A Brief Survey on Gatomorphisms.

1. A069770 - Swap Binary Tree Sides
2. A072796 - Exchange Two Leftmost Branches on General Trees
3. A057163 - Reflect of binary trees and polygon triangulations
4. A069767/A069768 - Recursive Swap of the other side of binary tree, including the root node
5. A073286/A073287 - Recursive Swap of the other side of binary tree, but leaving the root node intact
6. A073288/A073289 - "Gatomorphism A073288/A073289" (recursive application of the above one)
7. A057511/A057512 - Deep Rotation of general trees and parenthesizations
8. A057509/A057510 - Shallow Rotation of general trees and parenthesizations
9. A057164 - Deep Reverse of general trees and parenthesizations
10. A057508 - Shallow Reverse of general trees and parenthesizations
11. A057501/A057502 - Rotation of Non-crossing chords (handshake) arrangements; root rotation of the general trees
12. A057503/A057504 - Gatomorphism A057503/A057504
13. A057505/A057506 - Donaghey's M
14. A071661/A071662 - Donaghey's M²
15. A071663/A071664 - Donaghey's M³
16. A071665/A071666 - Donaghey's M⁴
17. A071667/A071668 - Donaghey's M⁵
18. A071669/A071670 - Donaghey's M⁶
19. A057161/A057162 - Rotation of polygon triangulations
20. A074679/A074680 - Gatomorphism A074679: rotate binary tree left, if possible, otherwise swap binary tree sides
21. A071655/A071656 - Gatomorphism A071655: If robr not possible, apply swap, otherwise rotate binary tree right and recurse down on both branches
22. A071659/A071660 - Gatomorphism A071659: If robr not possible, apply swap, otherwise recurse down on both branches and after that rotate binary tree right
23. A074681/A074682 - Gatomorphism A074681/A074682
24. A074683/A074684 - Gatomorphism A074683/A074684
25. A069787 - The car/cdr-flipped conjugate of deep reverse
26. A069769 - The car/cdr-flipped conjugate of shallow reverse
27. A069888 - Reflect non-crossing chords by the axis through corners (WM)
28. A069889 - Reflect non-crossing chords by the axis through corners
29. A069771 - Rotate non-crossing chords by 180 degrees
30. A069772 - Reflect non-crossing chords by X-axis
31. A072088/A072089 - The breadth-first <-> depth-first conversion of general trees
32. A057117/A057118 - Meeussen's breadth-first <-> depth-first conversion of binary trees

A069770

Description:

Swap Binary Tree Sides

Fixes:

- binary trees whose both sides are identical.

Counted by 1,1,0,1,0,2,0,5,0,14,0,42,... [aerated Catalans, A000108]

Cycles correspond to:

- rooted planar binary trees up to left-right swap

- necklaces of n+1 white beads and n-1 black beads. [Correspondence with above requires Raney's lemma.]

Counted by A007595: 1,1,1,3,7,22,66,217,715,2438,8398,...

Max cycle lengths and L.C.M.s of all cycle lengths:

Given by 1,1,2,2,2,2,2,2,2,2,2,... [because involution.]

L-word permuting:

No.

Telescoping:

No.

Occurs in A073200:

for the first time as the row 0. [by definition.]

Scheme functions implementing this gatomorphism on parenthesizations:

The constructive variant:

(define (gmA069770 s)
  (cond ((not (pair? s)) s)
        (else (cons (cdr s) (car s)))
  )
)

The destructive variant:

(define (gmA069770! s)
  (let ((ex-car (car s)))
     (set-car! s (cdr s))
     (set-cdr! s ex-car)
     s
  )
)

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

[()()()()]	[(()()())]		[()()(())]	[(()(()))]		[()(())()]	[((())())]
40000	13000		30010	12010		30100	12100
[()(()())]	[((()()))]		[()((()))]	[(((())))]		[(())()()]	[(()())()]
20200	11200		20110	11110		31000	22000
[(())(())]	[((()))()]
21010	21100

A072796

Description:

Exchange the two leftmost branches of general trees if the root degree > 1, otherwise keep the tree intact.

Fixes:

General plane trees which are either empty, or whose root degree is either 1 or the two leftmost branches of the root are identical.
Counted by A073190: 1,1,2,3,8,20,60,181,584,1916,6476,...

