improved Egg Notation - Analysis

Let me know if theres some specific numbers or another notation you guys would like me to add., i don't promise to add every suggestion but i'll take ideas.

The right side counts the red eggs upon applying the notation to the left side

Improved Egg Notation Analysis Table

iEN	FGH
[]()()()	$f_0(3)$
[()]()()()()	$f_{1}(4)$
[()()]()()()()()	$f_{2}(5)$
[][()()()]()()()()()()	$f_{0}(f_{3}(6))$
[()][()()()()]()()()()()()()	$f_{1}(f_{4}(7))$
[()()][()()()()()]()()()()()()()()	$f_{2}(f_{5}(8))$
[[]()()()]()()()()()()	$f_{f_{0}(3)}(6)$
[[()]()()()()]()()()()()()()	$f_{f_{1}(4)}(7)$
[[()()]()()()()()]()()()()()()()()	$f_{f_{2}(5)}(8)$
[{(),}]()()()	$f_\omega(3)$
[{(),()}]()()()()	$f_{\omega+1}(4)$
[{(),()()}]()()()()()	$f_{\omega+2}(5)$
[{(),}][{(),}]()()()	$f_\omega(f_\omega(3))$
[{(),}][{(),()}]()()()()	$f_{\omega}(f_{\omega+1}(4))$
[{(),}][{(),()()}]()()()()()	$f_{\omega}(f_{\omega+2}(5))$
[{(),}][{(),()()()}]()()()()()()	$f_{\omega}(f_{\omega+3}(6))$
[{(),()}][{(),()()()()}]()()()()()()()	$f_{\omega+1}(f_{\omega+4}(7))$
[{(),()()}][{(),()()()()()}]()()()()()()()()	$f_{\omega+2}(f_{\omega+5}(8))$
[{(),{(),}}]()()()()()()	$f_{\omega+\omega}(6)$
[{(),{(),()()()()}}][()()()()()()()]()()()	$f_{\omega+\omega+4}(f_{7}(3))$
[{()(),}]()()()()()()	$f_{\omega^2}(6))$
[()()(),()]()()()()()()()	$f_{\omega^3+1}(7))$
[{()()()(),()()()()()}]()()()()()()()()	$f_{\omega^4+5}(8))$
[{()()()()(),{()()()()(),()}}]()()()()	$f_{\omega^5+\omega^5+1}(4))$
[{()()()()()()(),{()(),()()()()}}]()()()	$f_{\omega^7+\omega^2+4}(3))$
[{{(),},()()()()}]()()()()	$f_{\omega^{\omega}+4}(6))$
[{{()()(),},{()(),()()()()()}}]()()()()()()()	$f_{\omega^{\omega^3}+\omega^2+5}(7))$
[{{()()()(),()},}]()()()	$f_{\omega^{(\omega^4+1)}}(3))$
[{{{(),},},()()()()()()()()()()}]()()()()	$f_{\omega^{\omega^\omega}+10}(4))$
[{{{(),},()()},}]()()()()()	$f_{\omega^{(\omega^\omega+2)}}(5))$
[{{{(),},{()(),}},}]()()()()	$f_{\omega^{(\omega^\omega+\omega^2)}}(4))$
[{{{(),()},},}]()()()	$f_{\omega^{\omega^{(\omega+1)}}}(3))$
[{{{(),()()()()()()},{()()(),}},()()()()}]()()()()()()()	$f_{\omega^{(\omega^{\omega+6}+\omega^3)}+4}(7))$
[{{{{(),},},},}]()()()()()()()()()()()	$f_{\omega^{\omega^{\omega^\omega}}}(11))$
[{{{{{(),},},},},}][{(),()()}]()()()()	$f_{\omega^{\omega^{\omega^{\omega^\omega}}}}(f_{\omega+2}(4)))$