Cheers,
15 new permutations of natural numbers (all induced
by various "Catalan" automorphisms),
namely: A069766-A069776, A069787, A069888-A069889, A070041
------------------------------------------------------------------------
First, here's an edition of the index entry:
http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation
That is, I transferred all "Catalan automorphism" -induced
permutations to their own subheading, called:
"permutations, of the integers, induced by Catalan automorphisms"
So here is the list of current "permutations ..." index entry, sans all
combinatorially Catalan related (that I know) permutations:
(so there are now some lines with "holes" nibbed from them,
just tell me if you want them rearranged so that they are
evenly justified.)
permutations, of the positive (or nonnegative) integers , (start):
permutations, of the integers, self-inverse: (1) A000027 A002251 A003100
A019444 A026243 A054429 A054430 A064429 A064505 A064614 A004442 A011262
permutations, of the integers, self-inverse: (2) A020703 A038722 A061579
A014681 A065190 A059893 A056023 A056011 A048647
permutations, of the integers, self-inverse: (3) A026239
permutations, of the integers, conjectured: A064389
permutations, of the integers, each paired with its inverse: ( 1)
A003188-A006068 A004484-A064206 A004485-A064207 A004486-A064208
A004487-A064211 A029654-A064360
permutations, of the integers, each paired with its inverse: ( 2)
A064413-A064664 A032447-A064275 A035312-A035358 A035506-A064357
A035513-A064274 A047708-A048850
permutations, of the integers, each paired with its inverse: ( 3)
A048672-A064273 A048673-A064216 A048679-A048680 A052330-A064358
A059900-A059884
permutations, of the integers, each paired with its inverse: ( 4)
A052331-A064359 A054238-A054239 A054424-A054426 A054427-A054428
A064706-A064707 A034175-A064928
permutations, of the integers, each paired with its inverse: ( 5)
A064929-A064930 A057027-A064578 A054082-A064579 A065164-A065168
A065165-A065169 A065166-A065170
permutations, of the integers, each paired with its inverse: ( 6)
A065171-A065172 A065174-A065175 A065181-A065182 A065183-A065184
A065186-A065187 A065188-A065189
permutations, of the integers, each paired with its inverse: ( 7)
A006368-A006369 A057114-A057115 A054084-A064786 A053212-A064787
permutations, of the integers, each paired with its inverse: ( 8)
A060736-A064788 A054068-A054069 A057028-A064789
permutations, of the integers, each paired with its inverse: ( 9)
A060734-A064790 A064537-A064791 A064736-A064745
A065249-A065250 A065259-A065260
permutations, of the integers, each paired with its inverse: (10)
A065263-A065264 A065265-A065266 A065269-A065270 A065271-A065272
A065275-A065276 A065277-A065278
permutations, of the integers, each paired with its inverse: (11)
A065281-A065282 A065283-A065284 A065287-A065288 A065289-A065290
A065253-A065254 A065306-A065307
permutations, of the integers, each paired with its inverse: (12)
A004515-A065256 A065257-A065258 A064417-A064956 A064418-A064958
A064419-A064959 A036552-A065037
permutations, of the integers, each paired with its inverse: (13)
A065649-A065650 A065627-A065628 A065629-A065630 A065631-A065632
A065633-A065634 A065635-A065636
permutations, of the integers, each paired with its inverse: (14)
A065637-A065638 A065639-A065640 A065660-A065661 A065662-A065663
A065664-A065665 A065666-A065667
permutations, of the integers, each paired with its inverse: (15)
A065668-A065669 A065670-A065671 A065672-A065673 A065561-A065578
A065562-A065579 A065934-A065935
permutations, of the integers, each paired with its inverse: (16)
A066248-A066249 A066250-A066251 A067587-A066884 A068225-A068226
THEN THE NEW SUBSECTION, AFTER THESE:
permutations, of the integers, induced by Catalan automorphisms, involutions:
A057163 A057164 A057508 A069766 A069769 A069770 A069771 A069772 A069787 A069888 A069889
permutations, of the integers, induced by Catalan automorphisms, each paired with its inverse:
A038776-A070041 A057117-A057118 A057161-A057162 A057501-A057502 A057503-A057504 A057505-A057506
A057509-A057510 A057511-A057512 A069767-A069768 A069773-A069774 A069775-A069776
---------------------------------------------------------------------------
Then additions to the sequences A038776, A057501, A057502, A057508, A057509, A057510 & A061855
seq: 1,2,4,5,3,9,10,13,14,12,6,7,8,11,23,24,27,28,26,36,37,41,42,40,32,
%Y A038776 Inverse of A070041. Cf. also A038774, A038775. If "expanded" produces A057117. Max cycle lengths: A057542.
