Cheers,

what follows:

AN ADDITION TO AN INDEX ENTRY,
COMPLETE RE-EDITS OF TWO SEQUENCES, A057505 & A057506,
MINOR CHANGES TO A057161 & A057162:
AND 24 PRE-NUMBERED NEW SEQUENCES A071651 - A071674.


First, to the end of:
http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation
to the end of these lines:
permutations, of the integers, induced by Catalan automorphisms, each paired with its inverse:
A071661-A071662, A071663-A071664, A071665-A071666, A071667-A071668, A071669-A071670
A071655-A071656, A071657-A071658, A071659-A071660

and yet new section, after the Catalan automorphisms:
permutations, of the integers, induced by Catalan rerankings, each paired with its inverse:
A071651-A071652, A071653-A071654

by "reranking" I mean that there are at least two different
ranking/unranking schemes, and switching between these.
In this case (A071651-A071654) it is hard to view them
as automorphisms because those permutations escape from the usual
"boundaries" set by A014137, i.e. the partial sums of Catalans:
1 2 4 9 23 65 197 626 2056 6918 23714 ...
(and I have some other ideas as well for these kind of rerankings.)


-------------------------------------------------------------------------------

Complete RE-EDITS of A057505 & A057506:

%I A057505
%S A057505 0,1,3,2,8,7,5,6,4,22,21,18,20,17,13,12,15,19,16,10,11,14,9,64,63,59,
%T A057505 62,58,50,49,55,61,57,46,48,54,45,36,35,32,34,31,41,40,52,60,56,43,47,
%U A057505 53,44,27,26,29,33,30,38,39,51,42,24,25,28,37,23,196,195,190,194,189
%N A057505 Permutation of natural numbers induced by the automorphism DonagheysM (or DeepRotateTriangularization) acting on the parenthesizations encoded by A014486.
%C A057505 This is equivalent to map M given by Donaghey on page 81 of his paper "Automorphisms on ...", and also equivalent to the transformation procedure depicted in the picture (23) of Donaghey-Shapiro paper.
%C A057505 This can be also considered as a "deeply recursive" variant of A057501 or a recursive variant of A057161.
%H A057505 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A057505 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%D A057505 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%D A057505 R. Donaghey & L. W. Shapiro, Motzkin numbers, J. Combin. Theory, Series A, 23 (1977), 291-301.
%Y A057505 Inverse permutation: A057506, and also its car/cdr-flipped conjugate, i.e. A0057505(n) = A057163(A057506(A057163(n))). Composition of A057163 & A057164, i.e. A057505(n) =  A057164(A057163(n)). 
%Y A057505 The 2nd, 3rd, 4th, 5th and 6th "power": A071661, A071663, A071665, A071667, A071669.
%Y A057505 Cycle counts: A057507. Maximum cycle lengths: A057545. LCM's of all cycles: A060114. See A057501 for other Maple procedures.
%K A057505 new,nonn
%O A057505 0,3
%A A057505 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Sep 03 2000
%p A057505 map(CatalanRankGlobal,map(DonagheysM,A014486)); or  map(CatalanRankGlobal,map(DeepRotateTriangularization,A014486));
%p A057505 DonagheysM := n -> pars2binexp(DonagheysMP(binexp2pars(n)));
%p A057505 DonagheysMP := h -> `if`((0 = nops(h)),h,[op(DonagheysMP(car(h))),DonagheysMP(cdr(h))]);
%p A057505 DeepRotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := DeepRotateTriangularization(BinTreeRightBranch(n))*2; z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;
%o A057505 (Two alternative Scheme functions implementing this automorphism on list-structures:)
%o A057505 (define (DonagheysM a) (cond ((null? a) a) (else (append (DonagheysM (car a)) (list (DonagheysM (cdr a)))))))
%o A057505 (define (DeepRotateTriangularization bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (car lt) (cons (DeepRotateTriangularization (cdr lt)) nt))))))



