First, to the end of: http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation to the section: permutations, of the integers, induced by Catalan rerankings, each paired with its inverse: A071651-A071652, A071653-A071654, A072634-A072635, A072636-A072637, A072656-A072657, A072658-A072659, A072646-A072647 (i.e. append onto the same line after A071651-A071652, A071653-A071654, and add the pair A072646-A072647 as a new line, whichever way it looks nice.) ---------------------------------------------------------------------------- And then 26 new PRE_NUMBERED sequences, A072634 - A072650 & A072654 - A072662 %I A072634 %S A072634 0,1,3,2,11,9,4,6,5,139,131,33,41,35,12,10,8,70,66,7,17,21,18,32907,32779,2051,2179,2059,161, %T A072634 137,129,8233,8201,43,515,547,521,140,132,34,42,36,16,14,72,16454,16390,68,1026,1090,1030,15, %U A072634 13,19,81,69,23,65,4117,4101,20,22,258,274,261,2147516555,2147483787,8388619,8421387,8388747 %N A072634 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using packA054238 as the packing bijection NxN -> N. %R A072634 %H A072634 A. Karttunen, Gatomorphisms %H A072634 Index entries for sequences that are permutations of the natural numbers %O A072634 0,3 %K A072634 nonn,new %A A072634 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %D A072634 %Y A072634 Inverse permutation: A072635. C.f. also A014486, A000695, A054238, A071651, A072636, A072646, A072656, A072658, A072644. %o A072634 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.) %o A072634 (define A072634 (lexrank->arithrank-bijection packA054238)) %o A072634 (define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun))) %o A072634 (define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun)))))) %o A072634 (define (packA054238 x y) (+ (A000695 x) (* 2 (A000695 y)))) %I A072635 %S A072635 0,1,3,2,6,8,7,19,16,5,15,4,14,52,43,51,42,20,22,53,60,21,61,56,179,155,178,154,177,164,557, %T A072635 163,556,11,39,13,41,151,123,153,125,12,40,33,117,152,124,471,381,477,553,479,555,505,1797, %U A072635 507,1799,478,554,1536,1791,506,1798,1626,5969,57,18,165,47,55,17,159,44,509,136,508,135,490 %N A072635 Inverse permutation of A072634. %R A072635 %H A072635 A. Karttunen, Gatomorphisms %H A072635 Index entries for sequences that are permutations of the natural numbers %O A072635 0,3 %K A072635 nonn,new %A A072635 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072635 A072644 gives the size of the corresponding parenthesizations, i.e. A072644(n) = A036377(A014486(A072635(n))+1)/2 [A036377(n+1) gives the binary width of n]. %Y A072635 A072635(n) = A057163(A072637(n)). C.f. also A014486, A059905, A059906, A071652, A072647, A072637. %D A072635 %o A072635 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.) %o A072635 (define A072635 (arithrank->lexrank-bijection A059905 A059906)) %o A072635 (define (arithrank->lexrank-bijection pr1 pr2) (lambda (n) (CatalanRankGlobal (parenthesization->binexp (unrank-bintree n pr1 pr2))))) %o A072635 (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 A072635 (define (A059905 n) (if (zero? n) n (+ (modulo n 2) (* 2 (A059905 (floor->exact (/ n 4))))))) %o A072635 (define (A059906 n) (A059905 (floor->exact (/ n 2)))) %I A072636 %S A072636 0,1,2,3,5,6,4,9,11,18,21,17,66,70,7,8,10,35,41,12,33,131,139,261,274,258,4101,4117,22,65,69, %T A072636 1030,1090,81,1026,16390,16454,20,23,19,68,72,13,14,36,521,547,42,515,8201,8233,15,16,34,43, %U A072636 129,132,137,2059,2179,140,161,2051,32779,32907,65554,65797,65541,16777234,16777490,277,4098 %N