22 PRENUMBERED NEW SEQUENCES: A123492-A123503, A123694-A123696, A123713-A123719 FOLLOW:
%I A123492
%S A123492 0,1,2,3,4,8,6,7,5,9,10,20,21,22,14,19,16,17,18,15,11,12,13,23,24,25,
%T A123492 26,27,54,55,57,58,59,61,62,63,64,37,38,53,56,60,42,51,44,45,46,47,48,
%U A123492 49,50,43,52,39,28,29,40,30,31,32,41,33,34,35,36,65,66,67,68,69,70,71
%N A123492 An involution of non-negative integers: signature permutation of a nonrecursive Catalan automorphism which swaps the sides of a binary tree if the left subtree of either the left or right hand side toplevel subtree is not empty, and otherwise keeps the binary tree intact.
%C A123492 This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C and D refer to arbitrary subtrees located on those nodes.)
%C A123492 .....B...C.....B...C...........A...B.............A...B...
%C A123492 ......\./.......\./.............\./...............\./....
%C A123492 .......x...D.....x...D...........x...C.............x...C.
%C A123492 ........\./.......\./.............\./...............\./..
%C A123492 .....A...x...-->...x...A...........x...D...-->...D...x...
%C A123492 ......\./...........\./.............\./...........\./....
%C A123492 .......x.............x...............x.............x.....
%H A123492 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123492 Index entries for signature-permutations of Catalan automorphisms
%Y A123492 Row 79361 of A089840. Used to construct A123493, A123494, A123715 and A123716. Cf. A069770.
%K A123492 nonn
%O A123492 0,3
%A A123492 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123492 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123492! s) (cond ((null? s) s) ((and (pair? (cdr s)) (pair? (cadr s))) (*A069770! s)) ((and (pair? (car s)) (pair? (caar s))) (*A069770! s))) s)
%I A123493
%S A123493 0,1,2,3,4,8,6,7,5,9,13,20,21,18,14,19,16,17,22,15,11,12,10,23,27,34,
%T A123493 35,32,54,61,57,58,64,55,48,49,46,37,41,53,56,47,42,51,44,45,50,60,62,
%U A123493 63,59,43,52,39,28,33,40,30,31,36,38,29,25,26,24,65,69,76,77,74,96
%N A123493 Signature permutation of a Catalan automorphism: Row 79361 of table A122201.
%C A123493 This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123492 with the recursion schema FORK (defined in A122201). See further comments at A123494.
%H A123493 Index entries for signature-permutations of Catalan automorphisms
%Y A123493 Inverse: A123494. Row 79361 of A122201.
%K A123493 nonn
%O A123493 0,3
%A A123493 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123493 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define *A123493! (!FORK *A123492!))
%I A123494
%S A123494 0,1,2,3,4,8,6,7,5,9,22,20,21,10,14,19,16,17,13,15,11,12,18,23,64,62,
%T A123494 63,24,54,61,57,58,27,55,25,26,59,37,60,53,56,38,42,51,44,45,36,41,34,
%U A123494 35,46,43,52,39,28,33,40,30,31,50,47,29,48,49,32,65,196,194,195,66
%N A123494 Signature permutation of a Catalan automorphism: Row 79361 of table A122202.
%C A123494 This is the signature-permutation of Catalan automorphism which is derived from the automorphism *A123492 with the recursion schema KROF (defined in A122202). Like automorphisms *A057163 and *A069767/*A069768 these automorphisms are closed with respect to the subset of "zigzagging" binary trees (i.e. those binary trees where there are no nodes with two non-empty branches, or equivalently, those ones for which Stanley's interpretation (c) forms a non-branching line), and thus induce a permutation of binary strings. That is, starting from the root of such a binary tree, the turns taken by non-empty branches are interpreted as binary digits 0 or 1, depending on whether the tree grows to the left or right. In this manner, the Catalan automorphisms *A123494 and *A123493 induce the Binary Reflected Gray Code (see A003188 and A006068).
%H A123494 Index entries for signature-permutations of Catalan automorphisms
%Y A123494 Inverse: A123493. Row 79361 of A122202. See also A123715 and A123716.
%K A123494 nonn
%O A123494 0,3
%A A123494 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123494 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define *A123494! (!KROF *A123492!))