Cycles correspond to:

- ???

Counted by A073191: 1,1,2,4,11,31,96,305,1007,3389,11636,...

Max cycle lengths and L.C.M.s of all cycle lengths:

Given by 1,1,2,2,2,2,2,2,2,2,2,... [because involution.]

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 constructive variant:

(define (gmA072796 s)
  (cond ((not (pair? s)) s)
        ((not (pair? (cdr s))) s)
        (else (cons (cadr s) (cons (car s) (cddr s))))
  )
)

The destructive variant:

(define (gmA072796! s)
  (cond ((not (pair? s)) s)
        ((not (pair? (cdr s))) s)
        (else (swap! s)
              (robr! s)
              (swap! (cdr s))
              s
        )
  )
)

The effect of this gatomorphism on the forest Cat[4] viewed as general trees, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/ \/\	/\ / \/\/\		/\/\ /\/ \	/\/\ / \/\		/\ / \ /\/ \	/\ / \ / \/\
[()(())()]	[(())()()]		[()(()())]	[(()())()]		[()((()))]	[((()))()]
30100	31000		20200	22000		20110	21100
/\/\/\/\		/\ /\/\/ \		/\ /\ / \/ \		/\/\/\ / \
[()()()()]		[()()(())]		[(())(())]		[(()()())]
40000		30010		21010		13000
/\ /\/ \ / \		/\ / \/\ / \		/\/\ / \ / \		/\ / \ / \ / \
[(()(()))]		[((())())]		[((()()))]		[(((())))]
12010		12100		11200		11110

A057163

Description:

Reflect binary trees and polygon triangulations

Fixes:

- binary trees whose left and right side are mirror images of each other

Counted by 1,1,0,1,0,2,0,5,0,14,0,42,... [aerated Catalans, A000108]

Cycles correspond to:

- rooted planar binary trees up to reflection

- necklaces of n+1 white beads and n-1 black beads. [Correspondence with above requires Raney's lemma.]

Counted by A007595: 1,1,1,3,7,22,66,217,715,2438,8398,...

Max cycle lengths and L.C.M.s of all cycle lengths:

Given by 1,1,2,2,2,2,2,2,2,2,2,... [because involution.]

L-word permuting:

No.

Telescoping:

No.

Occurs in A073200:

for the first time as the row 2.

Scheme functions implementing this gatomorphism on parenthesizations:

The constructive variant:

(define (gmA057163 s)
  (cond ((not (pair? s)) s)
        (else (cons (gmA057163 (cdr s))
                    (gmA057163 (car s)))
        )
  )
)

The destructive variant:

(define (gmA057163! s)
  (cond ((not (pair? s)))
        (else
              (swap! s)
              (gmA057163! (car s))
              (gmA057163! (cdr s))
        )
  )
  s
)

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

[()()()()]	[(((())))]		[()()(())]	[((()()))]		[()(())()]	[((())())]
40000	11110		30010	11200		30100	12100
[()(()())]	[(()(()))]		[()((()))]	[(()()())]		[(())()()]	[((()))()]
20200	12010		20110	13000		31000	21100
[(())(())]	[(()())()]
21010	22000

A069767/A069768

Description:

Recursive swap of the other side of binary tree.

Fixes:

- "complete" binary trees [with leaves all on the same level].
Counted by A036987: 1,1,0,1,0,0,0,1,0,0,0,0,0,0,0,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,...

Cycles correspond to:

- ???
Counted by A073431: 1,1,1,2,3,6,12,28,65,160,408,...

Max cycle lengths and L.C.M.s of all cycle lengths:

Counted by A011782: 1,1,2,4,8,16,32,64,128,256,512,...

L-word permuting:

No.

Telescoping:

No.

Occurs in A073200:

for the first time as the row 6.

