Cheers,
an addition to the index entry,
complete re-edits of three sequences A023359, A072796 & A072797,
and 47 new PRE_COMPUTED sequences
A073190 - A073210, A073265 - A073270, A073280-A073299.
------------------------------------------------------------------------------
Additions to the index entry
http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation
Continued from the line (5) of subsection "Catalan automorphisms":
permutations, of the integers, induced by Catalan automorphisms, each
paired with its inverse: (5) A072088-A072089, A073269-A073270, A073282-A073283, A073284-A073285, A073286-A073287, A073288-A073289
permutations, of the integers, induced by Catalan automorphisms, each
paired with its inverse: (6) A073194-A073195, A073196-A073197, A073198-A073199, A073205-A073206, A073207-A073208, A073209-A073210
permutations, of the integers, induced by Catalan automorphisms, each
paired with its inverse: (7) A073290-A073291, A073292-A073293, A073294-A073295, A073296-A073297, A073298-A073299
and the two involutions, A073280 & A073281 appended to the end of this line:
permutations, of the integers, induced by Catalan automorphisms,
involutions: (2) A072796 A072797 A073280 A073281
and table A073200 added to the end of this line:
permutations, of the integers, tables of: A003987 A054081 A065167 A073200
------------------------------------------------------------------------------
Complete re-edits of A023359, A072796, A072797.
%I A023359
%S A023359 0,1,2,3,6,10,18,31,56,98,174,306,542,956,1690,2983,5272,9310,16448,
%T A023359 29050,51318,90644,160118,282826,499590,882468,1558798,2753448,4863696,
%U A023359 8591212,15175514,26805983,47350056,83639030,147739848,260967362
%N A023359 Ordered partitions of n into powers of 2.
%H A023359 N. J. A. Sloane, Transforms
%F A023359 INVERT transform of characteristic function of powers of 2.
%F A023359 a(n) = (n=2^m) + sum(a(n-2^k),k=0...) - Len Smiley (smiley@math.uaa.alaska.edu), May 07 2001
%K A023359 nonn,easy,nice
%O A023359 0,3
%A A023359 David W. Wilson (davidwwilson@attbi.com)
%Y A023359 The column sums of the table A073265. Cf. also A073267, A073202, A073288.
%I A072796
%S A072796 0,1,2,3,4,6,5,7,8,9,10,14,16,19,11,15,12,17,18,13,20,21,22,23,24,25,
%T A072796 26,27,37,38,42,44,47,51,53,56,60,28,29,39,43,52,30,40,31,45,46,32,48,
%U A072796 49,50,33,41,34,54,55,35,57,58,59,36,61,62,63,64,65,66,67,68,69,70,71
%N A072796 Self-inverse permutation of natural numbers induced by the gatomorphism gmA072796! acting on the parenthesizations encoded by A014486.
%C A072796 This gatomorphism swaps the two leftmost branches of the general plane trees if the degree of the root > 1, and keeps the tree intact if it's planted (root's degree = 1). Together with the gatomorphism A069770 it is used to recursively compose the whole infinite sequence of gatomorphisms of simple type B given in the array A073200.
%H A072796 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A072796 Index entries for sequences that are permutations of the natural numbers
%o A072796 (Scheme function implementing this automorphism on list-structures, first the constructive variant:)
%o A072796 (define (gmA072796 s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (cons (cadr s) (cons (car s) (cddr s))))))
%o A072796 (THEN the destructive variant:)
%o A072796 (define (gmA072796! s) (cond ((not (pair? s)) s) ((not (pair? (cdr s))) s) (else (swap! s) (robr! s) (swap! (cdr s)) s)))
%o A072796 (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 A072796 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
%Y A072796 The car/cdr-flipped conjugate of A072797, i.e. A072796(n) = A057163(A072797(A057163(n))). Occurs first time in A073200 as the row 1. Fixed point counts and cycle counts are given in A073190 and A073191.
%K A072796 nonn
%O A072796 0,3
%A A072796 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%I A072797
%S A072797 0,1,2,3,4,5,7,6,8,9,10,11,12,13,17,18,16,14,15,20,19,21,22,23,24,25,
%T A072797 26,27,28,29,30,31,32,33,34,35,36,45,46,48,49,50,44,47,42,37,38,43,39,
%U A072797 40,41,54,55,53,51,52,57,56,58,59,61,60,62,63,64,65,66,67,68,69,70,71
%N A072797 Self-inverse permutation of natural numbers induced by the gatomorphism gmA072797! acting on the parenthesizations encoded by A014486.
%H A072797 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A072797 Index entries for sequences that are permutations of the natural numbers
%o A072797 (Scheme function implementing this automorphism on list-structures:)
%o A072797 (define (gmA072797! s) (cond ((not (pair? s)) s) ((not (pair? (car s))) s) (else (swap! s) (robl! s) (swap! (car s)) s)))
%o A072797 (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 A072797 (define (swap! s) (let ((ex-car (car s))) (set-car! s (cdr s)) (set-cdr! s ex-car) s))
%Y A072797 The car/cdr-flipped conjugate of A072796, i.e. A072797(n) = A057163(A072796(A057163(n))). Occurs first time in A073200 as the row 179. Fixed point counts and cycle counts are given in A073190 and A073191.
%K A072797 nonn
%O A072797 0,3
%A A072797 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
------------------------------------------------------------------------------
Please don't freak out with the longish comment lines I use in A073200-A073204,
as I will frequently add more "occurs first time as the row N" %Y-lines
to these entries, so I can also clean them otherwise also some
later time. You can suggest shorter/better phrasings if you wish.
%I A073200
%S A073200 0,1,0,3,1,0,2,2,1,0,7,3,3,1,0,8,4,2,3,1,0,6,6,8,2,3,1,0,4,5,7,7,2,3,1,0,5,7,6,6,
%T A073200 8,2,3,1,0,17,8,5,8,7,7,2,2,1,0,18,9,4,4,6,8,7,3,3,1,0,20,10,22,5,5,5,8,4,2,2,1,0,21,14,
%U A073200 21,17,4,4,6,5,8,3,3,1,0
%N A073200 Simple gatomorphisms of type B.
%C A073200 Each row is a permutation of non-negative integers induced by a gatomorphism (constructed as explained below) acting on the parenthesizations/plane binary trees as encoded and ordered by A014486/A063171.
%C A073200 The construction process is akin to the constructive mapping of primitive recursive functions to N: we have two basic primitives, A069770 (row 0) and A072796 (row 1), of which the former swaps the left and the right subtree of a binary tree, and the latter exchanges the positions of the two leftmost subtrees of plane general trees, unless the tree's degree is less than 2, in which case it just fixes it. From then on, the even rows are constructed recursively from any other gatomorphism in this table, using one of the five allowed recursion types:
%C A073200 0 - Apply the given gatomorphism and then recurse down to both subtrees of the new binary tree obtained. (last decimal digit of the row number = 2)
%C A073200 1 - First recurse down to both subtrees of the old binary tree, and only after that apply the given gatomorphism. (last digit = 4)
%C A073200 2 - Apply the given gatomorphism and then recurse down to the right subtree of the new binary tree obtained. (last digit = 6)
%C A073200 3 - First recurse down to the right subtree of old binary tree, and only after that apply the given gatomorphism. (last digit = 8)
%C A073200 4 - First recurse down to the left subtree of old binary tree, after that apply the given gatomorphism, and then recurse down to the right subtree of the new binary tree. (last digit = 0)
%C A073200 The odd rows > 2 are compositions of the rows 0, 1, 2, 4, 6, 8, ... (i.e. either one of the primitives A069770 or A072796, or one of the recursive compositions) at the left hand side, and any gatomorphism from the same array at the right hand side. See the scheme-functions index-for-recursive-sgtb and index-for-composed-sgtb how to compute the positions of the recursive and ordinary compositions in this table.