%I A057506
%S A057506 0,1,3,2,8,6,7,5,4,22,19,20,15,14,21,16,18,13,11,17,12,10,9,64,60,61,
%T A057506 52,51,62,53,55,41,39,54,40,38,37,63,56,57,43,42,59,47,50,36,33,48,34,
%U A057506 29,28,58,44,49,35,30,46,32,27,25,45,31,26,24,23,196,191,192,178,177
%N A057506 Permutation of natural numbers induced by the automorphism DeepRotateTriangularizationInv acting on the parenthesizations encoded by A014486. Inverse permutation of A057505.
%H A057506 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A057506 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%p A057506 map(CatalanRankGlobal,map(DonagheysMR,A014486));
%p A057506 DonagheysMR := n -> pars2binexp(deepreverse(DonagheysMP(deepreverse(binexp2pars(n)))));
%Y A057506 Cf. A057502 (for deepreverse), A057501 (for other procedures), A057503, A057507 (cycle counts).
%Y A057506 Inverse permutation: A057505, and also its car/cdr-flipped conjugate, i.e. A0057506(n) = A057163(A057505(A057163(n))). Composition of A057164 & A057163, i.e. A057506(n) =  A057163(A057164(n)). 
%Y A057506 The 2nd, 3rd, 4th, 5th and 6th "power": A071662, A071664, A071666, A071668, A071670.
%K A057506 new,nonn
%O A057506 0,3
%A A057506 Antti.Karttunen@iki.fi (karttu@megabaud.fi) Sep 03 2000
%o A057506 (Scheme function implementing this automorphism on list-structures:)
%o A057506 (define (DeepRotateTriangularizationInv bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (cdr lt) (cons nt (DeepRotateTriangularizationInv (car lt))))))))

-------------------------------------------------------------------------------
CHANGES TO A057161 & A057162:

From A057161 and A057162, EITHER just change the clockwise
and counter-clockwise in their names, i.e. they should be:


%I A057161
%S A057161 0,1,3,2,7,8,5,6,4,17,18,20,21,22,12,13,15,16,19,10,11,14,9,45,46,48,
%N A057161 Permutation of natural numbers induced by counter-clockwise rotation of the triangularizations of polygons encoded by A014486.

%I A057162
%S A057162 0,1,3,2,8,6,7,4,5,22,19,20,14,15,21,16,17,9,10,18,11,12,13,64,60,61,
%N A057162 Permutation of natural numbers induced by clockwise rotation of the triangularizations of polygons encoded by A014486.

(I got the directions wrong last time, and of course they depend
from many arbitrary "left-or-right-side first decisions" viz-a-viz
how to map parenthesizations to binary trees, and binary trees to
polygons triangularizations (and how to draw those polygons on the
screen or paper!), but in the system I have chosen
(the Gatomorphisms page), the directions go that way, A057161
counter-clockwise, and A057162 clockwise.)

OR: if you want to avoid these kind of arbitrary fixings of the direction,
the titles could as well be:

%I A057161
%S A057161 0,1,3,2,7,8,5,6,4,17,18,20,21,22,12,13,15,16,19,10,11,14,9,45,46,48,
%N A057161 Permutation of natural numbers induced by rotation of the triangularizations of polygons encoded by A014486.

%I A057162
%S A057162 0,1,3,2,8,6,7,4,5,22,19,20,14,15,21,16,17,9,10,18,11,12,13,64,60,61,
%N A057162 Permutation of natural numbers induced by rotation of the triangularizations of polygons encoded by A014486 (to the opposite direction of that of A057161)

(Or how to put that better in English, I don't know...)



-------------------------------------------------------------------------------

24 new sequences A071651 - A071674, of which
all but just the last four are also permutations
of the non-negative integers:


%I A071651
%S A071651 0,1,2,3,4,7,5,6,10,11,29,16,22,56,8,12,9,15,36,14,21,28,66,67,436,137,254,1597,37,79,46,121,
%T A071651 667,106,232,407,2212,17,38,23,30,68,13,18,20,78,465,44,153,276,1653,19,25,27,45,91,35,55,136,
%U A071651 703,77,120,253,435,2278,2279,95267,9454,32386,1276004,704,3161,1082,7382,222779,5672,27029
%N A071651 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A061579 as the packing bijection NxN -> N.
%R A071651
%H A071651 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A>
%H A071651 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071651 0,3
%K A071651 new,nonn
%A A071651 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071651 Inverse permutation: A071652. C.f. also A014486, A061579, A071653, A071654.
%D A071651
%o A071651 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.)
%o A071651 (define lexrank->arithrankA061579 (lexrank->arithrank-bijection packA061579))
%o A071651 (define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun)))
%o A071651 (define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun))))))
%o A071651 (define (packA061579 x y) (/ (+ (expt (+ x y) 2) (* 3 x) y) 2))