%H A038776 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
seq: 1,3,2,7,8,5,4,6,17,18,20,21,22,12,13,10,9,11,15,14,16,19,45,46,48,49,50,54,55,57,58,59
%H A057501 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
REPLACE THE CURRENT Y-lines with these two lines:
%Y A057501 Inverse of A057502 and the car/cdr-flipped conjugate of A069773, i.e. A057501(n) = A057163(A069773(A057163(n))).
%Y A057501 Max cycle lengths: A057543. Cf. A057503, A057505, A057508, A057509, A057511, A057517, A057161, A069771, A069772.
%o A057501 (Scheme function implementing this automorphism on list-structures:) (define (RotateHandshakes a) (cond ((null? a) (list)) (else (append (car a) (list (cdr a))))))
%o A057501 (Destructive variant:) (define (RotateHandshakes! s) (cond ((not (pair? s))) ((not (pair? (car s))) (swap! s)) (else (robr! s) (RotateHandshakes! (cdr s)))) s)
%o A057501 (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 A057501 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
seq: 1,3,2,7,6,8,4,5,17,16,18,14,15,20,19,21,9,10,22,11,12,13,45,44,46,42,43,48,47,49,37,38
%H A057502 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%Y A057502 Inverse of A057501 and the car/cdr-flipped conjugate of A069774, i.e. A057502(n) = A057163(A069774(A057163(n))). Cf. also A057507, A057510, A057513, A069771, A069772.
%o A057502 (Scheme function implementing this automorphism on list-structures:) (define (RotateHandshakesInv! s) (cond ((not (pair? s))) ((not (pair? (cdr s))) (swap! s)) (else (RotateHandshakesInv! (cdr s)) (robl! s))) s)
%o A057502 (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 A057502 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
seq: 1,2,3,4,6,5,7,8,9,14,11,16,19,10,15,12,17,18,13,20,21,22,23,37,28,42,51,25,39,30,44,47
%H A057508 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%Y A057508 The car/cdr-flipped conjugate of A069769, i.e. A057508(n) = A057163(A069769(A057163(n))). Cf also A057164, A057509, A057510, A033538.
%o A057508 (Scheme function implementing this automorphism on list-structures:) reverse
%o A057508 (Destructive variant, see A057509 for Rol!) (define (Rev1! s) (cond ((pair? s) (Rev1! (cdr s)) (Rol! s))) s)
%o A057508 (Another variant, see A057510 for Ror!) (define (Rev2! s) (cond ((pair? s) (Ror! s) (Rev2! (cdr s)))) s)
seq: 1,2,3,4,6,5,7,8,9,11,14,16,19,10,15,12,17,18,13,20,21,22,23,25,28,30,33,37,39,42,44,47
%H A057509 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%Y A057509 Inverse of A057510 and the car/cdr-flipped conjugate of A069775, and also composition of A069770 & A057501, i.e. A057509(n) = A057163(A069775(A057163(n))) = A057501(A069770(n)).
%Y A057509 Cycle counts given by A003239. Cf. also A057511.
%o A057509 (Scheme function implementing this automorphism on list-structures:) (define (Rol s) (cond ((not (pair? s)) s) (else (append (cdr s) (list (car s))))))
%o A057509 (Destructive variant, see A057501 for RotateHandshakes! and swap!) (define (Rol! s) (cond ((pair? s) (swap! s) (RotateHandshakes! s))) s)
seq: 1,2,3,4,6,5,7,8,9,14,10,16,19,11,15,12,17,18,13,20,21,22,23,37,24,42,51,25,38,26,44,47
%H A057510 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%Y A057510 Inverse of A057509 and the car/cdr-flipped conjugate of A069776, and also composition of A057502 & A069770, i.e. A057510(n) = A057163(A069776(A057163(n))) = A069770(A057502(n)).
%Y A057510 Cycle counts given by A003239. Cf. also A057512, A057513.
%o A057510 (Scheme function implementing this automorphism on list-structures, see A057502 for RotateHandshakes! and swap!:) (define (Ror! s) (cond ((pair? s) (RotateHandshakesInv! s) (swap! s))) s)
seq: 0,2,10,12,42,52,56,170,178,204,212,232,240,682,722,738,812,
%Y A061855 Obtained by "reflecting" the terms of A061854. Cf. also A035928 (ReflectBinSeq), A061856, A069766.
---------------------------------------------------------------------------
Then, complete re-edits of A057161 - A057164
(now I use only the term "triangularization of polygon"
in the first two, not "rotation of binary trees",
because the latter has wholly other established meaning).
