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Signature-permutation of Catalan bijection "Knack".
+10
32
0, 1, 3, 2, 8, 7, 6, 4, 5, 22, 21, 20, 17, 18, 19, 16, 14, 9, 10, 15, 11, 12, 13, 64, 63, 62, 58, 59, 61, 57, 54, 45, 46, 55, 48, 49, 50, 60, 56, 53, 44, 47, 51, 42, 37, 23, 24, 38, 25, 26, 27, 52, 43, 39, 28, 29, 40, 30, 31, 32, 41, 33, 34, 35, 36, 196, 195, 194, 189, 190
COMMENTS
This automorphism of binary trees first swaps the left and right subtree of the root and then proceeds recursively to the (new) left subtree, to do the same operation there. This is one of those Catalan bijections which extend to a unique automorphism of the infinite binary tree, which in this case is A153142. See further comments there and in A153141. This bijection, Knack, is a ENIPS-transformation of the simple swap: ENIPS(* A069770) (i.e., row 1 of A122204). Furthermore, Knack and Knick (the inverse, A069767) have a special property, that FORK and KROF transforms (explained in A122201 and A122202) transform them to their own inverses, i.e., to each other: FORK(Knick) = KROF(Knick) = Knack and FORK(Knack) = KROF(Knack) = Knick, thus this occurs also as row 1 in A122288 and naturally, the double-fork fixes both, e.g., FORK(FORK(Knack)) = Knack. Note: the name in Finnish is "Naks".
REFERENCES
A. Karttunen, paper in preparation.
PROG
(Scheme implementations of this automorphism. These act on S-expressions, i.e. list-structures:)
(CONSTRUCTIVE VERSION:) (define (* A069768 s) (cond ((not (pair? s)) s) (else (cons (* A069768 (cdr s)) (car s)))))
Signature permutations of SPINE-transformations of Catalan automorphisms in table A122204.
+10
27
0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 8, 3, 2, 1, 0, 6, 7, 4, 3, 2, 1, 0, 7, 6, 6, 5, 3, 2, 1, 0, 8, 5, 5, 4, 5, 3, 2, 1, 0, 9, 4, 7, 6, 6, 6, 3, 2, 1, 0, 10, 22, 8, 7, 4, 5, 6, 3, 2, 1, 0, 11, 21, 9, 8, 7, 4, 4, 4, 3, 2, 1, 0, 12, 20, 14, 13, 8, 7, 5, 5, 4, 3, 2, 1, 0, 13, 17, 11, 12, 13
COMMENTS
Row n is the signature permutation of the Catalan automorphism which is obtained from the n-th automorphism in the table A122204 with the recursion scheme "SPINE", or equivalently row n is obtained as SPINE(ENIPS(n-th row of A089840)). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A122285. This table contains also all the rows of A122203 and A089840.
REFERENCES
A. Karttunen, paper in preparation, draft available by e-mail.
CROSSREFS
The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A082347, 2: A057508, 3: A131142, 4: A131148, 5: A131146, 6: A131144, 7: A131173, 8: A131170, 9: A131154, 10: A131152, 11: A131150, 12: A057504, 13: A131164, 14: A131166, 15: A069767, 16: A131168, 17: A131172, 18: A131156, 19: A131158, 20: A131162, 21: A131160. Other rows: row 169: A130359, 3608: A130339, 3617: A057509, 65167: A074685.
Permutation of natural numbers induced by the Catalan bijection gma082345 acting on the parenthesizations encoded by A014486/ A063171.
+10
8
0, 1, 3, 2, 7, 8, 6, 4, 5, 17, 18, 20, 21, 22, 16, 19, 14, 9, 10, 15, 11, 13, 12, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 44, 47, 53, 56, 60, 42, 51, 37, 23, 24, 38, 25, 27, 26, 43, 52, 39, 28, 29, 41, 33, 35, 36, 40, 30, 34, 32, 31, 129, 130, 132, 133, 134
PROG
(Scheme functions implementing this automorphism on list-structures:)
(define (gma082345! s) (cond ((pair? s) (gma069767! s) (gma082345! (cdr s)))) s)
Permutation of natural numbers induced by the Catalan bijection gma082346 acting on the parenthesizations encoded by A014486/ A063171.
+10
7
0, 1, 3, 2, 7, 8, 6, 4, 5, 17, 18, 20, 22, 21, 16, 19, 14, 9, 10, 15, 11, 12, 13, 45, 46, 48, 50, 49, 54, 55, 61, 64, 63, 57, 62, 58, 59, 44, 47, 53, 60, 56, 42, 51, 37, 23, 24, 38, 25, 26, 27, 43, 52, 39, 28, 29, 40, 30, 31, 32, 41, 33, 34, 35, 36, 129, 130, 132, 134, 133
PROG
(Scheme functions implementing this automorphism on list-structures:)
(define (gma082346! s) (cond ((pair? s) (gma082346! (cdr s)) (gma069768! s))) s)
Permutation of natural numbers induced by the Catalan bijection gma082348 acting on the parenthesizations encoded by A014486/ A063171.
+10
5
0, 1, 3, 2, 8, 7, 6, 5, 4, 22, 21, 20, 18, 17, 19, 16, 15, 12, 13, 14, 11, 10, 9, 64, 63, 62, 59, 58, 61, 57, 55, 49, 50, 54, 48, 46, 45, 60, 56, 53, 47, 44, 52, 43, 40, 31, 32, 41, 34, 35, 36, 51, 42, 39, 30, 33, 38, 29, 26, 27, 37, 28, 25, 24, 23, 196, 195, 194, 190, 189
PROG
(Scheme functions implementing this automorphism on list-structures:)
(define (gma082348! s) (cond ((pair? s) (gma082348! (cdr s)) (gma069767! s))) s)
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