%I A123495
%S A123495 0,1,3,2,6,7,8,4,5,14,15,16,17,18,21,22,19,9,10,20,11,12,13,37,38,39,
%T A123495 40,41,42,43,44,45,46,47,48,49,50,58,59,62,63,64,56,60,51,23,24,52,25,
%U A123495 26,27,57,61,53,28,29,54,30,31,32,55,33,34,35,36,107,108,109,110,111
%N A123495 Signature permutation of a nonrecursive Catalan automorphism: Row 65518 of table A089840.
%H A123495 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123495 Index entries for signature-permutations of Catalan automorphisms
%Y A123495 Inverse: A123496. a(n) = A082351(A069770(n)). Row 65518 of A089840. Used to construct automorphism *A082357. Cf. A069770 and A074679.
%K A123495 nonn
%O A123495 0,3
%A A123495 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123495 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123495! s) (cond ((null? s) s) ((and (pair? (car s)) (pair? (cdr s))) (*A069770! s) (*A074679! s)) (else (*A074679! s))) s)
%I A123496
%S A123496 0,1,3,2,7,8,4,5,6,17,18,20,21,22,9,10,11,12,13,16,19,14,15,45,46,48,
%T A123496 49,50,54,55,57,58,59,61,62,63,64,23,24,25,26,27,28,29,30,31,32,33,34,
%U A123496 35,36,44,47,53,56,60,42,51,37,38,43,52,39,40,41,129,130,132,133,134
%N A123496 Signature permutation of a nonrecursive Catalan automorphism: Row 65796 of table A089840.
%H A123496 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123496 Index entries for signature-permutations of Catalan automorphisms
%Y A123496 Inverse: A123495. a(n) = A069770(A082352(n)). Row 65796 of A089840. Used to construct automorphism *A082358. Cf. A069770 and A074680.
%K A123496 nonn
%O A123496 0,3
%A A123496 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123496 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123496! s) (cond ((null? s) s) ((and (pair? (car s)) (pair? (caar s))) (*A074680! s) (*A069770! s)) (else (*A074680! s))) s)
%I A123497
%S A123497 0,1,2,3,4,6,7,5,8,9,10,14,16,19,17,18,12,11,13,20,15,21,22,23,24,25,
%T A123497 26,27,37,38,42,44,47,51,53,56,60,45,46,48,49,50,31,32,30,28,29,34,33,
%U A123497 35,36,54,55,40,39,41,57,43,58,59,61,52,62,63,64,65,66,67,68,69,70,71
%N A123497 Signature permutation of a nonrecursive Catalan automorphism: Row 1655089 of table A089840.
%H A123497 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123497 Index entries for signature-permutations of Catalan automorphisms
%Y A123497 Inverse: A123498. Row 1655089 of A089840. Used to construct automorphism *A123501. A074680(n) = A083927(a(A057123(n))).
%K A123497 nonn
%O A123497 0,3
%A A123497 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123497 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123497! s) (cond ((null? s) s) ((and (pair? (car s)) (pair? (cdar s))) (*A074680! s) (let ((old-cddr-s (cddr s))) (set-cdr! (cdr s) (cdadr s)) (set-cdr! (cadr s) old-cddr-s))) ((pair? (car s)) (*A072797! s)) ((pair? (cdr s)) (*A072796! s))) s)
%I A123498
%S A123498 0,1,2,3,4,7,5,6,8,9,10,17,16,18,11,20,12,14,15,13,19,21,22,23,24,25,
%T A123498 26,27,45,46,44,42,43,48,47,49,50,28,29,54,53,55,30,57,31,37,38,32,39,
%U A123498 40,41,33,61,34,51,52,35,56,58,59,36,60,62,63,64,65,66,67,68,69,70,71
%N A123498 Signature permutation of a nonrecursive Catalan automorphism: Row 1654249 of table A089840.
%H A123498 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123498 Index entries for signature-permutations of Catalan automorphisms
%Y A123498 Inverse: A123497. Row 1654249 of A089840. Used to construct automorphism *A123502. A074679(n) = A083927(a(A057123(n))).