A072636 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using packA054238tr as the packing bijection NxN -> N. %R A072636 %H A072636 A. Karttunen, Gatomorphisms %H A072636 Index entries for sequences that are permutations of the natural numbers %O A072636 0,3 %K A072636 nonn,new %A A072636 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072636 Inverse permutation: A072637. C.f. also A014486, A000695, A054238, A071651, A072634, A072646, A072656, A072658, A072644. %D A072636 %o A072636 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.) %o A072636 (define A072636 (lexrank->arithrank-bijection packA054238tr)) %o A072636 (define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun))) %o A072636 (define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun)))))) %o A072636 (define (packA054238tr x y) (+ (A000695 y) (* 2 (A000695 x)))) %I A072637 %S A072637 0,1,2,3,6,4,5,14,15,7,16,8,19,42,43,51,52,11,9,39,37,10,28,38,112,123,121,151,149,122,376, %T A072637 150,466,20,53,17,44,154,155,126,127,18,47,54,156,135,136,480,481,477,475,387,385,476,1531, %U A072637 386,1234,415,413,1542,1540,414,1324,1541,5095,29,12,113,40,30,13,114,41,377,124,467,152,378 %N A072637 Inverse permutation of A072636. %R A072637 %H A072637 A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence) %H A072637 Index entries for sequences that are permutations of the natural numbers %O A072637 0,3 %K A072637 nonn,new %A A072637 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072637 A072644 gives the size of the corresponding parenthesizations, i.e. A072644(n) = A036377(A014486(A072637(n))+1)/2 [A036377(n+1) gives the binary width of n]. %Y A072637 A072637(n) = A057163(A072635(n)). C.f. also A014486, A059905, A059906, A071654, A072646. %D A072637 %o A072637 (Scheme code otherwise as for A072635, but with "transposed coordinates":) %o A072637 (define A072637 (arithrank->lexrank-bijection A059906 A059905)) %I A072638 %S A072638 0,1,3,10,66,2278,2598060,3374961778891,5695183504492614029263278,16217557574922386301420536972254869595782763547560, %T A072638 131504586847961235687181874578063117114329409897615188504091716162522225834932122128288032336298141 %N A072638 Maximum position in A071673 where the value n occurs. %R A072638 %O A072638 0,3 %K A072638 nonn,new %A A072638 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072638 Binary width of each term: A072641. Cf. A072639, A072640, A072654. %D A072638 %p A072638 %I A072639 %S A072639 0,1,3,11,139,32907,2147516555,9223372039002292363,170141183460469231740910675754886398091, %T A072639 57896044618658097711785492504343953926805133516280751251469702679711451218059,6703903964971298549787012499102923063739682910296196688861780721860882015036831384445555807181163499337520273047169830560393222157224754675896535954260107 %N A072639 a(0) = 0, a(n) = sum_{i=0}^{n-1} 2^((2^i)-1) %C A072639 Maximum position in A072644 where the value n occurs. %R A072639 %O A072639 0,3 %K A072639 nonn,new %A A072639 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072639 Binary width of each term: A000079. Cf. A072638, A072640, A072654. %D A072639 %p A072639 A072639 := proc(n) local i; add(2^((2^i)-1),i=0..