%I A071652
%S A071652 0,1,2,3,4,6,7,5,14,16,8,9,15,42,19,17,11,37,43,51,44,20,12,39,121,52,126,53,21,10,40,123,149,
%T A071652 127,154,56,18,28,38,124,151,385,155,163,47,54,30,112,122,152,387,475,164,135,156,57,13,114,
%U A071652 376,150,388,477,503,136,480,165,22,23,41,378,466,386,478,505,413,481,508,60,45,29,107,125
%N A071652 Inverse permutation of A071651.
%R A071652
%H A071652 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A>
%H A071652 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071652 0,3
%K A071652 new,nonn,tabl
%A A071652 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071652 Inverse permutation: A071651. A071671 gives the corresponding parenthesizations (from the term 1 onward) encoded as binary numbers, i.e. A071671(n) = A063171(A071652(n)) for n >= 1.
%Y A071652 A071652(n) = A057163(A071654(n)). C.f. also A014486, A002262, A025581, A071653, A071654.
%D A071652
%o A071652 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.)
%o A071652 (define arithrankA061579->lexrank (arithrank->lexrank-bijection A002262 A025581))
%o A071652 (define (arithrank->lexrank-bijection pr1 pr2) (lambda (n) (CatalanRankGlobal (parenthesization->binexp (unrank-bintree n pr1 pr2)))))
%o A071652 (define (unrank-bintree rank pr1 pr2) (cond ((zero? rank) (list)) (else (cons (unrank-bintree (pr1 (-1+ rank)) pr1 pr2) (unrank-bintree (pr2 (-1+ rank)) pr1 pr2)))))
%o A071652 (define (A002262 n) (- n (binomial_n_2 (floor->exact (+ (/ 1 2) (sqrt (* 2 (1+ n))))))))
%o A071652 (define (A025581 n) (- (binomial_n_2 (1+ (floor->exact (+ (/ 1 2) (sqrt (* 2 (1+ n))))))) (1+ n)))





%I A071653
%S A071653 0,1,3,2,10,6,5,7,4,66,28,21,36,15,14,9,12,56,22,8,16,29,11,2278,435,253,703,136,120,55,91,
%T A071653 1653,276,45,153,465,78,77,35,27,44,20,25,18,68,2212,407,30,232,667,121,19,13,23,106,46,38,
%U A071653 79,1597,254,17,37,137,436,67,2598060,95266,32385,248160,9453,7381,1596,4278,1368685,38503
%N A071653 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A001477 as the packing bijection NxN -> N.
%R A071653
%H A071653 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A>
%H A071653 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071653 0,3
%K A071653 new,nonn
%A A071653 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071653 Inverse permutation: A071654. C.f. also A014486, A001477, A071651, A071652.
%D A071653
%o A071653 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.)
%o A071653 (define lexrank->arithrankA001477 (lexrank->arithrank-bijection packA001477))
%o A071653 (define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun)))
%o A071653 (define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun))))))
%o A071653 (define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))




%I A071654
%S A071654 0,1,3,2,8,6,5,7,19,15,4,22,16,52,14,13,20,60,43,51,41,11,18,53,178,42,153,39,10,21,47,155,
%T A071654 177,125,151,38,12,61,56,136,154,555,123,150,40,33,55,179,164,135,479,553,122,152,117,29,17,
%U A071654 159,557,163,417,477,552,124,471,113,9,64,44,490,556,507,415,476,554,381,467,37,36,57,191,127
%N A071654 Inverse permutation of A071653.
%R A071654
%H A071654 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A>
%H A071654 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071654 0,3
%K A071654 new,nonn,tabl
%A A071654 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071654 Inverse permutation: A071653. A071672 gives the corresponding parenthesizations (from the term 1 onward) encoded as binary numbers, i.e. A071672(n) = A063171(A071654(n)) for n >= 1.
%Y A071654 A071654(n) = A057163(A071652(n)). C.f. also A014486, A002262, A025581, A071651, A071652.
%D A071654
%o A071654 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.)
%o A071654 (define arithrankA001477->lexrank (arithrank->lexrank-bijection A025581 A002262))
%o A071654 (define (arithrank->lexrank-bijection pr1 pr2) (lambda (n) (CatalanRankGlobal (parenthesization->binexp (unrank-bintree n pr1 pr2)))))
%o A071654 (define (unrank-bintree rank pr1 pr2) (cond ((zero? rank) (list)) (else (cons (unrank-bintree (pr1 (-1+ rank)) pr1 pr2) (unrank-bintree (pr2 (-1+ rank)) pr1 pr2)))))
%o A071654 (define (A002262 n) (- n (binomial_n_2 (floor->exact (+ (/ 1 2) (sqrt (* 2 (1+ n))))))))
%o A071654 (define (A025581 n) (- (binomial_n_2 (1+ (floor->exact (+ (/ 1 2) (sqrt (* 2 (1+ n))))))) (1+ n)))