%R A073200
%H A073200 A. Karttunen, Gatomorphisms (With the complete source and explanation)
%O A073200 0,4
%K A073200 nonn,tabl,new
%A A073200 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073200 Four other tables giving the corresponding cycle-counts: A073201, counts of the fixed elements: A073202, the lengths of the largest cycles: A073203, the LCM's of all the cycles: A073204. The ordinary compositions are encoded using the NxN->N bijection A054238 (which in turn uses the bit-interleaving function A000695).
%Y A073200 The first 21 rows of this table:
%Y A073200 Row 0: A069770. Row 1: A072796. Row 2: A057163. Row 3: A073269, Row 4: A057163 (duplicate), Row 5: A073270, Row 6: A069767, Row 7: A001477 (identity perm.), Row 8: A069768, Row 9: A073280.
%Y A073200 Row 10: A069770 (dupl.), Row 11: A072796 (dupl.), Row 12: A057511, Row 13: A073282, Row 14: A057512, Row 15: A073281, Row 16: A057509, Row 17: A073280 (dupl.), Row 18: A057510, Row 19: A073283, Row 20: A073284.
%Y A073200 Other gatomorphism-induced EIS-permutations which occur in this table. Only the first known occurrence is given. Involutions are marked with *, others paired with their inverse:
%Y A073200 Row 164: A057164*, Row 168: A057508*, Row 179: A072797*
%Y A073200 Row 41: A073286 - Row 69: A073287. Row 105: A073290 - Row 197: A073291. Row 416: A073288 - Row 696: A073289.
%Y A073200 Row 261: A057501 - Row 521: A057502. Row 2618: A057503 - Row 5216: A057504. Row 2614: A057505 - Row 5212: A057506
%Y A073200 Row 10435: A073292 - Row ...: A073293. Row 17517: A057161 - Row ...: A057162.
%D A073200
%o A073200 (Scheme functions showing how to compute the row where either the recursive composition of foo (with rectype 0-4) or an ordinary composition of lhs and rhs occur in this table, where foo, lhs and rhs are also indices to this table):
%o A073200 (define (index-for-recursive-sgtb foo rectype) (+ 2 (* 10 foo) (* 2 rectype)))
%o A073200 (define (index-for-composed-sgtb lhs rhs) (let ((new-lhs (cond ((< lhs 2) lhs) ((even? lhs) (1+ (/ lhs 2))) (else (error "Only the primitive gatomorphisms A069770 (0) & A072796 (1) or one of the recursively composed gatomorphisms (even numbers >= 2) can occur at the left side of the composition. Odd number not allowed: " lhs))))) (1+ (packA054238 (* 2 new-lhs) rhs))))
%o A073200 (define (packA054238 x y) (+ (A000695 x) (* 2 (A000695 y))))
%o A073200 (define (A000695 n) (if (zero? n) n (+ (modulo n 2) (* 4 (A000695 (floor->exact (/ n 2)))))))
%I A073201
%S A073201 1,1,1,1,1,1,3,2,1,1,7,4,1,1,1,22,11,3,1,1,1,66,31,7,2,1,1,1,217,96,22,4,3,1,1,1,715,305,66,
%T A073201 11,7,2,1,1,1,2438,1007,217,30,22,4,2,2,1,1,8398,3389,715,93,66,11,3,5,1,1,1,29414,11636,2438,
%U A073201 292,217,30,6,14,2,2,1,1,104006,40498,8398,965,715,93,12,42,3,3,1,1,1
%N A073201 Array of cycle count sequences for the table A073200.
%C A073201 Each row of this table gives the counts of separate orbits/cycles to which the gatomorphism given in the corresponding row of A073200 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
%C A073201 Note that for involutions (self-inverse gatomorphisms) this is always (A000108(n)+Affffff(n))/2, where Affffff is the corresponding "fix-count sequence" from the table A073202.
%R A073201
%H A073201 A. Karttunen, Gatomorphisms (With the complete source and explanation)
%O A073201 0,7
%K A073201 nonn,tabl,new
%A A073201 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073201 Few EIS-sequences which occur in this table. Only the first known occurrence(s) given (marked with ? if not yet proven/unclear):
%Y A073201 Rows 0, 2, 4, etc.: A007595, Row 1: A073191, Row 7: A000108, Rows 12, 14, 20, ...: A057513, Rows 16, 18, ...: A003239, Row 57, ..., 164: A007123?, Row 168: A073193, Row 261: A002995, Row 2614: A057507, Row 2618 (?), row 17517: A001683
%D A073201
%p A073201
%I A073202
%S A073202 1,1,1,0,1,1,1,2,1,1,0,3,0,1,1,2,8,1,0,1,1,0,20,0,0,0,1,1,5,60,2,0,1,0,1,1,0,181,0,0,
%T A073202 0,0,0,1,1,14,584,5,0,2,0,1,2,1,1,0,1916,0,0,0,0,0,5,0,1,1,42,6476,14,0,5,0,0,14,1,2,1,1,0,
%U A073202 22210,0,0,0,0,0,42,0,1,0,1,1
%N A073202 Array of fix-count sequences for the table A073200.
%C A073202 Each row of this table gives the counts of elements fixed by the gatomorphism (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
%R A073202
%H A073202 A. Karttunen, Gatomorphisms (With the complete source and explanation)
%O A073202 0,8
%K A073202 nonn,tabl,new
%A A073202 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073202 Cf. also A073201, A073203.
%Y A073202 Few EIS-sequences which occur in this table. Only the first known occurrence(s) given (marked with ? if not yet proven/unclear):
%Y A073202 Rows 0, 2, 4, etc.: "Aerated Catalans" shifted right and prepended with 1 (Cf. A000108), Row 1: A0073190, Rows 3, 5, 261, 2614, 2618, 17517, etc: A019590 but with offset 0 instead of 1 (means that the gatomorphisms like A073269, A073270, A057501, A057505, A057503 and A057161 never fix any Catalan structure of size larger than 1)
&Y A073202 Row 6: A036987, Row 7: A000108, Rows 12, 14, 20, ...: A057546, Rows 16, 18: A034731, Row 41: A073268, Row 105: essentially A073267, Row 57, ..., 164: A001405, Row 168: A073192, Row 416: essentially A023359 ?, Row 10435: also A036987.
%D A073202
%p A073202
%I A073203
%S A073203 1,1,1,2,1,1,2,1,1,1,2,2,2,1,1,2,2,2,2,1,1,2,2,2,3,2,1,1,2,2,2,6,2,2,1,1,2,2,2,8,
%T A073203 2,3,2,1,1,2,2,2,10,2,6,4,1,1,1,2,2,2,12,2,8,8,1,2,1,1,2,2,2,14,2,10,16,1,4,1,1,1,2,2,
%U A073203 2,16,2,12,32,1,8,2,2,1,1
%N A073203 Array of max. cycle length sequences for the table A073200.
%C A073203 Each row of this table gives the longest cycle/orbit produced by the gatomorphism (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
%R A073203
%H A073203 A. Karttunen, Gatomorphisms (With the complete source and explanation)
%O A073203 0,4
%K A073203 nonn,tabl,new
%A A073203 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073203 Cf. also A073201, A073202, A073204.
%Y A073203 Few EIS-sequences which occur in this table. Only the first known occurrence(s) given:
%Y A073203 Rows 6 & 8: A011782, Row 7: A000012, Row 12, 14: A000793 (shifted right and prepended with 1), Row 261: A057543, Row 2614: A057545, Rows 2618, 17517: A057544.
