Rule 5. Otherwise:
$$E@a\#b = E@(E@a\#(b-1))$$

## Examples

• E10#^#3 = E10###10
• E10#^#10#^#3 = E10#^#10###10
• E10#^#*#3 = E10#^#10#^#10
• E10#^##3 = E10#^#*#^#*#^#10
• E10#^#^#3 = E10#^(###)10
• E100#^#1 = E100#100

## Sources

1. Saibian, SbiisCascading-E Notation. Retrieved 2013-05-02.