%I A071655
%S A071655 0,1,3,2,7,8,5,4,6,17,18,20,21,22,12,13,10,9,11,15,14,19,16,45,46,48,49,50,54,55,57,58,59,61,
%T A071655 62,63,64,31,32,34,35,36,26,27,24,23,25,29,28,33,30,40,41,38,37,39,52,51,56,60,43,42,47,44,
%U A071655 53,129,130,132,133,134,138,139,141,142,143,145,146,147,148,157,158,160,161,162,166,167,169
%N A071655 Permutation of natural numbers induced by the automorphism robr_car_cdr! acting on the parenthesizations encoded by A014486.
%R A071655
%H A071655 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071655 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071655 0,3
%K A071655 new,nonn
%A A071655 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071655 Inverse permutation: A071656. The car/cdr-flipped conjugate of A071660, i.e. A071655(n) = A057163(A071660(A057163(n))). Cf. also A071657, A071658.
%D A071655
%o A071655 (Scheme function implementing this automorphism on list-structures:)
%o A071655 (define (robr_car_cdr! s) (cond ((not (pair? s))) ((not (pair? (car s))) (swap! s)) (else (robr! s) (robr_car_cdr! (car s)) (robr_car_cdr! (cdr s)))) s)
%o A071655 (define (robr! s) (let ((ex-cdr (cdr s))) (set-cdr! s (caar s)) (set-car! (car s) ex-cdr) (swap! (car s)) (swap! s) s))
%o A071655 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))



%I A071656
%S A071656 0,1,3,2,7,6,8,4,5,17,16,18,14,15,20,19,22,9,10,21,11,12,13,45,44,46,42,43,48,47,50,37,38,49,
%T A071656 39,40,41,54,53,55,51,52,61,60,63,23,24,62,25,26,27,57,56,64,28,29,58,30,31,32,59,33,34,35,
%U A071656 36,129,128,130,126,127,132,131,134,121,122,133,123,124,125,138,137,139,135,136,145,144,147
%N A071656 Permutation of natural numbers induced by the automorphism car_cdr_robl! acting on the parenthesizations encoded by A014486.
%R A071656
%H A071656 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071656 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071656 0,3
%K A071656 new,nonn
%A A071656 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071656 Inverse permutation: A071655. The car/cdr-flipped conjugate of A071659, i.e. A071656(n) = A057163(A071659(A057163(n))). Cf. also A071657, A071658.
%D A071656
%o A071656 (Scheme function implementing this automorphism on list-structures:)
%o A071655 (define (car_cdr_robl! s) (cond ((not (pair? s))) ((not (pair? (cdr s))) (swap! s)) (else (car_cdr_robl! (car s)) (car_cdr_robl! (cdr s)) (robl! s))) s)
%o A071656 (define (robl! s) (let ((ex-car (car s))) (set-car! s (cddr s)) (set-cdr! (cdr s) ex-car) (swap! (cdr s)) (swap! s) s))
%o A071656 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))



%I A071657
%S A071657 0,1,3,2,7,8,5,6,4,17,18,20,21,22,12,13,15,16,19,10,9,14,11,45,46,48,49,50,54,55,57,58,59,61,
%T A071657 62,63,64,31,32,34,35,36,40,41,43,44,47,52,53,56,60,26,27,24,28,25,38,37,42,51,29,33,30,39,
%U A071657 23,129,130,132,133,134,138,139,141,142,143,145,146,147,148,157,158,160,161,162,166,167,169
%N A071657 Permutation of natural numbers induced by the automorphism car_robr_cdr! acting on the parenthesizations encoded by A014486.
%R A071657
%H A071657 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071657 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071657 0,3
%K A071657 new,nonn
%A A071657 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071657 Inverse permutation: A071658, and also its car/cdr-flipped conjugate, i.e. A071657(n) = A057163(A071658(A057163(n))). Cf. also A071655, A071656, A071659, A071660.
%D A071657
%o A071657 (Scheme function implementing this automorphism on list-structures:)
%o A071657 (define (car_robr_cdr! s) (cond ((not (pair? s))) ((not (pair? (car s))) (swap! s)) (else (car_robr_cdr! (car s)) (robr! s) (car_robr_cdr! (cdr s)))) s)
%o A071657 (define (robr! s) (let ((ex-cdr (cdr s))) (set-cdr! s (caar s)) (set-car! (car s) ex-cdr) (swap! (car s)) (swap! s) s))
%o A071657 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))