%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,
%T A057161 49,50,54,55,57,58,59,61,62,63,64,31,32,34,35,36,40,41,43,44,47,52,53,
%U A057161 56,60,26,27,29,30,33,38,39,42,51,24,25,28,37,23,129,130,132,133,134
%N A057161 Permutation of natural numbers induced by clockwise rotation of the triangularizations of polygons encoded by A014486.
%H A057161 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A057161 Illustration of triangulations of polygons.
%H A057161 Index entries for sequences that are permutations of the natural numbers
%p A057161 a(n) = CatalanRankGlobal(RotateTriangularization(A014486[n]))
%p A057161 CatalanRankGlobal given in A057117 and the other Maple procedures in A038776.
%p A057161 NextSubBinTree := proc(nn) local n,z,c; n := nn; c := 0; z := 0; while(c < 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); od; RETURN(z); end;
%p A057161 BinTreeLeftBranch := n -> NextSubBinTree(floor(n/2));
%p A057161 BinTreeRightBranch := n -> NextSubBinTree(floor(n/(2^(1+binwidth(BinTreeLeftBranch(n))))));
%p A057161 RotateTriangularization := proc(nn) local n,s,z,w; n := binrev(nn); z := 0; w := 0; while(1 = (n mod 2)) do s := BinTreeRightBranch(n); z := z + (2^w)*s; w := w + binwidth(s); z := z + (2^w); w := w + 1; n := floor(n/2); od; RETURN(z); end;
%Y A057161 Inverse of A057162 and also its car/cdr-flipped conjugate, composition of A069769 & A069767, i.e. A057161(n) = A057163(A057162(A057163(n))) = A069767(A069769(n))
%Y A057161 Cf. also A057163, A057164, A057501, A057505. Max cycle lengths: A057544.
%K A057161 nonn
%O A057161 0,3
%A A057161 Antti Karttunen (my_firstname.my_surname@iki.fi) Aug 18 2000
%o A057161 (Scheme function implementing this automorphism on list-structures:) (define (RotateTriangularization bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (car lt) (cons (cdr lt) nt))))))
%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,
%T A057162 51,52,62,53,54,37,38,55,39,40,41,63,56,57,42,43,58,44,45,23,24,46,25,
%U A057162 26,27,59,47,48,28,29,49,30,31,32,50,33,34,35,36,196,191,192,177,178
%N A057162 Permutation of natural numbers induced by counter-clockwise rotation of the triangularizations of polygons encoded by A014486.
%C A057162 In A057161 and A057162, the cycles between A014138[n-1]+1-th and A014138[n]th term partition A000108[n] objects encoded by the corresponding terms of A014486 into A001683[n+2] equivalence classes of flexagons (or unlabeled plane boron trees), thus the latter sequence can be produced also with the Maple procedure RotBinTreePermutationCycleCounts given below.
%H A057162 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A057162 Index entries for sequences that are permutations of the natural numbers
%p A057162 a(n) = CatalanRankGlobal(RotateTriangularizationR(A014486[n]))
%p A057162 RotateTriangularizationR := n -> ReflectBinTree(RotateTriangularization(ReflectBinTree(n)));
%p A057162 with(group); RotBinTreePermutationCycleCounts := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,RotateTriangularization(CatalanUnrank(n,r)))]; od; a := [op(a),(`if`((n < 2),1,nops(convert(b,'disjcyc'))))]; od; RETURN(a); end;
%Y A057162 Inverse of A057161 and also its car/cdr-flipped conjugate, composition of A057508 & A069768, i.e. A057162(n) = A057163(A057161(A057163(n))) = A069768(A057508(n))
%K A057162 nonn
%O A057162 0,3
%A A057162 Antti Karttunen (my_firstname.my_surname@iki.fi) Aug 18 2000
%o A057162 (Scheme function implementing this automorphism on list-structures:) (define (RotateTriangularizationInv bt) (let loop ((lt bt) (nt (list))) (cond ((not (pair? lt)) nt) (else (loop (cdr lt) (cons nt (car lt)))))))
%I A057163
%S A057163 0,1,3,2,8,7,6,5,4,22,21,20,18,17,19,16,15,13,12,14,11,10,9,64,63,62,
%T A057163 59,58,61,57,55,50,49,54,48,46,45,60,56,53,47,44,52,43,41,36,35,40,34,
%U A057163 32,31,51,42,39,33,30,38,29,27,26,37,28,25,24,23,196,195,194,190,189
%N A057163 Self-inverse permutation of natural numbers:induced by reflections of the rooted plane binary trees and triangularizations of polygons encoded by A014486.