%D A073203
%p A073203
%I A073204
%S A073204 1,1,1,2,1,1,2,1,1,1,2,2,2,1,1,2,2,2,2,1,1,2,2,2,6,2,1,1,2,2,2,12,2,2,1,1,2,2,2,120,
%T A073204 2,6,2,1,1,2,2,2,120,2,12,4,1,1,1,2,2,2,840,2,120,8,1,2,1,1,2,2,2,840,2,120,16,1,4,1,1,1,2,
%U A073204 2,2,5040,2,840,32,1,8,2,2,1,1
%N A073204 Array of LCMs-of-cycle-lengths sequences for the table A073200.
%C A073204 Each row of this table gives the least common multiple of all cycle lengths produced by the gatomorphism (given in the corresponding row of A073200) when it acts on A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
%R A073204
%H A073204 A. Karttunen, Gatomorphisms (With the complete source and explanation)
%O A073204 0,4
%K A073204 nonn,tabl,new
%A A073204 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073204 Cf. also A073201-A073203.
%Y A073204 Few EIS-sequences which occur in this table. Only the first known occurrence(s) given:
%Y A073204 Rows 6 & 8: A011782, Row 7: A000012, Row 2614: A060114, Row 2618 (?), ..., 17517: A057544
%D A073204
%p A073204
%I A073190
%S A073190 1,1,2,3,8,20,60,181,584,1916,6476,22210,77416,272840,971640,3488925,12621168,45946156,168206604,
%T A073190 618853270,2286974856,8485246456,31596023208,118037654258,442287721872,1661790513944,6259494791096,
%U A073190 23632483675620,89415243261968,338983803980176,1287512933614960,4898640955695197,18668194640522976
%N A073190 Number of general plane trees which are either empty (the case a(0)), or whose root degree is either 1 (i.e. the planted trees) or the two leftmost subtrees (of the root node) are identical.
%C A073190 The gatomorphism A072796 fixes only these kinds of trees, so this occurs in the table A073202 as the row 1.
%F A073190 a(0)=1, a(n) = Cat(n-1) + Sum_{i=0..n-2, (n-i) is even} Cat((n-i-2)/2)*Cat(i), where Cat(n) is A000108(n).
%R A073190
%O A073190 0,2
%K A073190 nonn,new
%A A073190 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073190 Occurs first time in A073202 as the row 1. A073191(n) = (A000108(n)+A073190(n))/2. Cf. also A073192.
%D A073190
%p A073190 A073190 := proc(n) local d; Cat(n-1)+ add( (`mod`((n-d+1),2))*Cat((n-d-2)/2)*Cat(d) ,d=0..n-2); end;
%p A073190 Cat := n -> binomial(2*n,n)/(n+1);
%I A073191
%S A073191 1,1,2,4,11,31,96,305,1007,3389,11636,40498,142714,507870,1823040,6591885,23989419,87795473,322922652,
%T A073191 1193058230,4425547638,16475756738,61539293424,230548633954,866095934598,3261868457698,12313423931624,
%U A073191 46583017295812,176581597506164,670613010315772,2551249717853632,9721638497461053,37101129758785587
%N A073191 Number of separate orbits/cycles to which the gatomorphisms A072796/A072797 partition each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
%F A073191 a(n) = (A000108(n)+A073190(n))/2.
%R A073191
%O A073191 0,2
%K A073191 nonn,new
%A A073191 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073191 Occurs first time in A073201 as the row 1.
%D A073191
%I A073192
%S A073192 1,1,2,3,8,18,54,155,500,1614,5456,18630,64960,228740,814914,2926323,10589916,38561814,141219432,
%T A073192 519711666,1921142832,7129756188,26555149404,99228108222,371886574632,1397548389644,5265131346368,
%U A073192 19881692345964,75235854904256,285270716963592,1083654463297738,4123558601296067,15716393331538476
%N A073192 Number of general plane trees whose nth subtree from the left is equal with the nth subtree from the right, for all its subtrees (i.e. are palindromic in the shallow sense).
%C A073192 The gatomorphism A057508 fixes only these kinds of trees, so this occurs in the table A073202 as the row 168.
%F A073192 a(n) = Sum_{i=0..n, (n-i) is even} Gat((n-i)/2)*Gat(i-1), where Gat(-1) = 1, and otherwise like A000108(n).
%R A073192
%O A073192 0,2
%K A073192 nonn,new
%A A073192 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073192 Occurs first time in A073202 as the row 168. A073193(n) = (A000108(n)+A073192(n))/2. Cf. also A073190.
%D A073192
%p A073192 A073192 := proc(n) local d; add( (`mod`((n-d+1),2))*Cat((n-d)/2)*(`if`((0=d),1,Cat(d-1))) ,d=0..n); end;
%p A073192 Cat := n -> binomial(2*n,n)/(n+1);
%I A073193
%S A073193 1,1,2,4,11,30,93,292,965,3238,11126,38708,136486,485820,1744677,6310584,22973793,84103302,309429066,
%T A073193 1143487428,4242631626,15798011604,59018856522,221143860936,830895360978,3129747395548,11816242209260,
%U A073193 44707621630984,169491903327308,643756466807480,2449320482695021,9334097320261488,35625229104293337
%N A073193 Number of separate orbits/cycles to which the gatomorphism A057508 partitions each A000108(n) structures encoded in the range [A014137(n-1)..A014138(n-1)] of the sequence A014486/A063171.
%F A073193 a(n) = (A000108(n)+A073192(n))/2
%R A073193
%O A073193 0,2
%K A073193 nonn,new
%A A073193 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073193 Occurs first time in A073201 as the row 168.
%D A073193
%p A073193
%I A073265
%S A073265 1,1,0,0,1,0,1,2,0,0,0,1,1,0,0,0,2,3,0,0,0,0,2,3,1,0,0,0,1,0,4,4,0,0,0,0,0,1,6,6,
%T A073265 1,0,0,0,0,0,2,3,8,5,0,0,0,0,0,0,2,3,13,10,1,0,0,0,0,0,0,0,6,12,15,6,0,0,0,0,0,0,0,2,
%U A073265 6,10,25,15,1,0,0,0,0,0,0,0,0,4,16,31,26,7,0,0,0,0,0,0,0,0,0,6,18,30,45,21,1,0,0,0,0,0,0,0
%N A073265 Table T(n,k) (listed antidiagonalwise in order T(1,1), T(2,1), T(1,2), T(3,1), T(2,2), ...) giving an ordered partitions of n into exactly k powers of 2.
%F A073265 T(0,k) = T(n,0) = 0, T(n,k) = 0 if k > n, T(n,1) = 1 if n = 2^m, 0 otherwise, and in other cases T(n,k) = Sum_{i=0..[log2(n-1)]} T(n-(2^i), k-1).
%e A073265 T(6,3) = 4 because there are four ordered partitions of 6 into 3 powers of 2, namely: 4+1+1, 1+4+1, 1+1+4 and 2+2+2, and it's recursively computed from T(5,2)+T(4,2)+T(2,2) = 2+1+1.
%R A073265
%O A073265 0,8
%K A073265 nonn,tabl,new
%A A073265 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073265 The first row is equal to the characteristic function of A000079, i.e. A036987 with offset 1 instead of 0, and the second row is A073267. The column sums give A023359. A073266 gives the upper triangular region of this array.