Scheme functions implementing this gatomorphism on parenthesizations:

The destructive variants:

(define (gmA069767! s) (cond ((not (pair? s))) (else (swap! s) (gmA069767! (cdr s)) ) ) s )

(define (gmA069768! s) (cond ((not (pair? s))) (else (swap! s) (gmA069768! (car s)) ) ) s )

Notes:

In each forest of Cat[n] binary trees of n internal (branching nodes), there is a subset of 2^n-1 binary trees whose height (i.e. max depth) is equal to their size. This gatomorphism keeps that subset closed, and furthermore, it acts transitively on it, i.e. those trees form a single cycle of their own, as can be seen below. If we let the root node stand for the least significant bit, and the next-to-top node on those trees stand the most significant bit, and mark 0 when the next node upwards is at the right, and 1 when it is at left, we get the sequence of binary words (in this case, of three bits) shown below on the top of the eight binary trees belonging to that closed cycle.
It is easy to see that this gatomorphism induces the well-known wrap-around binary increment/decrement algorithm on the binary strings that are in bijective correspondence with such binary trees.

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

000	001	010	011	100	101	110	111
[()()()()]	[(()()())]	[()(()())]	[((()()))]	[()()(())]	[(()(()))]	[()((()))]	[(((())))]
40000	13000	20200	11200	30010	12010	20110	11110
[(())()()]	[(()())()]	[(())(())]	[((()))()]		[()(())()]	[((())())]
31000	22000	21010	21100		30100	12100

A073286/A073287

Description:

Recursive Swap of the other side of binary tree, but leaving the root node intact

Fixes:

some things
Counted by Affffff 1,1,2,3,8,20,58,179,576,1902,6426,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,4,10,26,74,222,698,2264,7536,...

Max cycle lengths:

Ammmmmm 1,1,1,2,4,8,16,32,64,128,256,...

L.C.M.s of cycles lengths:

Allllll 1,1,1,2,4,8,16,32,64,128,256,...

L-word permuting:

No.

Telescoping:

No.

Occurs in A073200:

for the first time as the row 41.

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.

[(()()())]	[((()()))]	[(()(()))]	[(((())))]		[(()())()]	[((()))()]
13000	11200	12010	11110		22000	21100
[()()()()]		[()()(())]		[()(())()]		[()(()())]
40000		30010		30100		20200
[()((()))]		[(())()()]		[(())(())]		[((())())]
20110		31000		21010		12100

A073288/A073289

Description:

Gatomorphism A073288/A073289

Fixes:

some things
Counted by Affffff 1,1,2,3,6,10,18,31,56,98,174,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,4,9,20,47,112,279,712,1868,...

Max cycle lengths:

Ammmmmm 1,1,1,2,4,8,16,32,64,128,256,...

L.C.M.s of cycles lengths:

Allllll 1,1,1,2,4,8,16,32,64,128,256,...

L-word permuting:

No.

Telescoping:

No.

Occurs in A073200:

for the first time as the row 416.

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.

[(()()())]	[((()()))]	[(()(()))]	[(((())))]		[()(()())]	[()((()))]
13000	11200	12010	11110		20200	20110
[(()())()]	[((()))()]		[()()()()]		[()()(())]		[()(())()]
22000	21100		40000		30010		30100
[(())()()]		[(())(())]		[((())())]
31000		21010		12100

A057511/A057512

Description:

Deep Rotation of general trees and parenthesizations

Fixes:

some things
Counted by Affffff 1,1,2,3,5,6,10,11,18,21,34,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,4,9,21,56,153,451,1357,4212,...

Max cycle lengths given by:

Ammmmmm 1,1,1,2,3,4,6,6,12,15,20,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,1,2,6,12,60,60,420,840,2520,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\ /\/ \/\	/\ / \/\/\		/\/\ /\/ \	/\/\ / \/\		/\/\/\/\
[()()(())]	[()(())()]	[(())()()]		[()(()())]	[(()())()]		[()()()()]
30010	30100	31000		20200	22000		40000
/\ / \ /\/ \	/\ / \ / \/\		/\ /\/ \ / \	/\ / \/\ / \		/\ /\ / \/ \
[()((()))]	[((()))()]		[(()(()))]	[((())())]		[(())(())]
20110	21100		12010	12100		21010
/\/\/\ / \		/\/\ / \ / \		/\ / \ / \ / \
[(()()())]		[((()()))]		[(((())))]
13000		11200		11110

A057509/A057510

Description:

Shallow Rotation of general trees and parenthesizations

Fixes:

some things
Counted by Affffff 1,1,2,3,7,15,46,133,436,1433,4878,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,4,10,26,80,246,810,2704,9252,...