%C A057163 Each A000108[n] n+2 side polygon triangularizations (and the corresponding rooted binary plane trees of 2n edges) can be reflected over n+2 axes of symmetry, which all can be generated by appropriate compositions of the permutations A057161/A057162 and A057163.
%H A057163 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A057163 Index entries for sequences that are permutations of the natural numbers
%e A057163 Example: a(5)=7 and a(7)=5, A014486[5] = 44 (101100 in binary), A014486[7] = 52 (110100 in binary) and these encode the following rooted plane binary trees, which are reflections of each other:
%e A057163 .....0.0...............0.0....
%e A057163 ......1...0.........0...1.....
%e A057163 ..0.....1.............1.....0.
%e A057163 .....1...................1....
%p A057163 a(n) = CatalanRankGlobal(ReflectBinTree(A014486[n])) [For other Maple procedures, follow A057161.]
%p A057163 ReflectBinTree := n -> ReflectBinTree2(n)/2; ReflectBinTree2 := n -> (`if`((0 = n), n,ReflectBinTreeAux(binrev(n))));
%p A057163 ReflectBinTreeAux := proc(n) local a,b; a := ReflectBinTree2(BinTreeLeftBranch(n)); b := ReflectBinTree2(BinTreeRightBranch(n)); RETURN((2^(binwidth(b)+binwidth(a))) + (b * (2^(binwidth(a)))) + a); end;
%Y A057163 A057123(A057163(n)) = A057164(A057123(n)) for all n.
%Y A057163 Conjugates between the car/cdr-flipped variants of other automorphisms, e.g. A057162(n) = a(A057161(a(n))), A069768(n) = a(A069767(a(n))), A069769(n) = a(A057508(a(n))), A069773(n) = a(A057501(a(n))), A069774(n) = a(A057502(a(n))), A069775(n) = a(A057509(a(n))), A069776(n) = a(A057510(a(n))), A069787(n) = a(A057164(a(n))), where a(n) = A057163(n).
%K A057163 nonn
%O A057163 0,3
%A A057163 Antti Karttunen (my_firstname.my_surname@iki.fi) Aug 18 2000
%o A057163 (Scheme function implementing this automorphism on list-structures:) (define (ReflectBinTree bt) (cond ((not (pair? bt)) bt) (else (cons (ReflectBinTree (cdr bt)) (ReflectBinTree (car bt))))))
%o A057163 (Destructive variant:) (define (ReflectBinTree! s) (cond ((pair? s) (swap! s) (ReflectBinTree! (car s)) (ReflectBinTree! (cdr s)))) s)
%o A057163 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
%I A057164
%S A057164 0,1,2,3,4,6,5,7,8,9,14,11,16,19,10,15,12,17,20,13,18,21,22,23,37,28,
%T A057164 42,51,25,39,30,44,53,33,47,56,60,24,38,29,43,52,26,40,31,45,54,34,48,
%U A057164 57,61,27,41,32,46,55,35,49,58,62,36,50,59,63,64,65,107,79,121,149,70
%N A057164 Self-inverse permutation of natural numbers induced by reflections of the rooted plane trees and mountain ranges encoded by A014486.
%C A057164 CatalanRankGlobal given in A057117 and the other Maple procedures in A056539.
%H A057164 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A057164 Index entries for sequences that are permutations of the natural numbers
%e A057164 a(10)=14 and a(14)=10, A014486[10] = 172 (10101100 in binary), A014486[14] = 202 (11001010 in binary) and these encode the following mountain ranges (and the corresponding rooted plane trees), which are reflections of each other:
%e A057164 ...../\___________/\
%e A057164 /\/\/__\_________/__\/\/\
%e A057164 .
%e A057164 ...../...........\
%e A057164 ..\|/.............\|/
%p A057164 a(n) = CatalanRankGlobal(runcounts2binexp(reverse(binexp2runcounts(A014486[n])))) [i.e. reverse and complement the totally balanced binary sequences]
%Y A057164 A057123[A057163[n]] = A057164[A057123[n]] for all n. Also the car/cdr-flipped conjugate of A069787, i.e. A057164(n) = A057163(A069787(A057163(n))). Fixed terms are given by A061856. Cf. also A057508, A069772.