%D A073265
%p A073265
%I A073266
%S A073266 1,1,1,0,2,1,1,1,3,1,0,2,3,4,1,0,2,4,6,5,1,0,0,6,8,10,6,1,1,1,3,13,15,15,7,1,0,2,3,12,
%T A073266 25,26,21,8,1,0,2,6,10,31,45,42,28,9,1,0,0,6,16,30,66,77,64,36,10,1,0,2,4,18,40,76,126,126,
%U A073266 93,45,11,1,0,0,6,16,50,96,168,224,198,130,55,12,1,0,0,6,20,50,126,218,336,378,300,176,66,13
%N A073266 The upper triangular region of the table A073265.
%R A073266
%O A073266 0,5
%K A073266 nonn,tabl,new
%A A073266 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%D A073266
%p A073266
%I A073267
%S A073267 0,0,1,2,1,2,2,0,1,2,2,0,2,0,0,0,1,2,2,0,2,0,0,0,2,0,0,0,0,0,0,0,1,2,2,0,2,0,0,0,
%T A073267 2,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,2,2,0,2,0,0,0,2,0,0,0,0,0,0,0,
%U A073267 2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,2,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0
%N A073267 Ordered partitions of n into exactly two powers of 2.
%e A073267 For 2 there is only partition {1+1}, for 3 there is {1+2, 2+1}, for 4 {2+2}, for 5 {1+4, 4+1}, for 6 {2+4,4+2}, for 7 none, thus a(2)=1, a(3)=2, a(4)=1, a(5)=2, a(6)=2 and a(7)=0.
%R A073267
%O A073267 0,4
%K A073267 nonn,new
%A A073267 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073267 The second row of the table A073265. The essentially same sequence 1,1,2,1,2,2,0,1,... occurs first time in A073202 as the row 105 (the fix count sequence of A073290). The positions of 1's for n > 1 is given by the characteristic function of A000079, i.e. A036987 with offset 1 instead of 0, and the positions of 2's is given by A018900. Cf. also A023359.
%D A073267
%p A073267
%I A073268
%S A073268 1,1,2,3,8,20,58,179,576,1902,6426,22092,77026,271702,967840,3476555,12578728,45800278,167693698,
%T A073268 617037126,2280467586,8461771342,31510700712,117725789124,441141656810,1657559677646,6243810767912,
%U A073268 23574126409230,89197372463912,338167916077736,1284449006711960,4887105338968171,18624660432574568
%N A073268 Number of plane binary trees whose right (or respectively: left) subtree is a unique "complete" tree of (2^m)-1 nodes with all the leaf-nodes at the same depth m, and the left (or respectively: right) subtree is any plane binary tree of size n - 2^m + 1.
%C A073268 The gatomorphism A073286 fixes only these kinds of trees, so this occurs in A073202 as the row 41.
%F A073268 a(0)=1, a(n) = Sum_{i=0..[log2(n)]} Cat(n-(2^i))
%R A073268
%O A073268 0,3
%K A073268 nonn,new
%A A073268 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073268 Occurs first time in A073202 as the row 41.
%D A073268
%o A073268 (Mit Scheme) (define (A073268 n) (if (zero? n) 1 (let sumloop ((i (floor->exact (/ (log n) (log 2)))) (s 0)) (cond ((negative? i) s) (else (sumloop (-1+ i) (+ s (A000108 (- n (expt 2 i))))))))))
%p A073268 A073268 := proc(n) local i; if(0=n) then 1 else add(Cat(n-2^i),i=0..floor(evalf(log[2](n)))); fi; end;
%p A073268 Cat := n -> binomial(2*n,n)/(n+1);
%I A073194
%S A073194 0,1,2,3,4,6,7,5,8,9,11,14,16,19,17,20,10,12,15,21,13,18,22,23,25,28,30,33,37,39,42,44,47,51,
%T A073194 53,56,60,45,48,54,57,61,26,27,24,31,34,38,40,43,52,58,62,32,35,41,46,29,49,55,63,36,50,59,
%U A073194 64,65,67,70,72,75,79,81,84,86,89,93,95,98,102,107,109,112,114,117,121,123,126,128,131,135
%N A073194 Permutation of natural numbers induced by the gatomorphism gmA073194 acting on the parenthesizations as ordered by A014486.
%C A073194 Is there equivalent gatomorphism in A073200?
%H A073194 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073194 Index entries for sequences that are permutations of the natural numbers
%R A073194
%O A073194 0,3
%K A073194 nonn,new
%A A073194 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073194 Inverse permutation: A073195. The car/cdr-flipped conjugate of A073205, i.e. A073194(n) = A057163(A073205(A057163(n))). Cf. also A073196-A073199.
%Y A073194 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073194
%o A073194 (Scheme function implementing this automorphism on list-structures:)
%o A073194 (define (gmA073194 s) (cond ((pair? s) (gmA072797! s) (gmA073194 (car s)) (gmA072796! s) (gmA073194 (cdr s)))) s)
%I A073195
%S A073195 0,1,2,3,4,7,5,6,8,9,16,10,17,20,11,18,12,14,21,13,15,19,22,23,44,24,42,43,25,57,26,45,53,27,
%T A073195 46,54,61,28,47,29,48,55,30,49,31,37,56,32,38,58,62,33,50,34,39,59,35,40,51,63,36,41,52,60,
%U A073195 64,65,126,66,128,131,67,124,68,121,127,69,165,123,125,70,166,71,122,173,72,169,73,129,156
%N A073195 Permutation of natural numbers induced by the gatomorphism gmA073195 acting on the parenthesizations as ordered by A014486.
%C A073195 Is there equivalent gatomorphism in A073200?
%R A073195
%H A073195 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073195 Index entries for sequences that are permutations of the natural numbers
%O A073195 0,3
%K A073195 nonn,new
%A A073195 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073195 Inverse permutation: A073194. The car/cdr-flipped conjugate of A073206, i.e. A073195(n) = A057163(A073206(A057163(n))). Cf. also A073194-A073199.
%Y A073195 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073195
%o A073195 (Scheme function implementing this automorphism on list-structures:)
%o A073195 (define (gmA073195 s) (cond ((pair? s) (gmA073195 (cdr s)) (gmA072796! s) (gmA073195 (car s)) (gmA072797! s))) s)
%I A073196
%S A073196 0,1,2,3,4,6,7,5,8,9,10,14,16,19,17,21,12,11,15,18,13,20,22,23,24,25,26,27,37,38,42,44,53,51,
%T A073196 56,47,60,45,46,58,49,59,31,34,30,28,29,40,39,43,52,48,62,35,33,41,57,32,54,63,55,36,61,50,
%U A073196 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,107,108,109,110,111,121,122,126,128,131,154,156
%N A073196 Permutation of natural numbers induced by the gatomorphism gmA073196 acting on the parenthesizations as ordered by A014486.
%C A073196 Is there equivalent gatomorphism in A073200?
%R A073196
%H A073196 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073196 Index entries for sequences that are permutations of the natural numbers
%O A073196 0,3
%K A073196 nonn,new
%A A073196 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073196 Inverse permutation: A073197. The car/cdr-flipped conjugate of A073207, i.e. A073196(n) = A057163(A073207(A057163(n))). Cf. also A073194-A073199.
%Y A073196 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073196
%o A073196 (Scheme function implementing this automorphism on list-structures:)
%o A073196 (define (gmA073196 s) (cond ((pair? s) (gmA072797! s) (gmA073196 (car s)) (gmA072796! s) (gmA073196 (car s)))) s)
%I A073197
%S A073197 0,1,2,3,4,7,5,6,8,9,10,17,16,20,11,18,12,14,19,13,21,15,22,23,24,25,26,27,45,46,44,42,57,54,
%T A073197 43,53,61,28,29,48,47,55,30,49,31,37,38,35,51,40,63,33,50,32,58,60,34,56,39,41,36,62,52,59,
%U A073197 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,129,130,132,133,134,128,131,126,121,122,166,169
%N A073197 Permutation of natural numbers induced by the gatomorphism gmA073197 acting on the parenthesizations as ordered by A014486.
