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I will start with extended chain notation. For more extension, I will write numbers below the chain.

a\rightarrow_b c=a\xrightarrow[b]{} c

This looks like chemical equation, so I call {what is below the chain} {catalyst}. The catalyst makes number really bigger.

Catalyst must not be one. Some reactions require more than one catalysts.

a\xrightarrow[1,0]{} b = a\xrightarrow[a\xrightarrow[a\xrightarrow[\cdots a\xrightarrow[]{} a]{} a]{} a]{} a (b stories)

Next, using the same rule, we can define up to a\xrightarrow[1,x]{} b .

Then,

a\xrightarrow[2,0]{} b = a\xrightarrow[1,a\xrightarrow[1,a\xrightarrow[\cdots 1,a\xrightarrow[]{} a]{} a]{} a]{} a (b stories)

Keep going on until a\xrightarrow[x,y]{} b. At this stage, I think this is as big as f_{\omega ^4} (x).

Next, three catalysts come.

a\xrightarrow[1,0,0]{} b = a\xrightarrow[a\xrightarrow[a\xrightarrow[\cdots a\xrightarrow[]{} a,0]{} a,0]{} a,0]{} a (b stories)

See the pattern? As you know, definition can be up to n-entry.

This notation makes up to f_{\omega^\omega}(x).

If you have a question, ask in a comment.


For comment, I will write generalized formula.

If the last number under the chain is not zero, it is same as extended chain (so you can ignore what is the left).

a\xrightarrow[k,0]{} b = a\xrightarrow[k-1,a\xrightarrow[k-1,a\xrightarrow[\cdots k-1,a\xrightarrow[]{} a]{} a]{} a]{} a (b stories)

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