%I A071658
%S A071658 0,1,3,2,8,6,7,4,5,20,19,22,14,15,21,16,17,9,10,18,11,12,13,64,53,55,51,52,54,60,62,37,38,61,
%T A071658 39,40,41,57,56,63,42,43,58,44,45,23,24,46,25,26,27,59,47,48,28,29,49,30,31,32,50,33,34,35,
%U A071658 36,162,191,192,154,155,184,159,160,149,150,193,151,152,153,194,156,161,177,178,157,182,196
%N A071658 Permutation of natural numbers induced by the automorphism cdr_robl_car! acting on the parenthesizations encoded by A014486.
%R A071658
%H A071658 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071658 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071658 0,3
%K A071658 new,nonn
%A A071658 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071658 Inverse permutation: A071657, and also its car/cdr-flipped conjugate, i.e. A071658(n) = A057163(A071657(A057163(n))). Cf. also A071655, A071656, A071659, A071660.
%D A071658
%o A071658 (Scheme function implementing this automorphism on list-structures:)
%o A071658 (define (cdr_robl_car! s) (cond ((not (pair? s))) ((not (pair? (cdr s))) (swap! s)) (else (cdr_robl_car! (cdr s)) (robl! s) (cdr_robl_car! (car s)))) s)
%o A071658 (define (robl! s) (let ((ex-car (car s))) (set-car! s (cddr s)) (set-cdr! (cdr s) ex-car) (swap! (cdr s)) (swap! s) s))
%o A071658 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))



%I A071659
%S A071659 0,1,3,2,7,8,4,6,5,17,18,20,21,22,10,9,14,16,19,11,12,15,13,45,46,48,49,50,54,55,57,58,59,61,
%T A071659 62,63,64,26,27,23,25,24,38,37,42,44,47,51,53,56,60,29,28,30,32,31,39,40,43,52,33,34,35,41,
%U A071659 36,129,130,132,133,134,138,139,141,142,143,145,146,147,148,157,158,160,161,162,166,167,169
%N A071659 Permutation of natural numbers induced by the automorphism car_cdr_robr! acting on the parenthesizations encoded by A014486.
%R A071659
%H A071659 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071659 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071659 0,3
%K A071659 new,nonn
%A A071659 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071659 Inverse permutation: A071660. The car/cdr-flipped conjugate of A071656, i.e. A071659(n) = A057163(A071656(A057163(n))). Cf. also A071657, A071658.
%D A071659
%o A071659 (Scheme function implementing this automorphism on list-structures:)
%o A071659 (define (car_cdr_robr! s) (cond ((not (pair? s))) ((not (pair? (car s))) (swap! s)) (else (car_cdr_robr! (car s)) (car_cdr_robr! (cdr s)) (robr! s))) s)
%o A071659 (define (robr! s) (let ((ex-cdr (cdr s))) (set-cdr! s (caar s)) (set-car! (car s) ex-cdr) (swap! (car s)) (swap! s) s))
%o A071659 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))




%I A071660
%S A071660 0,1,3,2,6,8,7,4,5,15,14,19,20,22,16,21,17,9,10,18,11,12,13,39,41,40,37,38,52,51,53,55,54,60,
%T A071660 61,62,64,43,42,56,57,63,44,58,45,23,24,46,25,26,27,47,59,48,28,29,49,30,31,32,50,33,34,35,
%U A071660 36,113,112,117,118,120,114,119,115,107,108,116,109,110,111,151,153,152,149,150,155,154,159
%N A071660 Permutation of natural numbers induced by the automorphism robl_car_cdr! acting on the parenthesizations encoded by A014486.
%R A071660
%H A071660 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071660 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071660 0,3
%K A071660 new,nonn
%A A071660 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071660 Inverse permutation: A071659. The car/cdr-flipped conjugate of A071655, i.e. A071660(n) = A057163(A071655(A057163(n))). Cf. also A071657, A071658.
%D A071660
%o A071660 (Scheme function implementing this automorphism on list-structures:)
%o A071660 (define (robl_car_cdr! s) (cond ((not (pair? s))) ((not (pair? (cdr s))) (swap! s)) (else (robl! s) (robl_car_cdr! (car s)) (robl_car_cdr! (cdr s)))) s)
%o A071660 (define (robl! s) (let ((ex-car (car s))) (set-car! s (cddr s)) (set-cdr! (cdr s) ex-car) (swap! (cdr s)) (swap! s) s))
%o A071660 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))