(n-1)); end; %I A072640 %S A072640 0,1,3,11,100,2090,415729,15670134831,2990945138477441778,28553369883190773267638351857949, %T A072640 22025764139998888788032314837232820672093881927968648914 %N A072640 Maximum position in A072660 where the value n occurs. %R A072640 %O A072640 0,3 %K A072640 nonn,new %A A072640 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072640 Binary width of each term: A072642. Cf. A072638, A072639, A072654. %D A072640 %p A072640 %I A072641 %S A072641 0,1,2,4,7,12,22,42,83,164,326 %N A072641 Binary widths of the terms of A072638. %R A072641 %O A072641 0,3 %K A072641 nonn,new %A A072641 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072641 Cf. A072642, A072655. %D A072641 %p A072641 %I A072642 %S A072642 0,1,2,4,7,12,19,34,62,105,184 %N A072642 Binary widths of the terms of A072640. %R A072642 %O A072642 0,3 %K A072642 nonn,new %A A072642 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072642 Cf. A072641, A072655. %D A072642 %p A072642 %I A072643 %S A072643 0,1,2,2,3,3,3,3,3,4,4,4,4,4,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5, %T A072643 5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6, %U A072643 6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6,6 %N A072643 Half of the binary width of the terms of A014486, the number of digits in A063171(n)/2. %R A072643 %O A072643 0,3 %K A072643 nonn,new %A A072643 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072643 Each value v occurs A000108(v) times. The maximum position for value v to occur is A014138(v). Permutations: A071673, A072644, A072645, A072660. Cf. also A002024, A072649. %D A072643 %p A072643 %I A072644 %S A072644 0,1,2,2,3,3,3,4,4,3,4,3,4,5,5,5,5,4,4,5,5,4,5,5,6,6,6,6,6,6,7,6,7,4,5,4,5,6,6,6, %T A072644 6,4,5,5,6,6,6,7,7,7,7,7,7,7,8,7,8,7,7,8,8,7,8,8,9,5,4,6,5,5,4,6,5,7,6,7,6,7,6,7, %U A072644 6,5,6,6,7,6,6,7,7,7,8,7,8,8,8,8,8,8,7,8,7,8,7,8,7,8,7,9,8,8,7,9,8,8,9,8,9,9,9,9 %N A072644 The size of the parenthesizations obtained with the global ranking/unranking scheme A072634-A072637. %R A072644 %O A072644 0,3 %K A072644 nonn,new %A A072644 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072644 Cf. A072635 & A072637. A072644(n) = A036377(A014486(A072635(n))+1)/2 or = A036377(A014486(A072637(n))+1)/2 [A036377(n+1) gives the binary width of n]. %Y A072644 Each value v occurs A000108(v) times. The maximum position for value v to occur is A072639(v). Permutations: A071673, A072643, A072645, A072660. %D A072644 %p A072644 %I A072645 %S A072645 0,1,2,3,2,4,3,3,5,3,4,4,4,7,4,4,5,3,6,5,5,9,4,5,5,5,6,4,5,7,6,6,6,5,10,5,4,5,4,5, %T A072645 6,6,6,4,5,6,5,8,7,7,7,6,8,6,7,11,5,6,5,6,6,5,4,6,7,6,6,6,8,5,6,7,5,7,6,6,10,8,7,7, %U A072645 7,7,7,8,8,7,7,8,6,12,6,5,6,4,7,7,7,8,5,6,5,6,8,8,8,7,7,8,7,6,10,5,6,7,6,8,6,6,8,7 %N A072645 The size of the parenthesizations obtained with the global ranking/unranking scheme A072646/A072647. %R A072645 %O A072645 0,3 %K A072645 nonn,new %A A072645 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072645 Each value v occurs A000108(v) times. The maximum position for value v to occur is A072654(v). Permutations: A071673, A072643, A072644, A072660. %D A072645 %p A072645 %I A072646 %S A072646 0,1,2,4,6,7,3,9,17,15,27,14,36,62,12,5,10,38,43,11,22,93,161,150,159,91,993,1624,28,35,61, %T A072646 384,413,69,235,2591,4243,16,46,25,37,39,19,20,24,242,415,44,237,1606,2627,33,8,23,72,56,58, %U A072646 98,1004,1080,100,111,614,6782,11107,6780,11096,6771,317826,514388,260,612,1002,75052,76018 %N A072646 Composition of the permutations A048680 & A072636, i.e. a(n) = A048680(A072636(n)). %H A072646 A. Karttunen, Gatomorphisms %H A072646 Index entries for sequences that are permutations of the natural numbers %R A072646 %O A072646 0,3 %K A072646 nonn,new %A A072646 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072646 Inverse permutation A072647. %D A072646 %p