%C A073197 Is there equivalent gatomorphism in A073200?
%R A073197
%H A073197 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073197 Index entries for sequences that are permutations of the natural numbers
%O A073197 0,3
%K A073197 nonn,new
%A A073197 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073197 Inverse permutation: A073196. The car/cdr-flipped conjugate of A073208, i.e. A073197(n) = A057163(A073208(A057163(n))). Cf. also A073194-A073199.
%Y A073197 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073197
%o A073197 (Scheme function implementing this automorphism on list-structures:)
%o A073197 (define (gmA073197 s) (cond ((pair? s) (gmA073197 (car s)) (gmA072796! s) (gmA073197 (car s)) (gmA072797! s))) s)
%I A073198
%S A073198 0,1,2,3,4,6,7,5,8,9,14,16,11,19,17,18,10,12,15,20,13,21,22,23,37,42,30,51,44,47,28,25,39,53,
%T A073198 33,56,60,45,46,48,49,50,26,29,24,31,40,38,43,32,52,54,55,27,34,41,57,35,58,59,61,36,62,63,
%U A073198 64,65,107,121,70,149,126,135,86,84,114,154,95,163,177,128,131,137,140,144,72,89,79,67,109
%N A073198 Permutation of natural numbers induced by the gatomorphism gmA073198 acting on the parenthesizations as ordered by A014486.
%C A073198 Is there equivalent gatomorphism in A073200?
%R A073198
%H A073198 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073198 Index entries for sequences that are permutations of the natural numbers
%O A073198 0,3
%K A073198 nonn,new
%A A073198 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073198 Inverse permutation: A073199. The car/cdr-flipped conjugate of A073209, i.e. A073198(n) = A057163(A073209(A057163(n))). Cf. also A073194-A073197.
%Y A073198 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073198
%o A073198 (Scheme function implementing this automorphism on list-structures:)
%o A073198 (define (gmA073198 s) (cond ((pair? s) (gmA072797! s) (gmA073198 (cdr s)) (gmA072796! s) (gmA073198 (cdr s)))) s)
%I A073199
%S A073199 0,1,2,3,4,7,5,6,8,9,16,12,17,20,10,18,11,14,15,13,19,21,22,23,44,31,42,53,30,43,26,45,49,34,
%T A073199 54,57,61,24,47,32,46,55,25,48,28,37,38,29,39,40,41,27,50,33,51,52,35,56,58,59,36,60,62,63,
%U A073199 64,65,126,87,128,156,68,140,84,121,124,96,154,165,179,86,127,91,122,155,73,123,72,129,141
%N A073199 Permutation of natural numbers induced by the gatomorphism gmA073199 acting on the parenthesizations as ordered by A014486.
%C A073199 Is there equivalent gatomorphism in A073200?
%R A073199
%H A073199 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073199 Index entries for sequences that are permutations of the natural numbers
%O A073199 0,3
%K A073199 nonn,new
%A A073199 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073199 Inverse permutation: A073198. The car/cdr-flipped conjugate of A073210, i.e. A073199(n) = A057163(A073210(A057163(n))). Cf. also A073194-A073197.
%Y A073199 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073199
%o A073199 (Scheme function implementing this automorphism on list-structures:)
%o A073199 (define (gmA073199 s) (cond ((pair? s) (gmA073199 (cdr s)) (gmA072796! s) (gmA073199 (cdr s)) (gmA072797! s))) s)
%I A073205
%S A073205 0,1,2,3,4,7,5,6,8,9,12,17,16,18,10,21,11,14,15,13,19,20,22,23,26,31,30,32,45,57,44,42,43,46,
%T A073205 47,48,50,24,35,49,56,63,25,58,28,37,38,29,39,40,41,27,59,33,51,52,34,53,54,55,36,60,61,62,
%U A073205 64,65,68,73,72,74,87,99,86,84,85,88,89,90,92,129,141,166,169,167,128,165,126,121,122,127,123
%N A073205 Permutation of natural numbers induced by the gatomorphism gmA073205 acting on the parenthesizations as ordered by A014486.
%C A073205 Is there equivalent gatomorphism in A073200?
%R A073205
%H A073205 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073205 Index entries for sequences that are permutations of the natural numbers
%O A073205 0,3
%K A073205 nonn,new
%A A073205 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073205 Inverse permutation: A073206. The car/cdr-flipped conjugate of A073194, i.e. A073205(n) = A057163(A073194(A057163(n))). Cf. also A073207-A073210.
%Y A073205 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%o A073205 (Scheme function implementing this automorphism on list-structures:)
%o A073205 (define (gmA073205 s) (cond ((pair? s) (gmA072796! s) (gmA073205 (cdr s)) (gmA072797! s) (gmA073205 (car s)))) s)
%I A073206
%S A073206 0,1,2,3,4,6,7,5,8,9,14,16,10,19,17,18,12,11,13,20,21,15,22,23,37,42,24,51,44,47,26,25,27,53,
%T A073206 56,38,60,45,46,48,49,50,31,32,30,28,33,34,35,39,36,54,55,57,58,59,40,29,43,52,61,62,63,41,
%U A073206 64,65,107,121,66,149,126,135,68,67,69,154,163,108,177,128,131,137,140,144,73,74,72,70,75,76
%N A073206 Permutation of natural numbers induced by the gatomorphism gmA073206 acting on the parenthesizations as ordered by A014486.
%C A073206 Is there equivalent gatomorphism in A073200?
%R A073206
%H A073206 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073206 Index entries for sequences that are permutations of the natural numbers
%O A073206 0,3
%K A073206 nonn,new
%A A073206 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073206 Inverse permutation: A073205. The car/cdr-flipped conjugate of A073195, i.e. A073206(n) = A057163(A073195(A057163(n))). Cf. also A073207-A073210.
%Y A073206 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073206
%o A073206 (Scheme function implementing this automorphism on list-structures:)
%o A073206 (define (gmA073206 s) (cond ((pair? s) (gmA073206 (car s)) (gmA072797! s) (gmA073206 (cdr s)) (gmA072796! s))) s)
%I A073207
%S A073207 0,1,2,3,4,7,5,6,8,9,11,17,16,20,12,18,10,14,15,13,19,21,22,23,31,28,24,33,45,49,44,42,43,54,
%T A073207 53,57,61,30,29,46,47,55,25,48,26,37,40,32,38,39,41,34,50,27,51,52,35,56,58,59,36,60,62,63,
%U A073207 64,65,84,87,73,96,79,81,68,67,90,93,69,98,102,129,132,142,141,146,128,140,126,121,124,127
%N A073207 Permutation of natural numbers induced by the gatomorphism gmA073207 acting on the parenthesizations as ordered by A014486.
%C A073207 Is there equivalent gatomorphism in A073200?
%R A073207
%H A073207 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073207 Index entries for sequences that are permutations of the natural numbers
%O A073207 0,3
%K A073207 nonn,new
%A A073207 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073207 Inverse permutation: A073208. The car/cdr-flipped conjugate of A073196, i.e. A073207(n) = A057163(A073196(A057163(n))). Cf. also A073205-A073210.