%I A071661
%S A071661 0,1,2,3,4,6,7,5,8,9,14,16,11,19,17,20,12,10,15,21,18,13,22,23,37,42,28,51,44,53,30,25,39,56,
%T A071661 47,33,60,45,54,57,48,61,31,34,26,24,38,40,43,29,52,58,62,49,46,55,35,32,27,41,63,59,50,36,
%U A071661 64,65,107,121,79,149,126,154,84,70,112,163,135,93,177,128,156,165,137,179,86,95,72,67,109
%N A071661 Permutation A057505 applied twice ("squared").
%R A071661
%H A071661 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071661 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071661 0,3
%K A071661 new,nonn
%A A071661 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071661 Inverse permutation: A071662, and also its car/cdr-flipped conjugate, i.e. A071661(n) = A057163(A071662(A057163(n))) = A057505(A057505(n)). Cf. also A071663, A071665, A071667, A071669.
%D A071661
%p A071661



%I A071662
%S A071662 0,1,2,3,4,7,5,6,8,9,17,12,16,21,10,18,11,14,20,13,15,19,22,23,45,31,44,58,26,49,30,42,57,35,
%T A071662 43,56,63,24,46,32,47,59,25,48,28,37,54,34,40,53,62,27,50,29,38,55,33,39,51,61,36,41,52,60,
%U A071662 64,65,129,87,128,170,73,142,86,126,169,100,127,168,189,68,133,91,140,175,72,141,84,121,166
%N A071662 Permutation A057506 applied twice ("squared").
%R A071662
%H A071662 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071662 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071662 0,3
%K A071662 new,nonn
%A A071662 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071662 Inverse permutation: A071661, and also its car/cdr-flipped conjugate, i.e. A071662(n) = A057163(A071661(A057163(n))) = A057506(A057506(n)). Cf. also A071664, A071666, A071668, A071670.
%D A071662
%p A071662



%I A071663
%S A071663 0,1,3,2,8,5,6,7,4,22,13,15,18,10,19,11,20,21,12,14,16,17,9,64,36,41,50,27,52,29,55,59,32,38,
%T A071663 43,46,24,60,33,39,47,25,61,48,62,63,35,34,40,49,26,51,28,53,56,30,54,57,58,31,37,42,44,45,
%U A071663 23,196,106,120,148,78,153,83,162,176,92,111,125,134,69,178,94,113,136,71,181,139,185,190,101
%N A071663 Permutation A057505 applied three times ("cubed").
%R A071663
%D A071663 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%H A071663 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071663 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071663 0,3
%K A071663 new,nonn
%A A071663 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071663 Inverse permutation: A071664, and also its car/cdr-flipped conjugate, i.e. A071663(n) = A057163(A071664(A057163(n))) = A057505(A071661(n)). Cf. also A071665, A071667, A071669.
%D A071663
%p A071663



%I A071664
%S A071664 0,1,3,2,8,5,6,7,4,22,13,15,18,10,19,11,20,21,12,14,16,17,9,64,36,41,50,27,52,29,55,59,32,38,
%T A071664 47,46,24,60,33,39,48,25,61,34,62,63,35,40,43,49,26,51,28,53,56,30,54,57,58,31,37,42,44,45,
%U A071664 23,196,106,120,148,78,153,83,162,176,92,111,144,134,69,178,94,113,139,71,181,97,185,190,101
%N A071664 Permutation A057506 applied three times ("cubed").
%R A071664
%H A071664 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071664 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071664 0,3
%K A071664 new,nonn
%A A071664 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071664 Inverse permutation: A071663, and also its car/cdr-flipped conjugate, i.e. A071664(n) = A057163(A071663(A057163(n))) = A057506(A071662(n)). Cf. also A071666, A071668, A071670.
%D A071664
%p A071664


%I A071665
%S A071665 0,1,2,3,4,7,5,6,8,9,17,12,16,21,10,18,11,14,20,13,15,19,22,23,45,31,44,58,26,49,30,42,57,35,
%T A071665 40,56,63,24,46,32,43,59,25,47,28,37,54,48,34,53,62,27,50,29,38,55,33,39,51,61,36,41,52,60,
%U A071665 64,65,129,87,128,170,73,142,86,126,169,100,115,168,189,68,133,91,124,175,72,140,84,121,166
%N A071665 Permutation A057505 applied four times ("^4"), permutation A071661 squared.
%R A071665
%H A071665 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071665 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071665 0,3
%K A071665 new,nonn
%A A071665 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071665 Inverse permutation: A071666, and also its car/cdr-flipped conjugate, i.e. A071665(n) = A057163(A071666(A057163(n))) = A057505(A071663(n)) = A071661(A071661(n)). Cf. also A071667, A071669.
%D A071665
%p A071665