%K A057164 nonn
%O A057164 0,3
%A A057164 Antti Karttunen (my_firstname.my_surname@iki.fi) Aug 18 2000
%o A057164 (Scheme function implementing this automorphism on list-structures:) (define (DeepRev lista) (cond ((not (pair? lista)) lista) ((null? (cdr lista)) (cons (DeepRev (car lista)) (list))) (else (append (DeepRev (cdr lista)) (DeepRev (cons (car lista) (list)))))))
---------------------------------------------------------------------------
Then 15 new sequences A069766-A069776, A069787, A069888-A069889, A070041.
%I A069766
%S A069766 0,1,2,3,5,4,6,7,11,9,10,8,12,17,14,21,19,13,20,16,18,15,22,23,37,29,33,27,41,25,39,31,32,26,
%T A069766 40,35,36,24,38,30,34,28,42,57,47,72,64,44,68,54,61,51,76,59,49,74,66,43,67,53,60,50,75,70,
%U A069766 46,71,56,58,48,73,63,65,45,69,55,62,52,77,78,126,98,112,92,142,84,134,104,108,88,138,118,122
%N A069766 Self-inverse permutation of natural numbers induced by the automorphism RotateHandshakes180 (A069771) or xReflectHandshakes (A069772) acting on the symmetric parenthesizations encoded by A061855.
%R A069766
%H A069766 Index entries for sequences that are permutations of the natural numbers
%O A069766 0,3
%K A069766 nonn
%A A069766 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069766 Cf. A061855, A069771, A069772.
%D A069766
%p A069766
%I A069767
%S A069767 0,1,3,2,7,8,6,5,4,17,18,20,21,22,16,19,15,12,13,14,11,10,9,45,46,48,49,50,54,55,57,58,59,61,
%T A069767 62,63,64,44,47,53,56,60,43,52,40,31,32,41,34,35,36,42,51,39,30,33,38,29,26,27,37,28,25,24,
%U A069767 23,129,130,132,133,134,138,139,141,142,143,145,146,147,148,157,158,160,161,162,166,167,169
%N A069767 Permutation of natural numbers induced by the automorphism SwapDownCar! acting on the rooted planar binary trees encoded by A014486. (I.e. flip all the branches on the leftmost path of the tree)
%R A069767
%H A069767 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069767 Index entries for sequences that are permutations of the natural numbers
%O A069767 0,3
%K A069767 nonn
%A A069767 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069767 Inverse permutation: A069768. Also its car/cdr-flipped conjugate, i.e. A069767(n) = A057163(A069768(A057163(n))). Cf. also A057161.
%D A069767
%o A069767 (Scheme function implementing this automorphism on list-structures:) (define (SwapDownCar! s) (cond ((pair? s) (swap! s) (SwapDownCar! (cdr s)))) s)
%o A069767 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
%I A069768
%S A069768 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,
%T A069768 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,
%U A069768 36,196,195,194,189,190,193,188,184,170,171,185,174,175,176,192,187,183,169,173,180,166,157
%N A069768 Permutation of natural numbers induced by the automorphism SwapDownCdr! acting on the rooted planar binary trees encoded by A014486. (I.e. flip all the branches on the rightmost path of the tree)
%R A069768
%H A069768 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069768 Index entries for sequences that are permutations of the natural numbers
%O A069768 0,3
%K A069768 nonn
%A A069768 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%D A069768
%Y A069768 Inverse permutation: A069767. Also its car/cdr-flipped conjugate, i.e. A069768(n) = A057163(A069767(A057163(n))). Cf. also A057162.
%o A069768 (Scheme function implementing this automorphism on list-structures:) (define (SwapDownCdr! s) (cond ((pair? s) (swap! s) (SwapDownCdr! (car s)))) s)
%I A069769
%S A069769 0,1,2,3,4,5,7,6,8,9,10,11,12,13,17,18,16,14,15,21,20,19,22,23,24,25,26,27,28,29,30,31,32,33,
%T A069769 34,35,36,45,46,48,49,50,44,47,42,37,38,43,39,40,41,58,59,57,54,55,56,53,51,52,63,62,61,60,
%U A069769 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93
%N A069769 Self-inverse permutation of natural numbers induced by the automorphism Rev1CarSide! acting on the parenthesizations encoded by A014486.
%R A069769
%H A069769 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069769 Index entries for sequences that are permutations of the natural numbers
%O A069769 0,3
%K A069769 nonn
%A A069769 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069769 The car/cdr-flipped conjugate of A057508, i.e. A069769(n) = A057163(A057508(A057163(n))). Cf. also A069787, A057161.