%Y A073207 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073207
%o A073207 (Scheme function implementing this automorphism on list-structures:)
%o A073207 (define (gmA073207 s) (cond ((pair? s) (gmA072796! s) (gmA073207 (cdr s)) (gmA072797! s) (gmA073207 (cdr s)))) s)
%I A073208
%S A073208 0,1,2,3,4,6,7,5,8,9,16,10,14,19,17,18,12,11,15,20,13,21,22,23,26,42,44,53,25,38,37,24,47,27,
%T A073208 51,56,60,45,48,49,46,50,31,32,30,28,39,40,43,29,52,54,55,34,33,41,57,35,58,59,61,36,62,63,
%U A073208 64,65,126,73,72,76,121,137,128,68,110,154,156,165,179,70,122,71,109,150,66,135,107,67,140
%N A073208 Permutation of natural numbers induced by the gatomorphism gmA073208 acting on the parenthesizations as ordered by A014486.
%C A073208 Is there equivalent gatomorphism in A073200?
%R A073208
%H A073208 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073208 Index entries for sequences that are permutations of the natural numbers
%O A073208 0,3
%K A073208 nonn,new
%A A073208 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073208 Inverse permutation: A073207. The car/cdr-flipped conjugate of A073197, i.e. A073208(n) = A057163(A073197(A057163(n))). Cf. also A073205-A073210.
%Y A073208 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073208
%o A073208 (Scheme function implementing this automorphism on list-structures:)
%o A073208 (define (gmA073208 s) (cond ((pair? s) (gmA073208 (cdr s)) (gmA072797! s) (gmA073208 (cdr s)) (gmA072796! s))) s)
%I A073209
%S A073209 0,1,2,3,4,7,5,6,8,9,10,17,16,18,11,21,12,14,20,13,15,19,22,23,24,25,26,27,45,46,44,42,49,48,
%T A073209 43,47,50,28,29,58,56,63,30,57,31,37,38,32,54,53,62,33,59,34,39,61,35,40,51,55,36,41,52,60,
%U A073209 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,129,130,132,133,134,128,131,126,121,122,141,142
%N A073209 Permutation of natural numbers induced by the gatomorphism gmA073209 acting on the parenthesizations as ordered by A014486.
%C A073209 Is there equivalent gatomorphism in A073200?
%R A073209
%H A073209 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073209 Index entries for sequences that are permutations of the natural numbers
%O A073209 0,3
%K A073209 nonn,new
%A A073209 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073209 Inverse permutation: A073210. The car/cdr-flipped conjugate of A073198, i.e. A073209(n) = A057163(A073198(A057163(n))). Cf. also A073205-A073208.
%Y A073209 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073209
%o A073209 (Scheme function implementing this automorphism on list-structures:)
%o A073209 (define (gmA073209 s) (cond ((pair? s) (gmA072796! s) (gmA073209 (car s)) (gmA072797! s) (gmA073209 (car s)))) s)
%I A073210
%S A073210 0,1,2,3,4,6,7,5,8,9,10,14,16,19,17,20,12,11,13,21,18,15,22,23,24,25,26,27,37,38,42,44,47,51,
%T A073210 53,56,60,45,46,54,57,61,31,34,30,28,29,35,33,32,36,58,62,49,48,59,40,43,39,52,63,55,50,41,
%U A073210 64,65,66,67,68,69,70,71,72,73,74,75,76,77,78,107,108,109,110,111,121,122,126,128,131,135,137
%N A073210 Permutation of natural numbers induced by the gatomorphism gmA073210 acting on the parenthesizations as ordered by A014486.
%C A073210 Is there equivalent gatomorphism in A073200?
%R A073210
%H A073210 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073210 Index entries for sequences that are permutations of the natural numbers
%O A073210 0,3
%K A073210 nonn,new
%A A073210 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073210 Inverse permutation: A073209. The car/cdr-flipped conjugate of A073199, i.e. A073210(n) = A057163(A073199(A057163(n))). Cf. also A073205-A073208.
%Y A073210 The scheme functions gma072796! and gma072797! referred to below are given in A072796 and A072797.
%D A073210
%o A073210 (Scheme function implementing this automorphism on list-structures:)
%o A073210 (define (gmA073210 s) (cond ((pair? s) (gmA073210 (car s)) (gmA072797! s) (gmA073210 (car s)) (gmA072796! s))) s)
%I A073269
%S A073269 0,1,3,2,7,6,8,4,5,17,18,16,14,15,20,19,21,9,10,22,11,12,13,45,46,48,49,50,44,47,42,37,38,43,
%T A073269 39,40,41,54,55,53,51,52,57,56,58,23,24,59,25,26,27,61,60,62,28,29,63,30,31,32,64,33,34,35,
%U A073269 36,129,130,132,133,134,138,139,141,142,143,145,146,147,148,128,131,137,140,144,126,135,121
%N A073269 Permutation of natural numbers induced by the composition of the gatomorphisms A069770 & A072796.
%F A073269 a(n) = A069770(A072796(n)).
%R A073269
%H A073269 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073269 Index entries for sequences that are permutations of the natural numbers
%O A073269 0,3
%K A073269 nonn,new
%A A073269 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073269 Inverse permutation: A073270. Occurs first time in A073200 as the row 3.
%D A073269
%p A073269
%I A073270
%S A073270 0,1,3,2,7,8,5,4,6,17,18,20,21,22,12,13,11,9,10,15,14,16,19,45,46,48,49,50,54,55,57,58,59,61,
%T A073270 62,63,64,31,32,34,35,36,30,33,28,23,24,29,25,26,27,40,41,39,37,38,43,42,44,47,52,51,53,56,
%U A073270 60,129,130,132,133,134,138,139,141,142,143,145,146,147,148,157,158,160,161,162,166,167,169
%N A073270 Permutation of natural numbers induced by the composition of the gatomorphisms A072796 & A069770.
%F A073270 a(n) = A072796(A069770(n)).
%R A073270
%H A073270 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073270 Index entries for sequences that are permutations of the natural numbers
%O A073270 0,3
%K A073270 nonn,new
%A A073270 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073270 Inverse permutation: A073269. Occurs first time in A073200 as the row 5.
%D A073270
%p A073270
%I A073280
%S A073280 0,1,2,3,5,4,6,8,7,13,12,11,10,9,15,14,19,22,21,16,20,18,17,36,35,34,32,31,33,30,29,27,26,28,
%T A073280 25,24,23,41,40,39,38,37,52,51,60,64,63,56,62,59,58,43,42,53,61,57,47,55,50,49,44,54,48,46,
%U A073280 45,106,105,104,101,100,103,99,97,92,91,96,90,88,87,102,98,95,89,86,94,85,83,78,77,82,76,74
%N A073280 Self-inverse permutation of natural numbers induced by the composition of the gatomorphisms A069770 & A057163.
%F A073280 a(n) = A069770(A057163(n)) = A057163(A069770(n)).
%R A073280
%H A073280 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073280 Index entries for sequences that are permutations of the natural numbers
%O A073280 0,3
%K A073280 nonn,new
%A A073280 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073280 Occurs first time in A073200 as the row 9.
%D A073280
%p A073280
%I A073281
%S A073281 0,1,3,2,7,5,8,4,6,17,18,12,11,15,20,13,21,9,10,22,14,16,19,45,46,48,49,50,31,32,30,28,29,40,
%T A073281 39,43,52,54,55,34,33,41,57,35,58,23,24,59,25,26,27,61,36,62,37,38,63,42,44,47,64,51,53,56,
%U A073281 60,129,130,132,133,134,138,139,141,142,143,145,146,147,148,87,88,90,91,92,86,89,84,79,80,85
%N A073281 Self-inverse permutation of natural numbers induced by the composition of the gatomorphisms A072796 & A073269.