%K A123498 nonn
%O A123498 0,3
%A A123498 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123498 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123498! s) (cond ((null? s) s) ((and (pair? (cdr s)) (pair? (cadr s))) (let ((old-cddr-s (cddr s))) (set-cdr! (cdr s) (cdadr s)) (set-cdr! (cadr s) old-cddr-s)) (*A074679! s)) ((pair? (cdr s)) (*A072796! s)) ((pair? (car s)) (*A072797! s))) s)
%I A123499
%S A123499 0,1,3,2,6,7,8,5,4,14,15,16,17,18,19,20,21,12,13,22,11,9,10,37,38,39,
%T A123499 40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,31,32,59,34,
%U A123499 35,36,60,61,62,30,33,63,28,23,24,64,29,25,26,27,107,108,109,110,111
%N A123499 Signature permutation of a nonrecursive Catalan automorphism: rotate a binary tree left if possible, otherwise apply *A089863.
%C A123499 This automorphism is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes, and () stands for an implied terminal node.
%C A123499 ...B...C...............A...B...........A...B.............B...A
%C A123499 ....\./.................\./.............\./...............\./.
%C A123499 .A...x........-->........x...C...........x..()...-->...()..x..
%C A123499 ..\./.....................\./.............\./...........\./...
%C A123499 ...x.......................x...............x.............x....
%C A123499 (a . (b . c)) --> ((a . b) . c) / ((a . b) . ()) --> (() . (b . a))
%C A123499 This automorphism cannot be represented as a composition of two smaller nonrecursive automorphisms. Cf. A123503.
%H A123499 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123499 Index entries for signature-permutations of Catalan automorphisms
%Y A123499 Inverse: A123500. Row 258 of A089840. Variant of A074679.
%K A123499 nonn
%O A123499 0,3
%A A123499 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123499 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123499! s) (cond ((null? s) s) ((pair? (cdr s)) (*A074679! s)) (else (*A089863! s))) s)
%I A123500
%S A123500 0,1,3,2,8,7,4,5,6,21,22,20,17,18,9,10,11,12,13,14,15,16,19,58,59,62,
%T A123500 63,64,57,61,54,45,46,55,48,49,50,23,24,25,26,27,28,29,30,31,32,33,34,
%U A123500 35,36,37,38,39,40,41,42,43,44,47,51,52,53,56,60,170,171,174,175,176
%N A123500 Signature permutation of a nonrecursive Catalan automorphism: rotate a binary tree right if possible, otherwise apply *A089859.
%C A123500 This automorphism is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes, and () stands for an implied terminal node.
%C A123500 .A...B...............B...C............B...C...........C...B...
%C A123500 ..\./.................\./..............\./.............\./....
%C A123500 ...x...C....-->....A...x............()..x......-->......x..().
%C A123500 ....\./.............\./..............\./.................\./..
%C A123500 .....x...............x................x...................x...
%C A123500 ((a . b) . c) --> (a . (b . c)) / (() . (b . c)) --> ((c . b) . ())
%C A123500 This automorphism cannot be represented as a composition of two smaller nonrecursive automorphisms. Cf. A123503.
%H A123500 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123500 Index entries for signature-permutations of Catalan automorphisms
%Y A123500 Inverse: A123499. Row 264 of A089840. Variant of A074680.
%K A123500 nonn
%O A123500 0,3
%A A123500 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123500 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123500! s) (cond ((null? s) s) ((pair? (car s)) (*A074680! s)) (else (*A089859! s))) s)
%I A123501
%S A123501 0,1,2,3,4,6,7,5,8,9,10,14,16,19,17,18,12,11,13,20,15,21,22,23,24,25,
%T A123501 26,27,37,38,42,44,47,51,53,56,60,45,46,48,49,50,31,34,30,28,29,35,33,
%U A123501 32,36,54,55,40,39,41,57,43,58,59,61,52,62,63,64,65,66,67,68,69,70,71
%N A123501 Signature permutation of a Catalan automorphism: apply *A123497 at the root, then recurse into the left subtree of the right hand side subtree of a binary tree.
%H A123501 Index entries for signature-permutations of Catalan automorphisms
%Y A123501 Inverse: A123502. A057501(n) = A083927(a(A057123(n))) = A083927(A085159(A057123(n))).