%D A069769
%I A069770
%S A069770 0,1,3,2,7,8,6,4,5,17,18,20,21,22,16,19,14,9,10,15,11,12,13,45,46,48,49,50,54,55,57,58,59,61,
%T A069770 62,63,64,44,47,53,56,60,42,51,37,23,24,38,25,26,27,43,52,39,28,29,40,30,31,32,41,33,34,35,
%U A069770 36,129,130,132,133,134,138,139,141,142,143,145,146,147,148,157,158,160,161,162,166,167,169
%N A069770 Self-inverse permutation of natural numbers induced by the automorphism SwapBinTree! acting on the rooted planar binary trees encoded by A014486. (I.e. swap the left and right subtrees at the root)
%R A069770
%H A069770 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069770 Index entries for sequences that are permutations of the natural numbers
%O A069770 0,3
%K A069770 nonn
%A A069770 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069770 Its own car/cdr-flipped conjugate, i.e. A069770(n) = A057163(A069770(A057163(n)))
%D A069770
%o A069770 (Scheme function implementing this automorphism on list-structures:) (define (SwapBinTree bt) (cond ((not (pair? bt)) bt) (else (cons (cdr bt) (car bt)))))
%o A069770 (Destructive variant:) (define (SwapBinTree! s) (cond ((pair? s) (swap! s))) s)
%o A069770 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
%I A069771
%S A069771 0,1,2,3,7,5,6,4,8,9,14,21,18,13,10,15,20,17,12,19,16,11,22,45,54,31,26,40,44,53,30,25,39,63,
%T A069771 59,50,36,46,55,32,27,41,42,51,28,23,37,62,58,49,35,43,52,29,24,38,61,57,48,34,60,56,47,33,
%U A069771 64,65,79,107,121,149,170,184,142,133,161,100,91,77,119,66,80,108,122,150,169,183,141,132,160
%N A069771 Self-inverse permutation of natural numbers induced by the automorphism RotateHandshakes180 acting on the parenthesizations encoded by A014486.
%C A069771 This automorphism rotates by 180 degrees the intepretation n (the non-crossing handshakes) of Stanley's exercise 19.
%R A069771
%H A069771 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069771 R. P. Stanley, Exercises on Catalan and Related Numbers
%H A069771 Index entries for sequences that are permutations of the natural numbers
%O A069771 0,3
%K A069771 nonn
%Y A069771 Cf. A057501, A069772, A069888, A069889.
%A A069771 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%D A069771
%o A069771 (Scheme function implementing this automorphism on list-structures:) (define (RotateHandshakes180 a) (RotateHandshakes_n_steps a (count-pars a)))
%o A069771 (define (count-pars a) (if (not (pair? a)) 0 (+ 1 (count-pars (car a)) (count-pars (cdr a)))))
%o A069771 (define (RotateHandshakes a) (if (null? a) (list) (append (car a) (list (cdr a)))))
%o A069771 (define (RotateHandshakes_n_steps a n) (if (zero? n) a (RotateHandshakes_n_steps (RotateHandshakes a) (-1+ n))))
%I A069772
%S A069772 0,1,2,3,7,6,5,4,8,9,10,21,20,19,14,15,18,17,16,13,12,11,22,45,46,44,42,43,31,32,30,28,29,63,
%T A069772 62,61,60,54,55,53,51,52,26,27,25,23,24,59,58,57,56,40,41,39,37,38,50,49,48,47,36,35,34,33,
%U A069772 64,65,67,66,68,69,170,171,169,166,167,168,165,163,164,107,109,108,110,111,142,143,141,138
%N A069772 Self-inverse permutation of natural numbers induced by the automorphism xReflectHandshakes acting on the parenthesizations encoded by A014486.
%C A069772 This automorphism reflects over the x-axis the intepretation n (the non-crossing handshakes) of Stanley's exercise 19.
%C A069772 Note that DeepRev (A057164) reflects over y-axis.
%C A069772 This transformation keeps palindromic parenthesizations/Dyck paths/rooted planar trees palindromic, but not necessarily same, meaning that this induces a permutation on the sequence A061855 (= A069766).
%R A069772
%H A069772 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069772 R. P. Stanley, Exercises on Catalan and Related Numbers
%H A069772 Index entries for sequences that are permutations of the natural numbers
%O A069772 0,3
%K A069772 nonn
%A A069772 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069772 Composition of A057164 and A069771 in either order, i.e. A069772(n) = A057164(A069771(n)) = A069771(A057164(n)). Cf. also A061855, A069766, A057501, A069888, A069889.