%F A073281 a(n) = A072796(A073269(n)) = A072796(A069770(A072796(n))).
%R A073281
%H A073281 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073281 Index entries for sequences that are permutations of the natural numbers
%O A073281 0,3
%K A073281 nonn,new
%A A073281 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073281 Occurs first time in A073200 as the row 15. This is a conjugate of A069770, so the fixed element and the cycle counts are same as for A069770: "Aerated Catalans" and A007595.
%D A073281
%p A073281
%I A073282
%S A073282 0,1,3,2,8,7,5,6,4,22,21,20,18,17,13,12,15,19,16,11,14,10,9,64,63,62,59,58,61,57,55,50,49,54,
%T A073282 48,46,45,36,35,34,32,31,41,40,52,60,56,43,53,47,44,33,30,39,51,42,29,38,27,26,28,37,25,24,
%U A073282 23,196,195,194,190,189,193,188,185,176,175,184,174,171,170,192,187,183,173,169,181,167,162
%N A073282 Permutation of natural numbers induced by the composition of the gatomorphisms A072796 & A057163.
%F A073282 a(n) = A072796(A057163(n))
%R A073282
%H A073282 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073282 Index entries for sequences that are permutations of the natural numbers
%O A073282 0,3
%K A073282 nonn,new
%A A073282 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073282 Inverse permutation: A073283. Occurs first time in A073200 as the row 13.
%D A073282
%p A073282
%I A073283
%S A073283 0,1,3,2,8,6,7,5,4,22,21,19,15,14,20,16,18,13,12,17,11,10,9,64,63,62,59,58,60,56,52,41,40,51,
%T A073283 39,38,37,61,57,53,43,42,55,47,50,36,35,49,34,32,31,54,44,48,33,30,46,29,27,26,45,28,25,24,
%U A073283 23,196,195,194,190,189,193,188,185,176,175,184,174,171,170,191,186,182,172,168,178,164,153
%N A073283 Permutation of natural numbers induced by the composition of the gatomorphisms A057163 & A072796.
%F A073283 a(n) = A057163(A072796(n))
%R A073283
%H A073283 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073283 Index entries for sequences that are permutations of the natural numbers
%O A073283 0,3
%K A073283 nonn,new
%A A073283 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073283 Inverse permutation: A073282. Occurs first time in A073200 as the row 19.
%D A073283
%p A073283
%I A073284
%S A073284 0,1,2,3,4,6,5,7,8,9,11,14,16,19,10,15,12,17,20,13,18,21,22,23,25,28,30,33,37,39,42,44,47,51,
%T A073284 53,56,60,24,29,38,43,52,26,40,31,45,48,32,54,57,61,27,41,34,46,55,35,49,58,62,36,50,59,63,
%U A073284 64,65,67,70,72,75,79,81,84,86,89,93,95,98,102,107,109,112,114,117,121,123,126,128,131,135
%N A073284 Permutation of natural numbers induced by the gatomorphism gma073284! acting on the parenthesizations encoded by A014486.
%R A073284
%H A073284 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073284 Index entries for sequences that are permutations of the natural numbers
%O A073284 0,3
%K A073284 nonn,new
%A A073284 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073284 Inverse permutation: A073285. Occurs first time in A073200 as the row 20.
%Y A073284 The scheme function gma072796! referred to below given in A072796.
%o A073284 (Scheme function implementing this automorphism on list-structures:)
%o A073284 (define (gmA073284! s) (cond ((pair? s) (gmA073284! (car s)) (gmA072796! s) (gmA073284! (cdr s)))) s)
%I A073285
%S A073285 0,1,2,3,4,6,5,7,8,9,14,10,16,19,11,15,12,17,20,13,18,21,22,23,37,24,42,51,25,38,26,44,47,27,
%T A073285 53,56,60,28,39,29,43,52,30,40,31,45,54,32,46,57,61,33,41,34,48,55,35,49,58,62,36,50,59,63,
%U A073285 64,65,107,66,121,149,67,108,68,126,154,69,135,163,177,70,109,71,122,150,72,110,73,128,137
%N A073285 Permutation of natural numbers induced by the gatomorphism gma073285! acting on the parenthesizations encoded by A014486.
%R A073285
%H A073285 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073285 Index entries for sequences that are permutations of the natural numbers
%O A073285 0,3
%K A073285 nonn,new
%A A073285 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073285 Inverse permutation: A073284.
%Y A073285 The scheme function gma072796! referred to below given in A072796.
%o A073285 (Scheme function implementing this automorphism on list-structures:)
%o A073285 (define (gmA073285! s) (cond ((pair? s) (gmA073285! (cdr s)) (gmA072796! s) (gmA073285! (car s)))) s)
%I A073286
%S A073286 0,1,2,3,4,5,6,8,7,9,10,11,12,13,14,15,19,21,22,16,20,18,17,23,24,25,26,27,28,29,30,31,32,33,
%T A073286 34,35,36,37,38,39,40,41,51,52,56,58,59,60,62,63,64,42,43,53,57,61,47,55,49,50,44,54,48,46,
%U A073286 45,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93
%N A073286 Permutation of natural numbers induced by the composition of the gatomorphisms A069770 & A069767.
%F A073286 a(n) = A069770(A069767(n)).
%R A073286
%H A073286 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073286 Index entries for sequences that are permutations of the natural numbers
%O A073286 0,3
%K A073286 nonn,new
%A A073286 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073286 Inverse permutation: A073287. Occurs first time in A073200 as the row 41. Counts of the fixed elements: A073268.
%D A073286
%p A073286
%I A073287
%S A073287 0,1,2,3,4,5,6,8,7,9,10,11,12,13,14,15,19,22,21,16,20,17,18,23,24,25,26,27,28,29,30,31,32,33,
%T A073287 34,35,36,37,38,39,40,41,51,52,60,64,63,56,62,58,59,42,43,53,61,57,44,54,45,46,47,55,48,49,
%U A073287 50,65,66,67,68,69,70,71,72,73,74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93
%N A073287 Permutation of natural numbers induced by the composition of the gatomorphisms A069768 & A069770.
%F A073287 a(n) = A069768(A069770(n)).
%R A073287
%H A073287 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073287 Index entries for sequences that are permutations of the natural numbers
%O A073287 0,3
%K A073287 nonn,new
%A A073287 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073287 Inverse permutation: A073286. Occurs first time in A073200 as the row 69. Counts of the fixed elements: A073268.
%D A073287
%p A073287
%I A073288
%S A073288 0,1,2,3,4,5,6,8,7,9,10,11,13,12,14,15,19,21,22,16,20,18,17,23,24,25,27,26,28,29,33,35,36,30,
%T A073288 34,32,31,37,38,39,41,40,51,52,56,58,59,60,62,63,64,42,43,53,57,61,47,55,49,50,44,54,48,46,
%U A073288 45,65,66,67,69,68,70,71,75,77,78,72,76,74,73,79,80,81,83,82,93,94,98,100,101,102,104,105,106
%N A073288 Permutation of natural numbers induced by the gatomorphism gma073288! acting on the parenthesizations encoded by A014486.
%R A073288
%H A073288 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073288 Index entries for sequences that are permutations of the natural numbers
%O A073288 0,3
%K A073288 nonn,new
%A A073288 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073288 Inverse permutation: A073289. Occurs first time in A073200 as the row 416.
%Y A073288 The scheme function gma073286! referred to below given in A073286.