%I A071666
%S A071666 0,1,2,3,4,6,7,5,8,9,14,16,11,19,17,20,12,10,15,21,18,13,22,23,37,42,28,51,44,53,30,25,39,56,
%T A071666 48,33,60,45,54,57,34,61,31,40,26,24,38,43,47,29,52,58,62,49,46,55,35,32,27,41,63,59,50,36,
%U A071666 64,65,107,121,79,149,126,154,84,70,112,163,138,93,177,128,156,165,95,179,86,114,72,67,109
%N A071666 Permutation A057506 applied four times ("^4"), permutation A071662 squared.
%R A071666
%H A071666 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071666 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071666 0,3
%K A071666 new,nonn
%A A071666 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071666 Inverse permutation: A071665, and also its car/cdr-flipped conjugate, i.e. A071666(n) = A057163(A071665(A057163(n))) = A057506(A071664(n)) = A071662(A071662(n)). Cf. also A071662, A071668, A071670.
%D A071666
%p A071666



%I A071667
%S A071667 0,1,3,2,8,6,7,5,4,22,19,20,15,14,21,16,18,13,11,17,12,10,9,64,60,61,52,51,62,53,55,41,39,54,
%T A071667 34,38,37,63,56,57,40,42,59,43,50,36,33,47,48,29,28,58,44,49,35,30,46,32,27,25,45,31,26,24,
%U A071667 23,196,191,192,178,177,193,179,181,153,151,180,103,150,149,194,182,183,118,154,185,155,162
%N A071667 Permutation A057505 applied five times ("^5").
%R A071667
%H A071667 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071667 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071667 0,3
%K A071667 new,nonn
%A A071667 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071667 Inverse permutation: A071668, and also its car/cdr-flipped conjugate, i.e. A071667(n) = A057163(A071668(A057163(n))) = A057505(A071665(n)). Cf. also A071661, A071663, A071669.
%D A071667
%p A071667


%I A071668
%S A071668 0,1,3,2,8,7,5,6,4,22,21,18,20,17,13,12,15,19,16,10,11,14,9,64,63,59,62,58,50,49,55,61,57,46,
%T A071668 34,54,45,36,35,32,40,31,41,43,52,60,56,47,48,53,44,27,26,29,33,30,38,39,51,42,24,25,28,37,
%U A071668 23,196,195,190,194,189,176,175,185,193,188,171,104,184,170,148,147,143,161,142,162,167,181
%N A071668 Permutation A057506 applied five times ("^5").
%R A071668
%H A071668 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071668 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071668 0,3
%K A071668 new,nonn
%A A071668 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071668 Inverse permutation: A071667, and also its car/cdr-flipped conjugate, i.e. A071668(n) = A057163(A071667(A057163(n))) = A057506(A071666(n)). Cf. also A071662, A071664, A071670.
%D A071668
%p A071668



%I A071669
%S A071669 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,
%T A071669 48,35,36,37,38,39,34,41,42,40,44,45,46,43,47,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,
%U A071669 64,65,66,67,68,69,70,71,72,73,74,75,132,77,78,79,80,81,90,83,84,82,86,87,88,85,89,91,92,93
%N A071669 Permutation A057505 applied six times ("^6"), permutation A071661 cubed, permutation A071663 squared.
%R A071669
%D A071669 R. Donaghey, Automorphisms on Catalan trees and bracketing, J. Combin. Theory, Series B, 29 (1980), 75-90.
%H A071669 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071669 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071669 0,3
%K A071669 new,nonn
%A A071669 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071669 Inverse permutation: A071670, and also its car/cdr-flipped conjugate, i.e. A071669(n) = A057163(A071670(A057163(n))) = A057505(A071667(n)) = A071661(A071661(A071661(n))) = A071663(A071663(n)). Cf. also A071665.
%D A071669
%p A071669