A072646 %I A072647 %S A072647 0,1,2,6,3,15,4,5,52,7,16,19,14,466,11,9,37,8,112,42,43,5095,20,53,44,39,127,10,28,481,123, %T A072647 121,149,51,17231,29,12,40,17,41,154,155,153,18,47,156,38,1512,477,475,385,151,1234,122,376, %U A072647 69829,55,159,56,113,164,30,13,178,377,124,152,126,1786,33,84,348,54,357,135,136,20253,1532 %N A072647 Composition of the permutations A072637 & A048679, i.e. a(n) = A072637(A048679(n)). %H A072647 A. Karttunen, Gatomorphisms %H A072647 Index entries for sequences that are permutations of the natural numbers %R A072647 %Y A072647 A072645 gives the size of the corresponding parenthesizations, i.e. A072645(n) = A036377(A014486(A072647(n))+1)/2 [A036377(n+1) gives the binary width of n]. %O A072647 0,3 %K A072647 nonn,new %A A072647 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %D A072647 %p A072647 %I A072648 %S A072648 1,3,3,4,5,5,5,5,6,6,6,6,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9, %T A072648 9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10, %U A072648 10,10,10,10,10,10,10,10,10,10,10,10,10,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11,11 %N A072648 a(n) = [log_{Phi}(n*sqrt(5))], where log_{Phi} is logarithm in the base Phi ( = (sqrt(5)+1)/2), and [] stands for the floor function. %C A072648 An approximate "inverse" of A000045 (of the Fibonacci numbers). %R A072648 %O A072648 1,2 %K A072648 nonn,new %A A072648 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072648 Used to construct A072649. %D A072648 %p A072648 [seq(floor(log[(sqrt(5)+1)/2](n*sqrt(5))),n=1..120)]; %o A072648 (Scheme function:) (define (A072648 n) (cond ((zero? n) n) (else (floor->exact (/ (log (* n *Sqrt5*)) *LogPhi*))))) %o A072648 (define *Sqrt5* (sqrt 5)) %o A072648 (define *Phi* (/ (1+ *Sqrt5*) 2)) %o A072648 (define *LogPhi* (log *Phi*)) %I A072649 %S A072649 1,2,3,3,4,4,4,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,7,7,7,7,7,7,7,8,8,8,8,8,8,8, %T A072649 8,8,8,8,8,8,8,8,8,8,8,8,8,8,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9,9, %U A072649 9,9,9,9,9,9,9,9,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10,10 %N A072649 n occurs A000045(n) times. %C A072649 First position where value n occurs is A000045(n+1), i.e. a(A000045(n)) = n-1, for n >= 2, and a((A000045(n)-1) = n-2, for n >= 3. %R A072649 %O A072649 1,2 %K A072649 nonn,new %A A072649 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072649 Used to construct A003714. Cf. also A002024, A072643, A072648, A072650. %D A072649 %o A072649 (MIT Scheme) (define (A072649 n) (let ((b (A072648 n))) (+ -1 b (floor->exact (/ n (A000045 (1+ b))))))) %I A072650 %S A072650 0,1,2,1,4,3,2,3,8,5,6,5,4,3,6,3,16,9,10,9,12,7,10,7,8,5,6,5,12,7,6,7,32,17,18,17,20,11,18, %T A072650 11,24,13,14,13,20,11,14,11,16,9,10,9,12,7,10,7,24,13,14,13,12,7,14,7,64,33,34,33,36,19,34, %U A072650 19,40,21,22,21,36,19,22,19,48,25,26,25,28,15,26,15,40,21,22,21,28,15,22,15,32,17,18,17,20 %N A072650 Starting from the right (the least significant end) rewrite 0 to 0 and x1 to 1 in the binary expansion of n. %e A072650 I.e. 23 is 10111 in binary, which after rewriting is 111, thus a(23) = 7, while 38 is 100110 in binary, which after the rewriting is 10010, i.e. a(38) = 18. %R A072650 %O A072650 0,3 %K A072650 nonn,new %A A072650 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072650 A048679(n) = A072650(A003714(n)). %D A072650 %o A072650 (MIT Scheme) (define (A072650 n) (let loop ((n n) (s 0) (i 0)) (cond ((zero? n) s) ((even? n) (loop (floor->exact (/ n 2)) s (1+ i))) (else (loop (floor->exact (/ n 4)) (+ s (expt 2 i)) (1+ i)))))) %I A072654 %S A072654 0,1,4,17,161,11107,39099276,308061560269405,11825896447872143037989337832,10770594215935749279482183269315609406974195761679603097, %T A072654 5521577958796399352092495436683350598124375604156506049598913887505725118345928241211637687725414588844260433 %N A072654 Maximum position in A072645 where the value n occurs. %R A072654 %O A072654 0,3 %K A072654 nonn,new %A A072654 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072654 Binary width of each term: A072655. Cf. A072638, A072639, A072640. %D A072654 %p A072654 %I A072655 %S A072655 0,1,3,5,8,14,26,49,94,183,362 %N A072655 Binary widths of the terms of A072654. %R A072655 %O A072655 0,3 %K A072655 nonn,new %A A072655 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072655 Cf. A072641, A072642. %D A072655 %p A072655 %I A072656 %S A072656 0,1,3,2,11,6,5,7,4,100,27,24,45,14,18,10,13,62,17,8,15,28,9,1988,477,179,1100,116,150,61,113, %T A072656 2090,147,35,189,302,58,162,44,39,73,23,34,20,102,1229,295,29,111,680,72,21,12,26,93,38,47, %U A072656 70,1292,91,16,22,117,187,36,415729,97407,12177,123059,3352,7025,2087,3341,318482,7009,621 %N A072656 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using packA048680oA054238 as the packing bijection NxN -> N. %H A072656 A. Karttunen, Gatomorphisms %H A072656 Index entries for sequences that are permutations of the natural numbers %R A072656 %Y A072656 Inverse permutation: A072657. C.f. also A014486, A000695, A054238, A048680, A071651, A072634, A072636, A072646, A072658, A072660. %O A072656 0,3 %K A072656 nonn,new %A A072656 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %D A072656 %o A072656 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.) %o A072656 (define A072656 (lexrank->arithrank-bijection packA048680oA054238)) %o A072656 (define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun))) %o A072656 (define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun)))))) %o A072656 (define (packA048680oA054238 x y) (A048680 (packA054238 x y))) %o A072656 (define (packA054238 x y) (+ (A000695 x) (* 2 (A000695 y)))) %I A072657 %S A072657 0,1,3,2,8,6,5,7,19,22,15,4,52,16,13,20,60,18,14,178,43,51,61,41,11,153,53,10,21,47,555,155, %T A072657 177,136,42,33,64,179,55,39,557,125,151,181,38,12,150,56,1799,479,553,417,154,552,164,135,192, %U A072657 117,36,471,191,29,17,159,1791,600,556,490,123,94,57,561,50,40,1810,122,152,5969,1950,1797 %N A072657 Inverse permutation of A072656. %H A072657 A. Karttunen, Gatomorphisms (With the complete source) %H A072657 Index entries for sequences that are permutations of the natural numbers %O A072657 0,3 %K A072657 nonn,new %A A072657 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072657 A072657(n) = A057163(A072659(n)). A072660 gives the size of the corresponding parenthesizations, i.e. A072660(n) = A036377(A014486(A072657(n))+1)/2 [A036377(n+1) gives the binary width of n]. %D A072657 %p A072657 %I A072658 %S A072658 0,1,2,3,4,7,5,6,11,9,28,15,17,62,8,13,10,14,45,18,24,27,100,36,187,117,91,1292,22,70,38,72, %T A072658 680,93,111,295,1229,16,47,26,29,102,12,20,23,58,302,73,189,147,2090,21,34,39,35,113,44,61, %U A072658 116,1100,162,150,179,477,1988,242,12228,3353,2879,81833,161,1004,394,632,31496,2887,5285,18349 %N A072658 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using packA048680oA054238tr as the packing bijection NxN -> N. %H A072658 A. Karttunen, Gatomorphisms %H A072658 Index entries for sequences that are