Max cycle lengths given by:

Ammmmmm 1,1,1,2,3,4,5,6,7,8,9,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,1,2,6,12,60,60,420,840,2520,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\ /\/ \/\	/\ / \/\/\		/\/\ /\/ \	/\/\ / \/\		/\/\/\/\
[()()(())]	[()(())()]	[(())()()]		[()(()())]	[(()())()]		[()()()()]
30010	30100	31000		20200	22000		40000
/\ / \ /\/ \	/\ / \ / \/\		/\ /\ / \/ \		/\/\/\ / \		/\ /\/ \ / \
[()((()))]	[((()))()]		[(())(())]		[(()()())]		[(()(()))]
20110	21100		21010		13000		12010
/\ / \/\ / \		/\/\ / \ / \		/\ / \ / \ / \
[((())())]		[((()()))]		[(((())))]
12100		11200		11110

A057164

Description:

Deep Reverse of general trees and parenthesizations

Fixes:

some things
Counted by Affffff 1,1,2,3,6,10,20,35,70,126,252,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,4,10,26,76,232,750,2494,8524,...

Max cycle lengths given by:

Ammmmmm 1,1,1,2,2,2,2,2,2,2,2,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,1,2,2,2,2,2,2,2,2,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\ / \/\/\		/\/\ /\/ \	/\/\ / \/\		/\ / \ /\/ \	/\ / \ / \/\
[()()(())]	[(())()()]		[()(()())]	[(()())()]		[()((()))]	[((()))()]
30010	31000		20200	22000		20110	21100
/\ /\/ \ / \	/\ / \/\ / \		/\/\/\/\		/\ /\/ \/\		/\ /\ / \/ \
[(()(()))]	[((())())]		[()()()()]		[()(())()]		[(())(())]
12010	12100		40000		30100		21010
/\/\/\ / \		/\/\ / \ / \		/\ / \ / \ / \
[(()()())]		[((()()))]		[(((())))]
13000		11200		11110

A057508

Description:

Shallow Reverse of general trees and parenthesizations

Fixes:

some things
Counted by Affffff 1,1,2,3,8,18,54,155,500,1614,5456,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,4,11,30,93,292,965,3238,11126,...

Max cycle lengths given by:

Ammmmmm 1,1,1,2,2,2,2,2,2,2,2,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,1,2,2,2,2,2,2,2,2,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\ / \/\/\		/\/\ /\/ \	/\/\ / \/\		/\ / \ /\/ \	/\ / \ / \/\
[()()(())]	[(())()()]		[()(()())]	[(()())()]		[()((()))]	[((()))()]
30010	31000		20200	22000		20110	21100
/\/\/\/\		/\ /\/ \/\		/\ /\ / \/ \		/\/\/\ / \
[()()()()]		[()(())()]		[(())(())]		[(()()())]
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 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,2,3,6,14,34,95,280,854,...

Max cycle lengths given by:

Ammmmmm 1,1,2,3,8,10,12,14,16,18,20,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,6,8,10,12,14,16,18,20,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\ /\/ \ / \	/\ /\/ \/\	/\ / \/\ / \	/\ / \/\/\	/\/\ /\/ \	/\/\ / \ / \	/\/\ / \/\
[()()(())]	[(()(()))]	[()(())()]	[((())())]	[(())()()]	[()(()())]	[((()()))]	[(()())()]
30010	12010	30100	12100	31000	20200	11200	22000
/\ / \ /\/ \	/\ / \ / \ / \	/\ / \ / \/\	/\ /\ / \/ \		/\/\/\/\	/\/\/\ / \
[()((()))]	[(((())))]	[((()))()]	[(())(())]		[()()()()]	[(()()())]
20110	11110	21100	21010		40000	13000

A057503/A057504

Description:

Gatomorphism A057503/A057504

Fixes:

some things
Counted by Affffff 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,1,4,6,19,49,150,442,1424,...

