A069770*


Description:
Swap Binary Tree Sides

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 0.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as binary trees, polygon triangulations, parenthesizations and Lukasiewicz-words.

                    
                    
[()()()()][(()()())]          [()()(())][(()(()))]          [()(())()][((())())]
4000013000          3001012010          3010012100

                    
                    
[()(()())][((()()))]          [()((()))][(((())))]          [(())()()][(()())()]
2020011200          2011011110          3100022000

[(())(())][((()))()]
2101021100


A072796*


Description:
Exchange of the two leftmost branches of general trees

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
Yes, the restriction to binary trees induces the gatomorphism A0?????.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 1.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as general trees, parenthesizations and Lukasiewicz-words.

                    
[()(())()][(())()()]          [()(()())][(()())()]          [()((()))][((()))()]
3010031000          2020022000          2011021100

                              
[()()()()]          [()()(())]          [(())(())]          [(()()())]
40000          30010          21010          13000

                              
[(()(()))]          [((())())]          [((()()))]          [(((())))]
12010          12100          11200          11110


A057163*


Description:
Reflect of binary trees and polygon triangulations

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 2.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as binary trees, polygon triangulations, parenthesizations and Lukasiewicz-words.

                    
                    
[()()()()][(((())))]          [()()(())][((()()))]          [()(())()][((())())]
4000011110          3001011200          3010012100

                    
                    
[()(()())][(()(()))]          [()((()))][(()()())]          [(())()()][((()))()]
2020012010          2011013000          3100021100

[(())(())][(()())()]
2101022000


A069767/A069768


Description:
Recursive Swap of the other side of binary tree

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 6.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as binary trees, polygon triangulations, parenthesizations and Lukasiewicz-words.

[()()()()][(()()())][()(()())][((()()))][()()(())][(()(()))][()((()))][(((())))]
4000013000202001120030010120102011011110

          
          
[(())()()][(()())()][(())(())][((()))()]          [()(())()][((())())]
31000220002101021100          3010012100


A057511/A057512


Description:
Deep Rotation of general trees and parenthesizations

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
Yes, the restriction to binary trees induces the gatomorphism A057163.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 12.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as general trees, parenthesizations and Lukasiewicz-words.

                    
[()()(())][()(())()][(())()()]          [()(()())][(()())()]          [()()()()]
300103010031000          2020022000          40000

                    
[()((()))][((()))()]          [(()(()))][((())())]          [(())(())]
2011021100          1201012100          21010

                    
[(()()())]          [((()()))]          [(((())))]
13000          11200          11110


A057509/A057510


Description:
Shallow Rotation of general trees and parenthesizations

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
Yes, the restriction to binary trees induces the gatomorphism A069770.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 16.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as general trees, parenthesizations and Lukasiewicz-words.

                    
[()()(())][()(())()][(())()()]          [()(()())][(()())()]          [()()()()]
300103010031000          2020022000          40000

                              
[()((()))][((()))()]          [(())(())]          [(()()())]          [(()(()))]
2011021100          21010          13000          12010

                    
[((())())]          [((()()))]          [(((())))]
12100          11200          11110


A057164*


Description:
Deep Reverse of general trees and parenthesizations

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
Yes, the restriction to binary trees induces the gatomorphism A057163.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 164.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
[()()(())][(())()()]          [()(()())][(()())()]          [()((()))][((()))()]
3001031000          2020022000          2011021100

                              
                              
[(()(()))][((())())]          [()()()()]          [()(())()]          [(())(())]
1201012100          40000          30100          21010

                    
                    
[(()()())]          [((()()))]          [(((())))]
13000          11200          11110


A057508*


Description:
Shallow Reverse of general trees and parenthesizations

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
Yes, the restriction to binary trees induces the gatomorphism A069770.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 168.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
[()()(())][(())()()]          [()(()())][(()())()]          [()((()))][((()))()]
3001031000          2020022000          2011021100

                              
                              
[()()()()]          [()(())()]          [(())(())]          [(()()())]
40000          30100          21010          13000

                              
                              
[(()(()))]          [((())())]          [((()()))]          [(((())))]
12010          12100          11200          11110


A057501/A057502


Description:
Rotation of Non-crossing chords (handshake) arrangements; root rotation of the general trees

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 261.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

[()()(())][(()(()))][()(())()][((())())][(())()()][()(()())][((()()))][(()())()]
3001012010301001210031000202001120022000

          
          
[()((()))][(((())))][((()))()][(())(())]          [()()()()][(()()())]
20110111102110021010          4000013000