%D A069772
%o A069772 (Scheme function implementing this automorphism on list-structures:) (define (xReflectHandshakes a) (DeepRev (RotateHandshakes180 a)))
%o A069772 (define (DeepRev lista) (cond ((not (pair? lista)) lista) ((null? (cdr lista)) (cons (DeepRev (car lista)) (list))) (else (append (DeepRev (cdr lista)) (DeepRev (cons (car lista) (list)))))))
%I A069773
%S A069773 0,1,3,2,6,8,7,4,5,14,15,19,20,22,16,21,17,9,10,18,11,12,13,37,38,39,40,41,51,52,53,54,55,60,
%T A069773 61,62,64,42,43,56,57,63,44,58,45,23,24,46,25,26,27,47,59,48,28,29,49,30,31,32,50,33,34,35,
%U A069773 36,107,108,109,110,111,112,113,114,115,116,117,118,119,120,149,150,151,152,153,154,155,156
%N A069773 Permutation of natural numbers induced by the automorphism RoblDownCar_et_Swap! acting on the parenthesizations encoded by A014486.
%R A069773
%H A069773 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069773 Index entries for sequences that are permutations of the natural numbers
%O A069773 0,3
%K A069773 nonn
%A A069773 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069773 Inverse of A069774, the car/cdr-flipped conjugate of A057501, i.e. A069773(n) = A057163(A057501(A057163(n))). Cf also A069775.
%D A069773
%o A069773 (Scheme function implementing this automorphism on list-structures:) (define (RoblDownCar_et_Swap! s) (cond ((not (pair? s))) ((not (pair? (cdr s))) (swap! s)) (else (robl! s) (RoblDownCar_et_Swap! (car s)))) s)
%o A069773 (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 A069773 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
%I A069774
%S A069774 0,1,3,2,7,8,4,6,5,17,18,20,21,22,9,10,14,16,19,11,12,15,13,45,46,48,49,50,54,55,57,58,59,61,
%T A069774 62,63,64,23,24,25,26,27,37,38,42,44,47,51,53,56,60,28,29,30,31,32,39,40,43,52,33,34,35,41,
%U A069774 36,129,130,132,133,134,138,139,141,142,143,145,146,147,148,157,158,160,161,162,166,167,169
%N A069774 Permutation of natural numbers induced by the automorphism RoblDownCar_et_SwapInv! acting on the parenthesizations encoded by A014486.
%R A069774
%H A069774 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069774 Index entries for sequences that are permutations of the natural numbers
%O A069774 0,3
%K A069774 nonn
%A A069774 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069774 Inverse of A069773, the car/cdr-flipped conjugate of A057502, i.e. A069774(n) = A057163(A057502(A057163(n))). Cf also A069776.
%D A069774
%o A069774 (Scheme function implementing this automorphism on list-structures:) (define (RoblDownCar_et_SwapInv! s) (cond ((not (pair? s))) ((not (pair? (car s))) (swap! s)) (else (RoblDownCar_et_SwapInv! (car s)) (robr! s))) s)
%o A069774 (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 A069774 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
%I A069775
%S A069775 0,1,2,3,4,5,7,6,8,9,10,11,12,13,17,18,16,14,15,21,19,20,22,23,24,25,26,27,28,29,30,31,32,33,
%T A069775 34,35,36,45,46,48,49,50,44,47,42,37,38,43,39,40,41,58,59,56,51,52,57,53,54,55,63,60,61,62,
%U A069775 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93
%N A069775 Permutation of natural numbers induced by the automorphism RolCarSide! acting on the parenthesizations encoded by A014486.
%R A069775
%H A069775 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069775 Index entries for sequences that are permutations of the natural numbers
%O A069775 0,3
%K A069775 nonn
%A A069775 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069775 Inverse of A069776, the car/cdr-flipped conjugate of A057509 and composition of A069770 & A069773, i.e. A069775(n) = A057163(A057509(A057163(n))) = A069773(A069770(n)). Cf. also A069787
%D A069775
%I A069776
%S A069776 0,1,2,3,4,5,7,6,8,9,10,11,12,13,17,18,16,14,15,20,21,19,22,23,24,25,26,27,28,29,30,31,32,33,
%T A069776 34,35,36,45,46,48,49,50,44,47,42,37,38,43,39,40,41,54,55,57,58,59,53,56,51,52,61,62,63,60,
%U A069776 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93
%N A069776 Permutation of natural numbers induced by the automorphism RorCarSide! acting on the parenthesizations encoded by A014486.
%R A069776
%H A069776 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069776 Index entries for sequences that are permutations of the natural numbers
%O A069776 0,3
%K A069776 nonn
%A A069776 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069776 Inverse of A069775, the car/cdr-flipped conjugate of A057510 and composition of A069774 & A069770, i.e. A069776(n) = A057163(A057510(A057163(n))) = A069770(A069774(n)).