Max cycle lengths given by:

Ammmmmm 1,1,2,5,6,7,8,9,10,11,12,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,5,6,7,8,9,10,11,12,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\/\/\/\	/\ / \ / \ / \	/\ / \ / \/\	/\ /\ / \/ \	/\/\ /\/ \	/\/\/\ / \
[()()()()]	[(((())))]	[((()))()]	[(())(())]	[()(()())]	[(()()())]
40000	11110	21100	21010	20200	13000
/\ /\/\/ \	/\/\ / \ / \	/\/\ / \/\		/\ / \ /\/ \	/\ / \/\ / \	/\ / \/\/\
[()()(())]	[((()()))]	[(()())()]		[()((()))]	[((())())]	[(())()()]
30010	11200	22000		20110	12100	31000
/\ /\/ \/\	/\ /\/ \ / \
[()(())()]	[(()(()))]
30100	12010

A057505/A057506

Description:

Donaghey's M

Fixes:

some things
Counted by Affffff 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,2,3,10,18,46,95,236,528,...

Max cycle lengths given by:

Ammmmmm 1,1,2,3,6,6,24,72,144,147,588,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,6,6,30,120,720,15120,1164240,15135120,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\/\ / \ / \	/\ / \/\/\	/\ / \ /\/ \	/\/\/\ / \	/\ / \ / \/\
[()()(())]	[((()()))]	[(())()()]	[()((()))]	[(()()())]	[((()))()]
30010	11200	31000	20110	13000	21100
/\ /\/ \/\	/\ /\/ \ / \	/\/\ / \/\	/\ /\ / \/ \	/\/\ /\/ \	/\ / \/\ / \
[()(())()]	[(()(()))]	[(()())()]	[(())(())]	[()(()())]	[((())())]
30100	12010	22000	21010	20200	12100
/\/\/\/\	/\ / \ / \ / \
[()()()()]	[(((())))]
40000	11110

A071661/A071662

Description:

Donaghey's M²

Fixes:

some things
Counted by Affffff 1,1,2,2,2,4,4,4,6,6,6,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,3,6,16,36,83,190,448,1056,...

Max cycle lengths given by:

Ammmmmm 1,1,1,3,3,5,12,36,72,147,294,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,1,3,3,15,60,360,7560,582120,7567560,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\ / \/\/\	/\/\/\ / \		/\ /\/ \/\	/\/\ / \/\	/\/\ /\/ \
[()()(())]	[(())()()]	[(()()())]		[()(())()]	[(()())()]	[()(()())]
30010	31000	13000		30100	22000	20200
/\ / \ /\/ \	/\ / \ / \/\	/\/\ / \ / \		/\ /\ / \/ \	/\ / \/\ / \	/\ /\/ \ / \
[()((()))]	[((()))()]	[((()()))]		[(())(())]	[((())())]	[(()(()))]
20110	21100	11200		21010	12100	12010
/\/\/\/\		/\ / \ / \ / \
[()()()()]		[(((())))]
40000		11110

A071663/A071664

Description:

Donaghey's M³

Fixes:

some things
Counted by Affffff 1,1,0,3,0,9,0,21,0,45,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,4,7,24,48,128,259,646,1426,...

Max cycle lengths given by:

Ammmmmm 1,1,2,2,2,5,20,24,48,49,196,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,2,2,10,40,240,5040,388080,5045040,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\/\/\/\	/\ / \ / \ / \		/\ /\/\/ \	/\ / \ /\/ \		/\ /\/ \/\	/\ /\ / \/ \
[()()()()]	[(((())))]		[()()(())]	[()((()))]		[()(())()]	[(())(())]
40000	11110		30010	20110		30100	21010
/\/\ /\/ \	/\ /\/ \ / \		/\ / \/\/\	/\ / \ / \/\		/\/\ / \/\	/\ / \/\ / \
[()(()())]	[(()(()))]		[(())()()]	[((()))()]		[(()())()]	[((())())]
20200	12010		31000	21100		22000	12100
/\/\/\ / \	/\/\ / \ / \
[(()()())]	[((()()))]
13000	11200

A071665/A071666

Description:

Donaghey's M⁴

Fixes:

some things
Counted by Affffff 1,1,2,2,2,4,4,4,10,6,10,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,3,6,16,40,93,226,540,1336,...