%K A123501 nonn
%O A123501 0,3
%A A123501 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123501 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123501! s) (*A123497! s) (cond ((and (pair? s) (pair? (cdr s))) (*A123501! (cadr s)))) s)
%I A123502
%S A123502 0,1,2,3,4,7,5,6,8,9,10,17,16,18,11,20,12,14,15,13,19,21,22,23,24,25,
%T A123502 26,27,45,46,44,42,49,48,43,47,50,28,29,54,53,55,30,57,31,37,38,32,39,
%U A123502 40,41,33,61,34,51,52,35,56,58,59,36,60,62,63,64,65,66,67,68,69,70,71
%N A123502 Signature permutation of a Catalan automorphism: first recurse into the left subtree of the right hand side subtree of a binary tree, and after that apply *A123498 at the root.
%H A123502 Index entries for signature-permutations of Catalan automorphisms
%Y A123502 Inverse: A123501. A057502(n) = A083927(a(A057123(n))) = A083927(A085160(A057123(n))).
%K A123502 nonn
%O A123502 0,3
%A A123502 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123502 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123502! s) (cond ((and (pair? s) (pair? (cdr s))) (*A123502! (cadr s)))) (*A123498! s) s)
%I A123503
%S A123503 0,1,2,3,4,6,5,8,7,9,10,14,16,19,11,15,12,21,22,13,20,17,18,23,24,25,
%T A123503 26,27,37,38,42,44,47,51,53,56,60,28,29,39,43,52,30,40,31,58,59,32,62,
%U A123503 63,64,33,41,34,57,61,35,54,45,46,36,55,48,49,50,65,66,67,68,69,70,71
%N A123503 An involution of non-negative integers: signature permutation of a nonrecursive Catalan automorphism, row 253 of table A089840.
%C A123503 This automorphism either swaps (if A057515(n) > 1) the first two toplevel elements (of a general plane tree, like *A072796 does), and otherwise (if n > 1, A057515(n)=1) swaps the sides of the left hand side subtree of the S-expression (when viewed as a binary tree, like *A089854 does). This is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes, and () stands for an implied terminal node.
%C A123503 ...B...C.............A...C............A...B...........B...A
%C A123503 ....\./...............\./..............\./.............\./
%C A123503 .A...x.....-->.....B...x................x..()....-->....x..()
%C A123503 ..\./...............\./..................\./.............\./
%C A123503 ...x....(A072796)....x....................x...(A089854)...x
%C A123503 (a . (b . c)) --> (b . (a . c)) / ((a . b) . ()) --> ((b . a) . ())
%C A123503 This is the first multiclause automorphism in table A089840 which cannot be represented as a composition of two smaller nonrecursive automorphisms, the property which is also shared by *A123499 and *A123500.
%H A123503 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123503 Index entries for signature-permutations of Catalan automorphisms
%Y A123503 Row 253 of A089840. Used to construct A123717 and A123718.
%K A123503 nonn
%O A123503 0,3
%A A123503 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123503 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123503! s) (cond ((null? s) s) ((pair? (cdr s)) (*A072796! s)) (else (*A089854! s))) s)
%I A123695
%S A123695 0,1,3,2,6,7,8,5,4,14,15,16,17,18,19,20,21,11,12,22,13,9,10,37,38,39,
%T A123695 40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,28,29,59,30,
%U A123695 31,32,60,61,62,33,34,63,35,23,24,64,36,25,26,27,107,108,109,110,111
%N A123695 Signature permutation of a nonrecursive Catalan automorphism: Row 1653002 of table A089840.
%C A123695 It is possible to recursively construct more of these kind of nonrecursive automorphisms, which by default (if A057515(n) > 1) work as *A074679, and otherwise apply the previous automorphism of this construction process (here *A074679 itself) to the left subtree of a binary tree, before the whole tree is swapped with *A069770. Do the associated cycle-count sequences converge to anything interesting?
%C A123695 This automorphism is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes, and () stands for an implied terminal node.
%C A123695 ...........................B...C........A...B..............................
%C A123695 ............................\./..........\./...............................
%C A123695 ..B...C.....A...B........A...x............x...C...A..()...............()..A
%C A123695 ...\./.......\./..........\./..............\./.....\./.................\./.
%C A123695 A...x....-->..x...C........x..()...-->..()..x.......x..()....-->....()..x..
%C A123695 .\./...........\./..........\./..........\./.........\./.............\./...
%C A123695 ..x.............x............x............x...........x...............x....
%H A123695 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123695 Index entries for signature-permutations of Catalan automorphisms
%Y A123695 Inverse: A123696. Row 1653002 of A089840. Variant of A074679.