A057503/A057504


Description:
Gatomorphism A057503

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 2618.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

[()()()()][(((())))][((()))()][(())(())][()(()())][(()()())]
400001111021100210102020013000

          
          
          
          
[()()(())][((()()))][(()())()]          [()((()))][((())())][(())()()]
300101120022000          201101210031000

[()(())()][(()(()))]
3010012010


A057505/A057506


Description:
Donaghey's M

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 2614.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

[()()(())][((()()))][(())()()][()((()))][(()()())][((()))()]
300101120031000201101300021100

[()(())()][(()(()))][(()())()][(())(())][()(()())][((())())]
301001201022000210102020012100

[()()()()][(((())))]
4000011110


A071661/A071662


Description:
Donaghey's M^2

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

          
          
          
          
[()()(())][(())()()][(()()())]          [()(())()][(()())()][()(()())]
300103100013000          301002200020200

          
          
          
          
[()((()))][((()))()][((()()))]          [(())(())][((())())][(()(()))]
201102110011200          210101210012010

          
          
          
          
[()()()()]          [(((())))]
40000          11110


A071663/A071664


Description:
Donaghey's M^3

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
                    
                    
[()()()()][(((())))]          [()()(())][()((()))]          [()(())()][(())(())]
4000011110          3001020110          3010021010

                    
                    
                    
                    
[()(()())][(()(()))]          [(())()()][((()))()]          [(()())()][((())())]
2020012010          3100021100          2200012100

[(()()())][((()()))]
1300011200


A071665/A071666


Description:
Donaghey's M^4

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

          
          
          
          
[()()(())][(()()())][(())()()]          [()(())()][()(()())][(()())()]
300101300031000          301002020022000

          
          
          
          
[()((()))][((()()))][((()))()]          [(())(())][(()(()))][((())())]
201101120021100          210101201012100

          
          
          
          
[()()()()]          [(((())))]
40000          11110


A071667/A071668


Description:
Donaghey's M^5

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

[()()(())][((()))()][(()()())][()((()))][(())()()][((()()))]
300102110013000201103100011200

[()(())()][((())())][()(()())][(())(())][(()())()][(()(()))]
301001210020200210102200012010

[()()()()][(((())))]
4000011110


A071669/A071670


Description:
Donaghey's M^6

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                              
                              
                              
                              
[()()()()]          [()()(())]          [()(())()]          [()(()())]
40000          30010          30100          20200

                              
                              
                              
                              
[()((()))]          [(())()()]          [(())(())]          [(()())()]
20110          31000          21010          22000

                              
                              
                              
                              
[(()()())]          [(()(()))]          [((()))()]          [((())())]
13000          12010          21100          12100

          
          
          
          
[((()()))]          [(((())))]
11200          11110


A057161/A057162


Description:
Rotation of polygon triangulations

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 17517.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as binary trees, polygon triangulations, parenthesizations and Lukasiewicz-words.

[()()()()][(()()())][(()())()][(())(())][()((()))][(((())))]
400001300022000210102011011110

          
          
[()()(())][(()(()))][((()))()]          [()(()())][((()()))][(())()()]
300101201021100          202001120031000

[()(())()][((())())]
3010012100


A074679/A074680


Description:
Gatomorphism A074679: rotate binary tree left, if possible

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
for the first time as the row 557243.

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

[()()()()][(())()()][((()))()][(((())))][()((()))][(()(()))][()()(())][(())(())][((())())][()(())()][(()())()][((()()))][()(()())][(()()())]
4000031000211001111020110120103001021010121003010022000112002020013000


A071655/A071656


Description:
Gatomorphism A071655: If robr not possible, apply swap, otherwise rotate binary tree right and recurse down on both branches

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
???

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

[()()(())][(()(()))][()(())()][((())())][(())()()][()(()())][((()()))][((()))()][(())(())][()((()))][(((())))][(()())()]
300101201030100121003100020200112002110021010201101111022000

[()()()()][(()()())]
4000013000


A071659/A071660


Description:
Gatomorphism A071659: If robr not possible, apply swap, otherwise recurse down on both branches and after that rotate binary tree right

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
???