Max cycle lengths given by:

Ammmmmm 1,1,1,3,3,5,6,18,45,147,147,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,1,3,3,15,30,180,3780,291060,3783780,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\/\/\ / \	/\ / \/\/\		/\ /\/ \/\	/\/\ /\/ \	/\/\ / \/\
[()()(())]	[(()()())]	[(())()()]		[()(())()]	[()(()())]	[(()())()]
30010	13000	31000		30100	20200	22000
/\ / \ /\/ \	/\/\ / \ / \	/\ / \ / \/\		/\ /\ / \/ \	/\ /\/ \ / \	/\ / \/\ / \
[()((()))]	[((()()))]	[((()))()]		[(())(())]	[(()(()))]	[((())())]
20110	11200	21100		21010	12010	12100
/\/\/\/\		/\ / \ / \ / \
[()()()()]		[(((())))]
40000		11110

A071667/A071668

Description:

Donaghey's M⁵

Fixes:

some things
Counted by Affffff 1,1,0,0,0,5,0,0,0,5,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,2,3,14,22,62,127,320,756,...

Max cycle lengths given by:

Ammmmmm 1,1,2,3,6,6,24,72,144,147,588,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,6,6,6,24,144,3024,232848,3027024,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\ / \ / \/\	/\/\/\ / \	/\ / \ /\/ \	/\ / \/\/\	/\/\ / \ / \
[()()(())]	[((()))()]	[(()()())]	[()((()))]	[(())()()]	[((()()))]
30010	21100	13000	20110	31000	11200
/\ /\/ \/\	/\ / \/\ / \	/\/\ /\/ \	/\ /\ / \/ \	/\/\ / \/\	/\ /\/ \ / \
[()(())()]	[((())())]	[()(()())]	[(())(())]	[(()())()]	[(()(()))]
30100	12100	20200	21010	22000	12010
/\/\/\/\	/\ / \ / \ / \
[()()()()]	[(((())))]
40000	11110

A071669/A071670

Description:

Donaghey's M⁶

Fixes:

some things
Counted by Affffff 1,1,2,5,14,37,88,193,402,813,1620,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,2,5,14,38,96,229,518,1222,2852,...

Max cycle lengths given by:

Ammmmmm 1,1,1,1,1,5,10,12,24,49,98,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,1,1,1,5,20,120,2520,194040,2522520,...

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, Dyck paths (mountain ranges), 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 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,1,4,6,19,49,150,442,1424,...

Max cycle lengths given by:

Ammmmmm 1,1,2,5,6,7,8,9,10,11,12,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,5,6,7,8,9,10,11,12,...

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.

[()()()()]	[(()()())]	[(()())()]	[(())(())]	[()((()))]	[(((())))]
40000	13000	22000	21010	20110	11110
[()()(())]	[(()(()))]	[((()))()]		[()(()())]	[((()()))]	[(())()()]
30010	12010	21100		20200	11200	31000
[()(())()]	[((())())]
30100	12100

A074679/A074680

Description:

Gatomorphism A074679: rotate binary tree left, if possible, otherwise swap binary tree sides

Fixes:

some things
Counted by Affffff 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,1,1,4,6,19,49,150,442,...

Max cycle lengths given by:

Ammmmmm 1,1,2,5,14,18,22,26,30,34,38,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,5,14,18,22,26,30,34,38,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\/\/\/\

/\ / \/\/\

/\ / \ / \/\

/\ / \ / \ / \

/\ / \ /\/ \

/\ /\/ \ / \

/\ /\/\/ \

/\ /\ / \/ \

/\ / \/\ / \

/\ /\/ \/\

/\/\ / \/\

/\/\ / \ / \

/\/\ /\/ \

/\/\/\ / \

[()()()()]

[(())()()]

[((()))()]

[(((())))]

[()((()))]

[(()(()))]

[()()(())]

[(())(())]

[((())())]

[()(())()]

[(()())()]

[((()()))]

[()(()())]

[(()()())]

40000

31000

21100

11110

20110

12010

30010

21010

12100

30100

22000

11200

20200

13000

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 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,2,2,4,4,5,5,11,11,...