%D A069776
%I A069787
%S A069787 0,1,2,3,4,5,7,6,8,9,10,12,11,13,17,18,16,14,15,21,20,19,22,23,24,26,25,27,31,32,30,28,29,35,
%T A069787 34,33,36,45,46,49,48,50,44,47,42,37,38,43,40,39,41,58,59,57,54,55,56,53,51,52,63,62,61,60,
%U A069787 64,65,66,68,67,69,73,74,72,70,71,77,76,75,78,87,88,91,90,92,86,89,84,79,80,85,82,81,83,100
%N A069787 Self-inverse permutation of natural numbers induced by the automorphism DeepRev1CarSide! acting on the parenthesizations encoded by A014486.
%R A069787
%H A069787 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069787 Index entries for sequences that are permutations of the natural numbers
%O A069787 0,3
%K A069787 nonn
%A A069787 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069787 The car/cdr-flipped conjugate of A057164, i.e. A069787(n) = A057163(A057164(A057163(n))).
%D A069787
%I A069888
%S A069888 0,1,3,2,7,5,8,4,6,17,12,20,10,15,18,13,21,9,14,22,11,16,19,45,31,54,26,40,48,34,57,24,38,61,
%T A069888 29,43,52,46,32,55,27,41,49,35,58,23,37,62,28,42,51,50,36,59,25,39,63,30,44,53,64,33,47,56,
%U A069888 60,129,87,157,73,115,138,96,166,68,110,180,82,124,152,132,90,160,76,118,141,99,169,66,108
%N A069888 Self-inverse permutation of natural numbers induced by the automorphism DeepReverse_et_RotateHandshakes! acting on the parenthesizations encoded by A014486.
%C A069888 This automorphism reflects non-crossing handshakes (the intepretation n of Stanley's exercise 19) over the diagonal that goes through corner at "1 o'clock".
%R A069888
%H A069888 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069888 R. P. Stanley, Exercises on Catalan and Related Numbers
%H A069888 Index entries for sequences that are permutations of the natural numbers
%O A069888 0,3
%K A069888 nonn
%A A069888 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002, suggested by Wouter Meeussen Dec 2001.
%Y A069888 Composition of A057164 and A057501, i.e. A069888(n) = A057501(A057164(n)). Cf. also A069889.
%D A069888
%I A069889
%S A069889 0,1,3,2,7,8,6,4,5,17,20,18,21,22,16,19,14,9,11,15,10,12,13,45,54,48,57,61,46,55,49,58,62,50,
%T A069889 59,63,64,44,53,47,56,60,42,51,37,23,28,39,25,30,33,43,52,38,24,29,40,26,31,34,41,27,32,35,
%U A069889 36,129,157,138,166,180,132,160,141,169,183,145,173,187,192,130,158,139,167,181,133,161,142
%N A069889 Self-inverse permutation of natural numbers induced by the automorphism RotateHandshakes_et_DeepReverse! acting on the parenthesizations encoded by A014486.
%C A069889 This automorphism reflects non-crossing handshakes (the intepretation n of Stanley's exercise 19) over the diagonal that goes through corner at "11 o'clock".
%R A069889
%H A069889 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A069889 R. P. Stanley, Exercises on Catalan and Related Numbers
%H A069889 Index entries for sequences that are permutations of the natural numbers
%O A069889 0,3
%K A069889 nonn
%A A069889 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A069889 Composition of A057501 and A057164, i.e. A069888(n) = A057164(A057501(n)). Cf. also A069888.
%D A069889
%I A070041
%S A070041 1,2,5,3,4,11,12,13,6,7,14,10,8,9,29,30,33,31,32,36,37,34,15,16,35,19,17,18,40,41,42,25,26,
%T A070041 38,27,20,21,39,28,24,22,23,85,86,89,87,88,95,96,97,90,91,98,94,92,93,104,105,112,108,109,106,
%U A070041 110,99,43,44,100,47,45,46,107,111,103,53,54,101,55,48,49,102,56,52,50,51,118,119,126,122,123
%N A070041 Permutation of natural numbers induced by the automorphism df->bf (from the depth-first to breadth-first encoding) acting on the rooted planar binary trees encoded by A014486. (with one-based indexing).
%H A070041 A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)
%H A070041 Index entries for sequences that are permutations of the natural numbers
%R A070041
%O A070041 1,2
%K A070041 nonn
%A A070041 Antti.Karttunen (my_firstname.my_surname@iki.fi) Apr 16 2002
%Y A070041 Inverse of A038776. Cf. also A057118.
%D A070041
------------------------------------------------------------------------
Yours,
Antti Karttunen