permutations of the natural numbers %R A072658 %Y A072658 Inverse permutation: A072659. C.f. also A014486, A000695, A054238, A048680, A071651, A072634, A072636, A072646, A072656, A072660. %O A072658 0,3 %K A072658 nonn,new %A A072658 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %D A072658 %o A072658 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.) %o A072658 (define A072656 (lexrank->arithrank-bijection packA048680oA054238)) %o A072658 (define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun))) %o A072658 (define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun)))))) %o A072658 (define (packA048680oA054238tr x y) (A048680 (packA054238tr x y))) %o A072658 (define (packA054238tr x y) (+ (A000695 y) (* 2 (A000695 x)))) %I A072659 %S A072659 0,1,2,3,4,6,7,5,14,9,16,8,42,15,17,11,37,12,19,121,43,51,28,44,20,126,39,21,10,40,385,123, %T A072659 149,124,52,54,23,112,30,53,376,127,154,84,56,18,163,38,1234,387,475,388,151,503,122,152,79, %U A072659 156,45,480,107,57,13,114,1540,371,466,378,155,166,29,348,31,47,1206,164,135,5095,1229,1531 %N A072659 Inverse permutation of A072658. %H A072659 A. Karttunen, Gatomorphisms (With the complete source) %H A072659 Index entries for sequences that are permutations of the natural numbers %O A072659 0,3 %K A072659 nonn,new %A A072659 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072659 A072659(n) = A057163(A072657(n)). A072660 gives the size of the corresponding parenthesizations, i.e. A072660(n) = A036377(A014486(A072659(n))+1)/2 [A036377(n+1) gives the binary width of n]. %D A072659 %I A072660 %S A072660 0,1,2,2,3,3,3,3,4,4,4,3,5,4,4,4,5,4,4,6,5,5,5,5,4,6,5,4,4,5,7,6,6,6,5,5,5,6,5,5, %T A072660 7,6,6,6,5,4,6,5,8,7,7,7,6,7,6,6,6,6,5,7,6,5,4,6,8,7,7,7,6,6,5,7,5,5,8,6,6,9,8,8, %U A072660 8,7,8,7,7,9,7,7,7,6,6,5,7,5,6,8,7,7,6,6,4,7,5,9,8,8,8,7,8,6,7,5,7,5,8,6,5,5,6,9 %N A072660 The size of the parenthesizations obtained with the global ranking/unranking scheme A072656-A072659. %R A072660 %O A072660 0,3 %K A072660 nonn,new %A A072660 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072660 Cf. A072657-A072659. Permutations: A071673, A072643, A072644, A072645. %D A072660 %p A072660 %I A072661 %S A072661 0,1,0,2,1,0,3,2,4,1,0,2,3,0,5,4,6,1,4,3,2,8,1,0,2,5,0,7,6,4,5,4,6,3,0,9,8,10,1,8, %T A072661 3,2,12,1,0,2,7,8,5,4,6,5,4,7,6,16,1,0,2,9,0,11,10,4,9,8,10,3,0,13,12,14,1,12,3,2,8,9,8,10, %U A072661 5,8,7,6,12,5,4,6,7,0,17,16,18,1,16,3,2,20,1,0,2,11,16,5,4,6,9,4,11,10,24,1,0,2,13,0,15,14 %N A072661 Composition of the A059905 & A048679, i.e. a(n) = A059905(A048679(n)) %R A072661 %O A072661 0,4 %K A072661 nonn,new %A A072661 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072661 The other side of this projection is A072662. Used to construct the permutations A072657 & A072659. %D A072661 %p A072661 %I A072662 %S A072662 0,0,1,0,1,2,0,1,0,2,3,2,1,4,0,1,0,3,2,2,3,0,4,5,4,1,6,0,1,4,2,3,2,3,8,0,1,0,5,2, %T A072662 4,5,0,6,7,6,1,4,4,5,4,3,6,2,3,0,8,9,8,1,10,0,1,8,2,3,2,5,12,0,1,0,7,2,6,7,8,4,5,4, %U A072662 5,6,4,5,4,6,7,6,3,16,0,1,0,9,2,8,9,0,10,11,10,1,4,8,9,8,3,10,2,3,0,12,13,12,1,14,0,1,12,2 %N A072662 Composition of the A059906 & A048679, i.e. a(n) = A059906(A048679(n)) %R A072662 %O A072662 0,6 %K A072662 nonn,new %A A072662 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 02 2002 %Y A072662 The other side of this projection is A072661. Used to construct the permutations A072657 & A072659. %D A072662 %p A072662 -------------------------------------------------------------------- Yours, Antti Karttunen