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

[()()()()][(()()())][(()())()][(())()()][()()(())][(()(()))][((()))()][()(())()][((())())][()(()())][((()()))][(())(())]
400001300022000310003001012010211003010012100202001120021010

[()((()))][(((())))]
2011011110


A069787*


Description:
The car/cdr-flipped conjugate of deep reverse

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
Yes, contraction gives the permutation A072799.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
                    
                    
[()(())()][()(()())]          [(())()()][(()()())]          [(())(())][(()(()))]
3010020200          3100013000          2101012010

                              
                              
                              
                              
[((()))()][((()()))]          [()()()()]          [()()(())]          [()((()))]
2110011200          40000          30010          20110

                    
                    
                    
                    
[(()())()]          [((())())]          [(((())))]
22000          12100          11110


A069769*


Description:
The car/cdr-flipped conjugate of shallow reverse

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
                    
                    
[(())()()][(()()())]          [(())(())][(()(()))]          [((()))()][((()()))]
3100013000          2101012010          2110011200

                              
                              
                              
                              
[()()()()]          [()()(())]          [()(())()]          [()(()())]
40000          30010          30100          20200

                              
                              
                              
                              
[()((()))]          [(()())()]          [((())())]          [(((())))]
20110          22000          12100          11110


A069888*


Description:
Reflect non-crossing chords through the corners 1 and 7 o'clock (WM)

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
                    
                    
[()()()()][(()()())]          [()()(())][()(()())]          [()(())()][((())())]
4000013000          3001020200          3010012100

                    
                    
                    
                    
[()((()))][(())(())]          [(())()()][(()(()))]          [(()())()][((()()))]
2011021010          3100012010          2200011200

[((()))()][(((())))]
2110011110


A069889*


Description:
Reflect non-crossing chords through the corners 11 and 5 o'clock

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
yes

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
                    
                    
[()()()()][(()()())]          [()()(())][((())())]          [()(())()][(()(()))]
4000013000          3001012100          3010012010

                    
                    
                    
                    
[()(()())][((()()))]          [()((()))][(((())))]          [(())()()][(()())()]
2020011200          2011011110          3100022000

[(())(())][((()))()]
2101021100


A069771*


Description:
Rotate non-crossing chords by 180 degrees

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
no?

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
                    
                    
[()()(())][(())()()]          [()(())()][((()()))]          [()(()())][(()(()))]
3001031000          3010011200          2020012010

                              
                              
                              
                              
[(()())()][((())())]          [()()()()]          [()((()))]          [(())(())]
2200012100          40000          20110          21010

                    
                    
                    
                    
[(()()())]          [((()))()]          [(((())))]
13000          21100          11110


A069772*


Description:
Reflect non-crossing chords by X-axis

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
No.

Occurs in A073200:
no?

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                    
                    
                    
                    
[()(())()][((()()))]          [()(()())][((())())]          [()((()))][((()))()]
3010011200          2020012100          2011021100

                              
                              
                              
                              
[(()())()][(()(()))]          [()()()()]          [()()(())]          [(())()()]
2200012010          40000          30010          31000

                    
                    
                    
                    
[(())(())]          [(()()())]          [(((())))]
21010          13000          11110


A072088/A072089


Description:
The breadth-first <-> depth-first conversion of general trees

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
Yes, the restriction to binary trees induces the gatomorphism A057117.

Telescoping:
Yes, contraction gives the permutation A072619.

Occurs in A073200:
no!?

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

                              
                              
                              
                              
[(())(())][((()))()]          [()()()()]          [()()(())]          [()(())()]
2101021100          40000          30010          30100

                              
                              
                              
                              
[()(()())]          [()((()))]          [(())()()]          [(()())()]
20200          20110          31000          22000

                              
                              
                              
                              
[(()()())]          [(()(()))]          [((())())]          [((()()))]
13000          12010          12100          11200

[(((())))]
11110


A057117/A057118


Description:
Wouter Meeussen's breadth-first <-> depth-first conversion of binary trees

Fixes:
some things
Counted by Affffff
Cycles correspond to:
some things
Counted by Acccccc
Max cycles lengths given by:
Ammmmmm
L-word permuting:
No.

Telescoping:
Yes, contraction gives the permutation A038776.

Occurs in A073200:
no!?

Compositions of:

Scheme functions implementing this gatomorphism on parenthesizations:


The effect of this gatomorphism on the forest Cat[4] viewed as polygon triangulations, binary trees, general trees, non-crossing chord arrangements, parenthesizations and Lukasiewicz-words.

          
          
          
          
[(())()()][(()()())][(((())))][((()))()]          [()(())()][()(()())][()((()))]
31000130001111021100          301002020020110

                    
                    
                    
                    
[(())(())][(()(()))][((())())]          [(()())()][((()()))]          [()()()()]
210101201012100          2200011200          40000

[()()(())]
30010