%K A123695 nonn
%O A123695 0,3
%A A123695 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123695 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123695! s) (cond ((null? s) s) ((pair? (cdr s)) (*A074679! s)) ((pair? (car s)) (*A074679! (car s)) (*A069770! s))) s)
%I A123696
%S A123696 0,1,3,2,8,7,4,5,6,21,22,17,18,20,9,10,11,12,13,14,15,16,19,58,59,62,
%T A123696 63,64,45,46,48,49,50,54,55,57,61,23,24,25,26,27,28,29,30,31,32,33,34,
%U A123696 35,36,37,38,39,40,41,42,43,44,47,51,52,53,56,60,170,171,174,175,176
%N A123696 Signature permutation of a nonrecursive Catalan automorphism: Row 1653063 of table A089840.
%C A123696 This automorphism is illustrated below, where letters A, B and C refer to arbitrary subtrees located on those nodes, and () stands for an implied terminal node.
%C A123696 ............................B...C.......C...D..............................
%C A123696 .............................\./.........\./...............................
%C A123696 .A...B.............B...C......x...D....B..x............()...C......C..()...
%C A123696 ..\./...............\./........\./......\./.............\./.........\./....
%C A123696 ...x...C..-->....A...x......()..x...-->..x..().......()..x....-->....x..().
%C A123696 ....\./...........\./........\./..........\./.........\./.............\./..
%C A123696 .....x.............x..........x............x...........x...............x...
%C A123696 See the comments at A123695.
%H A123696 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123696 Index entries for signature-permutations of Catalan automorphisms
%Y A123696 Inverse: A123695. Row 1653063 of A089840. Variant of A074680.
%K A123696 nonn
%O A123696 0,3
%A A123696 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123696 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123696! s) (cond ((null? s) s) ((pair? (car s)) (*A074680! s)) ((pair? (cdr s)) (*A074680! (cdr s)) (*A069770! s))) s)
%I A123713
%S A123713 0,1,2,3,4,5,7,8,6,9,10,11,12,13,17,18,20,21,22,16,19,14,15,23,24,25,
%T A123713 26,27,28,29,30,31,32,33,34,35,36,45,46,48,49,50,54,55,57,58,59,61,62,
%U A123713 63,64,44,47,53,56,60,42,51,37,38,43,52,39,40,41,65,66,67,68,69,70,71
%N A123713 Signature permutation of a nonrecursive Catalan automorphism: Row 1783367 of table A089840.
%H A123713 A. Karttunen, Table of n, a(n) for n = 0..6917
%H A123713 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123713 Index entries for signature-permutations of Catalan automorphisms
%Y A123713 Inverse: A123714. Row 1783367 of A089840. Differs from A089855 for the first time at n=102, where a(n)=103, while A089855(n)=102.
%K A123713 nonn
%O A123713 0,3
%A A123713 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123713 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123713! s) (cond ((not (pair? s)) s) ((pair? (car s)) (let ((org_cdar (cdar s))) (set-cdr! (car s) (cdr s)) (set-cdr! s (caar s)) (set-car! (car s) org_cdar) s)) ((and (pair? (cdr s)) (pair? (cadr s)) (pair? (caadr s)) (pair? (caaadr s))) (let ((org_b (car (caaadr s)))) (set-car! (caaadr s) (cdr (caaadr s))) (set-cdr! (caaadr s) (cdaadr s)) (set-cdr! (caadr s) (cdadr s)) (set-cdr! (cadr s) (cddr s)) (set-cdr! (cdr s) org_b) s)) (else s)))
%I A123714
%S A123714 0,1,2,3,4,5,8,6,7,9,10,11,12,13,21,22,19,14,15,20,16,17,18,23,24,25,
%T A123714 26,27,28,29,30,31,32,33,34,35,36,58,59,62,63,64,56,60,51,37,38,52,39,
%U A123714 40,41,57,61,53,42,43,54,44,45,46,55,47,48,49,50,65,66,67,68,69,70,71
%N A123714 Signature permutation of a nonrecursive Catalan automorphism: Row 1786785 of table A089840.
%C A123714 This automorphism is illustrated below, where letters A, B, C, D, E and F refer to arbitrary subtrees located on those nodes, and () stands for an implied terminal node.
%C A123714 .............................B...C............F...B......
%C A123714 ..............................\./..............\./.......
%C A123714 ...............................x...D............x...C....
%C A123714 ................................\./..............\./.....