Max cycle lengths given by:

Ammmmmm 1,1,2,3,12,29,97,378,775,2940,11368,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,6,12,870,13580,10962,20425900,1127689920,15057062823942180161400,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\ /\/\/ \	/\ /\/ \ / \	/\ /\/ \/\	/\ / \/\ / \	/\ / \/\/\	/\/\ /\/ \	/\/\ / \ / \	/\ / \ / \/\	/\ /\ / \/ \	/\ / \ /\/ \	/\ / \ / \ / \	/\/\ / \/\
[()()(())]	[(()(()))]	[()(())()]	[((())())]	[(())()()]	[()(()())]	[((()()))]	[((()))()]	[(())(())]	[()((()))]	[(((())))]	[(()())()]
30010	12010	30100	12100	31000	20200	11200	21100	21010	20110	11110	22000
/\/\/\/\	/\/\/\ / \
[()()()()]	[(()()())]
40000	13000

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 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,2,2,4,4,5,5,11,11,...

Max cycle lengths given by:

Ammmmmm 1,1,2,3,12,29,97,378,775,2940,11368,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,6,12,870,13580,10962,20425900,1127689920,15057062823942180161400,...

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\/\/\/\	/\/\/\ / \	/\/\ / \/\	/\ / \/\/\	/\ /\/\/ \	/\ /\/ \ / \	/\ / \ / \/\	/\ /\/ \/\	/\ / \/\ / \	/\/\ /\/ \	/\/\ / \ / \	/\ /\ / \/ \
[()()()()]	[(()()())]	[(()())()]	[(())()()]	[()()(())]	[(()(()))]	[((()))()]	[()(())()]	[((())())]	[()(()())]	[((()()))]	[(())(())]
40000	13000	22000	31000	30010	12010	21100	30100	12100	20200	11200	21010
/\ / \ /\/ \	/\ / \ / \ / \
[()((()))]	[(((())))]
20110	11110

A074681/A074682

Description:

Gatomorphism A074681/A074682

Fixes:

some things
Counted by Affffff 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,1,3,4,4,11,9,6,8,...

Max cycle lengths given by:

Ammmmmm 1,1,2,5,9,28,57,253,842,3753,10927,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,5,18,84,2793,211123440,140826255570,213340617315,156232599082560,...

L-word permuting:

No.

Telescoping:

No.

Occurs in A073200:

for the first time as the row 5572432.

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\/\/\/\	/\ /\ / \/ \	/\/\ / \ / \	/\ /\/\/ \	/\ / \/\/\	/\/\ / \/\	/\ /\/ \ / \	/\ / \ /\/ \	/\ / \ / \ / \
[()()()()]	[(())(())]	[((()()))]	[()()(())]	[(())()()]	[(()())()]	[(()(()))]	[()((()))]	[(((())))]
40000	21010	11200	30010	31000	22000	12010	20110	11110
/\ /\/ \/\	/\ / \ / \/\	/\/\/\ / \		/\/\ /\/ \	/\ / \/\ / \
[()(())()]	[((()))()]	[(()()())]		[()(()())]	[((())())]
30100	21100	13000		20200	12100

A074683/A074684

Description:

Gatomorphism A074683/A074684

Fixes:

some things
Counted by Affffff 1,1,0,0,0,0,0,0,0,0,0,...

Cycles correspond to:

some things
Counted by Acccccc 1,1,1,1,3,4,4,11,9,6,8,...

Max cycle lengths given by:

Ammmmmm 1,1,2,5,9,28,57,253,842,3753,10927,...

L.C.M.s of cycles lengths given by:

Allllll 1,1,2,5,18,84,2793,211123440,140826255570,213340617315,156232599082560,...

L-word permuting:

No.

Telescoping:

No.

Occurs in A073200:

for the first time as the row 5572434.

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, Dyck paths (mountain ranges), parenthesizations and Lukasiewicz-words.

/\/\/\/\	/\/\/\ / \	/\/\ /\/ \	/\ /\ / \/ \	/\ / \ / \/\	/\/\ / \ / \	/\ /\/\/ \	/\/\ / \/\	/\ / \ / \ / \
[()()()()]	[(()()())]	[()(()())]	[(())(())]	[((()))()]	[((()()))]	[()()(())]	[(()())()]	[(((())))]
40000	13000	20200	21010	21100	11200	30010	22000	11110
/\ / \ /\/ \	/\ / \/\/\	/\ / \/\ / \		/\ /\/ \/\	/\ /\/ \ / \
[()((()))]	[(())()()]	[((())())]		[()(())()]	[(()(()))]
20110	31000	12100		30100	12010