%I A071670
%S A071670 0,1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,20,21,22,23,24,25,26,27,28,29,30,31,32,33,
%T A071670 40,35,36,37,38,39,43,41,42,47,44,45,46,48,34,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,
%U A071670 64,65,66,67,68,69,70,71,72,73,74,75,110,77,78,79,80,81,85,83,84,89,86,87,88,90,82,91,92,93
%N A071670 Permutation A057506 applied six times ("^6"), permutation A071662 cubed, permutation A071664 squared.
%R A071670
%H A071670 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%H A071670 <a href="http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%O A071670 0,3
%K A071670 new,nonn
%A A071670 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071670 Inverse permutation: A071669, and also its car/cdr-flipped conjugate, i.e. A071670(n) = A057163(A071669(A057163(n))) = A057506(A071668(n)) = A071662(A071662(A071662(n))) = A071664(A071664(n)). Cf. also A071666.
%D A071670
%p A071670


%I A071671
%S A071671 10,1010,1100,101010,110010,110100,101100,11001010,11010010,111000,10101010,11001100,1101001010,
%T A071671 11100010,11010100,10110010,1100101010,1101001100,1110001010,1101010010,11100100,10110100,1100110010,
%U A071671 110100101010,1110001100,110101001010,1110010010,11101000,10101100,1100110100,110100110010
%N A071671 The binary encoding of parenthesizations given in a "global arithmetic order", using A061579 as the packing bijection NxN -> N.
%R A071671
%H A071671 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%O A071671 1,1
%K A071671 new,nonn,tabl
%A A071671 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071671 A071671(n) = A063171(A071652(n)). Permutation of A063171 and A071672. Particularly, applying the automorphism ReflectBinTree (A057163) to A071671(n) yields A071672(n). The length of each term / 2: A071673
%D A071671
%p A071671


%I A071672
%S A071672 10,1100,1010,111000,110010,101100,110100,11100010,11001100,101010,11110000,11010010,1110001100,
%T A071672 11001010,10111000,11100100,1111000010,1101001100,1110001010,1100111000,10110010,11011000,1110010010,
%U A071672 111100001100,1101001010,111000111000,1100110010,10101100,11101000,1101100010,111001001100
%N A071672 The binary encoding of parenthesizations given in a "global arithmetic order", using A001477 as the packing bijection NxN -> N.
%R A071672
%H A071672 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%O A071672 1,1
%K A071672 new,nonn,tabl
%A A071672 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071672 A071672(n) = A063171(A071654(n)). Permutation of A063171 and A071671. Particularly, applying the automorphism ReflectBinTree (A057163) to A071672(n) yields A071671(n). The length of each term / 2: A071673
%D A071672
%p A071672


[KEEP ALL 120 TERMS (i.e. full 15 rows) for the next two ones IF POSSIBLE:]


%I A071673
%S A071673 1,2,2,3,3,3,3,4,4,3,4,4,5,4,4,4,5,5,5,5,4,4,5,6,5,6,5,4,4,5,6,6,6,6,5,4,5,5,6,6,
%T A071673 7,6,6,5,5,5,6,6,6,7,7,6,6,6,5,4,6,7,6,7,7,7,6,7,6,4,5,5,7,7,7,7,7,7,7,7,5,5,5,6,
%U A071673 6,7,8,7,7,7,8,7,6,6,5,6,6,7,6,8,8,7,7,8,8,6,7,6,6,5,7,7,7,7,8,8,7,8,8,7,7,7,7,5
%N A071673 The size of each Catalan structure encoded by the corresponding terms in triangles A071671 & A071672. (i.e. the number of digits / 2)
%C A071673 Each value y (>= 1) occurs A000108(y) times.
%R A071673
%H A071673 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%O A071673 1,2
%K A071673 new,nonn,tabl
%A A071673 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%Y A071673 Same triangle computed modulo 2: A071674.
%D A071673
%p A071673


%I A071674
%S A071674 1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,0,1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0,0,0,1,0,1,1,0,0,
%T A071674 1,0,0,1,1,1,0,0,0,1,1,0,0,0,1,0,0,1,0,1,1,1,0,1,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,0,
%U A071674 0,1,0,1,1,1,0,1,0,0,1,0,0,1,0,0,0,1,1,0,0,0,1,0,0,1,1,1,1,1,0,0,1,0,0,1,1,1,1,1
%N A071674 Triangle A071673 computed modulo 2.
%R A071674
%H A071674 A. Karttunen, <A HREF="http://www.megabaud.fi/~karttu/matikka/Nekomorphisms/gatomorf.htm">Gatomorphisms</A> <I>(Includes the complete Scheme source for computing this sequence)</I>
%O A071674 1,1
%K A071674 new,nonn,tabl
%A A071674 Antti.Karttunen (my_firstname.my_surname@iki.fi) May 30 2002
%D A071674
%p A071674


----------------------------------------------------------------

Yours,

Antti Karttunen