%C A123714 .................................x...E............x...D..
%C A123714 ..................................\./.....-->......\./...
%C A123714 ..A...B.........C...A..............x...F............x...E
%C A123714 ...\./...........\./................\./..............\./.
%C A123714 ....x...C...-->...x...B..........()..x............()..x..
%C A123714 .....\./...........\./............\./..............\./...
%C A123714 ......x.............x..............x................x....
%C A123714 This is the last multiclause automorphism of total seven opened conses in the table A089840. The next nonrecursive automorphism, A089840[1786786], which consists of a single seven-node clause, swaps the first two toplevel elements (of a general plane tree, like *A072796 does), but only if A057515(n) > 6, and in other cases keeps the tree intact.
%H A123714 A. Karttunen, Table of n, a(n) for n = 0..6917
%H A123714 A. Karttunen, Prolog-program which illustrates the construction of this and other similar nonrecursive Catalan automorphisms.
%H A123714 Index entries for signature-permutations of Catalan automorphisms
%Y A123714 Inverse: A123713. Row 1786785 of A089840. Differs from A089857 for the first time at n=102, where a(n)=106, while A089857(n)=102.
%K A123714 nonn
%O A123714 0,3
%A A123714 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123714 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define (*A123714! s) (cond ((not (pair? s)) s) ((pair? (car s)) (let ((org_a (caar s))) (set-car! (car s) (cdr s)) (set-cdr! s (cdar s)) (set-cdr! (car s) org_a) s)) ((and (pair? (cdr s)) (pair? (cadr s)) (pair? (caadr s)) (pair? (caaadr s))) (let ((org_f (cddr s))) (set-cdr! (cdr s) (cdadr s)) (set-cdr! (cadr s) (cdaadr s)) (set-cdr! (caadr s) (cdr (caaadr s))) (set-cdr! (caaadr s) (car (caaadr s))) (set-car! (caaadr s) org_f) s)) (else s)))
%I A123715
%S A123715 0,1,2,3,4,8,6,7,5,9,13,20,21,22,14,19,16,17,18,15,11,12,10,23,27,34,
%T A123715 35,36,54,55,57,58,59,61,62,63,64,37,41,53,56,60,42,51,44,45,46,47,48,
%U A123715 49,50,43,52,39,28,33,40,30,31,32,38,29,25,26,24,65,69,76,77,78,96,97
%N A123715 Signature permutation of a Catalan automorphism: Row 79361 of table A122203.
%C A123715 This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123492 with the recursion schema SPINE (defined in A122203).
%H A123715 Index entries for signature-permutations of Catalan automorphisms
%Y A123715 Inverse: A123716. Row 79361 of A122203.
%K A123715 nonn
%O A123715 0,3
%A A123715 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123715 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define *A123715! (!SPINE *A123492!))
%I A123716
%S A123716 0,1,2,3,4,8,6,7,5,9,22,20,21,10,14,19,16,17,18,15,11,12,13,23,64,62,
%T A123716 63,24,54,61,57,58,59,55,25,26,27,37,60,53,56,38,42,51,44,45,46,47,48,
%U A123716 49,50,43,52,39,28,29,40,30,31,32,41,33,34,35,36,65,196,194,195,66
%N A123716 Signature permutation of a Catalan automorphism: Row 79361 of table A122204.
%C A123716 This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123492 with the recursion schema ENIPS (defined in A122204).
%H A123716 Index entries for signature-permutations of Catalan automorphisms
%Y A123716 Inverse: A123715. Row 79361 of A122204.
%K A123716 nonn
%O A123716 0,3
%A A123716 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123716 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define *A123716! (!ENIPS *A123492!))
%I A123717
%S A123717 0,1,2,3,4,6,5,8,7,9,11,14,16,19,10,15,13,21,22,12,20,17,18,23,25,28,
%T A123717 30,33,37,39,42,44,47,51,53,56,60,24,29,38,43,52,27,41,35,58,59,36,62,
%U A123717 63,64,26,40,34,57,61,31,54,45,46,32,55,48,49,50,65,67,70,72,75,79,81
%N A123717 Signature permutation of a Catalan automorphism: Row 253 of table A122203.
%C A123717 This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123503 with the recursion schema SPINE (defined in A122203).
%C A123717 The number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation begins as 1,1,2,1,3,1,4,1,8,1,16,1,47,..., the LCM of cycle sizes as 1,1,1,2,12,12,120,120,840,840,5040,5040,55440,... (cf. A089423) and the cycle-count sequence seems to be A045629. (To be proved.)
%D A123717 A. Karttunen, paper in preparation, draft available by e-mail.
%H A123717 Index entries for signature-permutations of Catalan automorphisms
%Y A123717 Inverse: A123718. a(n) = A057509(A089854(n)). Row 253 of A122203.
%K A123717 nonn
%O A123717 0,3
%A A123717 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123717 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define *A123717! (!SPINE *A123503!))
%I A123718
%S A123718 0,1,2,3,4,6,5,8,7,9,14,10,19,16,11,15,12,21,22,13,20,17,18,23,37,24,
%T A123718 51,42,25,38,26,56,60,27,53,44,47,28,39,29,52,43,30,40,31,58,59,32,62,
%U A123718 63,64,33,41,34,57,61,35,54,45,46,36,55,48,49,50,65,107,66,149,121,67
%N A123718 Signature permutation of a Catalan automorphism: Row 253 of table A122204.
%C A123718 This is the signature-permutation of Catalan automorphism which is derived from nonrecursive Catalan automorphism *A123503 with the recursion schema ENIPS (defined in A122204). See the comments at A123717.
%H A123718 Index entries for signature-permutations of Catalan automorphisms
%Y A123718 Inverse: A123717. a(n) = A089854(A057510(n)). Row 253 of A122204.
%K A123718 nonn
%O A123718 0,3
%A A123718 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123718 (Scheme function, destructive implementation of this automorphism acting on S-expressions:) (define *A123718! (!ENIPS *A123503!))
%I A123719
%S A123719 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,21,19,20,18,22,23,24,25,
%T A123719 26,27,28,29,30,31,35,33,34,32,36,37,38,39,40,41,42,43,44,45,58,56,54,
%U A123719 49,63,51,52,53,48,62,47,57,46,59,60,61,55,50,64,65,66,67,68,69,70,71
%N A123719 An involution of non-negative integers: signature permutation of Catalan automorphism which is obtained with recursion schema RIBS from automorphism *A085161
%C A123719 Recursion schema RIBS is defined in A122200. Number of fixed points in range [A014137(n-1)..A014138(n-1)] of this permutation is given by INVERT transform of A001405, appropriately shifted.
%H A123719 Index entries for signature-permutations of Catalan automorphisms
%Y A123719 a(n) = A085160(A085163(n)). A085163(n) = A085159(a(n)).
%K A123719 nonn
%O A123719 0,3
%A A123719 Antti Karttunen (His-Firstname.His-Surname(AT)gmail.com), Oct 11 2006
%o A123719 (Scheme function, constructive implementation of this automorphism acting on S-expressions:) (define *A123719 (RIBS *A085161))
%I A123694
%S A123694 0,7,91,92,93,94,95,114,115,116,117,118,4207,4209,4211,4214,4216,4299,
%T A123694 4301,4303,4305,4307,1228,1229,1230,1231,1232,1233,1234,1235,1236,1237,
%U A123694 1238,1239,1240,1241,1242,1243,1244,1245,1246,1247,1248,1249,1250,1347
%N A123694 a(n) gives the A089840-index of the nonrecursive Catalan automorphism which is formed from A089840[n] by applying it to the left subtree of a binary tree, and leaving the right-hand side subtree intact.
%e A123694 When A089840[1] = A069770 (swap binary tree sides) is applied to the left subtree of a binary tree, we get A089840[7] = A089854, thus a(1)=7. When A089840[12] = A074679 is applied to the left subtree of a binary tree, we get A089840[4207] = A089865, thus a(12)=4207.
%C A123694 If the count of fixed points of the automorphism A089840[n] is given by sequence f, then the count of fixed points of the automorphism A089840[A123694(n)] is given by CONV(f,A000108) (where CONV stands for convolution). See also the comments at A122200.
%H A123694 A. Karttunen, C-program for computing the initial terms of this sequence
%H A123694 A. Karttunen, Prolog-program which illustrates the nonrecursive Catalan automorphisms given on example-lines.
%K A123694 nonn
%O A123694 0,2
%A A123694 Antti Karttunen (His_Firstname.His_Surname(AT)gmail.com), Oct 11 2006