%o A073288 (Scheme function implementing this automorphism on list-structures:)
%o A073288 (define (gmA073288! s) (cond ((pair? s) (gmA073286! s) (gmA073288! (cdr s)))) s)
%I A073289
%S A073289 0,1,2,3,4,5,6,8,7,9,10,11,13,12,14,15,19,22,21,16,20,17,18,23,24,25,27,26,28,29,33,36,35,30,
%T A073289 34,31,32,37,38,39,41,40,51,52,60,64,63,56,62,58,59,42,43,53,61,57,44,54,45,46,47,55,48,49,
%U A073289 50,65,66,67,69,68,70,71,75,78,77,72,76,73,74,79,80,81,83,82,93,94,102,106,105,98,104,100,101
%N A073289 Permutation of natural numbers induced by the gatomorphism gma073289! acting on the parenthesizations encoded by A014486.
%R A073289
%H A073289 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A073289 Index entries for sequences that are permutations of the natural numbers
%O A073289 0,3
%K A073289 nonn,new
%A A073289 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073289 Inverse permutation: A073288. Occurs first time in A073200 as the row 696.
%D A073289
%Y A073289 The scheme function gma073287! referred to below given in A073287.
%o A073289 (Scheme function implementing this automorphism on list-structures:)
%o A073289 (define (gmA073289! s) (cond ((pair? s) (gmA073289! (cdr s)) (gmA073287! s))) s)
%I A073290
%S A073290 0,1,2,3,5,4,6,8,7,12,13,11,10,9,15,14,19,21,22,16,20,18,17,31,32,34,35,36,30,33,29,26,27,28,
%T A073290 25,24,23,40,41,39,38,37,52,51,56,58,59,60,62,63,64,43,42,53,57,61,47,55,49,50,44,54,48,46,
%U A073290 45,87,88,90,91,92,96,97,99,100,101,103,104,105,106,86,89,95,98,102,85,94,82,73,74,83,76,77
%N A073290 Permutation A069767 applied twice ("squared").
%F A073290 a(n) = A069767(A069767(n)).
%R A073290
%H A073290 Index entries for sequences that are permutations of the natural numbers
%O A073290 0,3
%K A073290 nonn,new
%A A073290 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073290 Inverse permutation: A073291. Occurs first time in A073200 as the row 105. A073267 gives essentially (apart from the first two terms) the counts of elements fixed. Cf. A073292-A073299.
%D A073290
%p A073290
%I A073291
%S A073291 0,1,2,3,5,4,6,8,7,13,12,11,9,10,15,14,19,22,21,16,20,17,18,36,35,34,31,32,33,30,28,23,24,29,
%T A073291 25,26,27,41,40,39,37,38,52,51,60,64,63,56,62,58,59,43,42,53,61,57,44,54,45,46,47,55,48,49,
%U A073291 50,106,105,104,100,101,103,99,96,87,88,97,90,91,92,102,98,95,86,89,93,84,79,65,66,80,67,68
%N A073291 Permutation A069768 applied twice ("squared").
%F A073291 a(n) = A069768(A069768(n)).
%R A073291
%H A073291 Index entries for sequences that are permutations of the natural numbers
%O A073291 0,3
%K A073291 nonn,new
%A A073291 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073291 Inverse permutation: A073290. Occurs first time in A073200 as the row 197. Cf. A073292-A073299.
%D A073291
%p A073291
> format(A0073292,73292);
%I A073292
%S A073292 0,1,3,2,8,7,6,4,5,21,22,20,18,17,19,16,14,10,9,15,11,13,12,58,59,62,63,64,57,61,55,49,50,54,
%T A073292 48,46,45,56,60,53,47,44,51,42,38,26,27,37,25,24,23,52,43,39,29,28,41,33,35,36,40,30,34,32,
%U A073292 31,170,171,174,175,176,184,185,188,189,190,193,194,195,196,169,173,183,187,192,167,181,161
%N A073292 Permutation A069767 applied three times ("cubed").
%F A073292 a(n) = A069767(A069767(A069767(n))).
%R A073292
%H A073292 Index entries for sequences that are permutations of the natural numbers
%O A073292 0,3
%K A073292 nonn,new
%A A073292 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073292 Inverse permutation: A073293. Occurs first time in A073200 as the row 10435. Cf. A073290-A073299.
%D A073292
%p A073292
> format(A0073293,73293);
%I A073293
%S A073293 0,1,3,2,7,8,6,5,4,18,17,20,22,21,16,19,15,13,12,14,11,9,10,50,49,48,45,46,55,54,61,64,63,57,
%T A073293 62,58,59,47,44,53,60,56,43,52,41,36,35,40,34,31,32,42,51,39,33,30,37,28,23,24,38,29,25,26,
%U A073293 27,148,147,146,142,143,145,141,138,129,130,139,132,133,134,162,161,160,157,158,181,180,192
%N A073293 Permutation A069768 applied three times ("cubed").
%F A073293 a(n) = A069768(A069768(A069768(n))).
%R A073293
%H A073293 Index entries for sequences that are permutations of the natural numbers
%O A073293 0,3
%K A073293 nonn,new
%A A073293 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073293 Inverse permutation: A073292. Cf. A073290-A073299.
%D A073293
%p A073293
> format(A0073294,73294);
%I A073294
%S A073294 0,1,2,3,4,5,6,7,8,10,9,11,13,12,14,15,16,18,17,19,20,22,21,26,27,25,24,23,29,28,33,35,36,30,
%T A073294 34,32,31,38,37,39,41,40,42,43,47,49,50,44,48,46,45,51,52,53,55,54,60,61,63,64,56,57,62,59,
%U A073294 58,73,74,76,77,78,72,75,71,68,69,70,67,66,65,82,83,81,80,79,94,93,98,100,101,102,104,105,106
%N A073294 Permutation A069767 applied four times or permutation A073290 applied twice.
%F A073294 a(n) = A069767(A069767(A069767(A069767(n)))) = A073290(A073290(n)).
%R A073294
%H A073294 Index entries for sequences that are permutations of the natural numbers
%O A073294 0,3
%K A073294 nonn,new
%A A073294 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073294 Inverse permutation: A073295. Cf. A073290-A073299. Occurs in A073200 as the row 142647371.
%D A073294
%p A073294
> format(A0073295,73295);
%I A073295
%S A073295 0,1,2,3,4,5,6,7,8,10,9,11,13,12,14,15,16,18,17,19,20,22,21,27,26,25,23,24,29,28,33,36,35,30,
%T A073295 34,31,32,38,37,39,41,40,42,43,47,50,49,44,48,45,46,51,52,53,55,54,60,61,64,63,56,57,62,58,
%U A073295 59,78,77,76,73,74,75,72,70,65,66,71,67,68,69,83,82,81,79,80,94,93,102,106,105,98,104,100,101
%N A073295 Permutation A069768 applied four times or permutation A073291 applied twice.
%F A073295 a(n) = A069768(A069768(A069768(A069768(n)))) = A073291(A073291(n)).
%R A073295
%H A073295 Index entries for sequences that are permutations of the natural numbers
%O A073295 0,3
%K A073295 nonn,new
%A A073295 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073295 Inverse permutation: A073294. Cf. A073290-A073299.
%D A073295
%p A073295
> format(A0073296,73296);
%I A073296
%S A073296 0,1,3,2,7,8,6,5,4,18,17,20,22,21,16,19,15,13,12,14,11,9,10,49,50,48,46,45,55,54,61,63,64,57,
%T A073296 62,59,58,47,44,53,60,56,43,52,41,35,36,40,34,32,31,42,51,39,33,30,37,28,24,23,38,29,25,27,
%U A073296 26,142,143,146,147,148,141,145,139,133,134,138,132,130,129,161,162,160,158,157,181,180,187
%N A073296 Permutation A069767 applied five times or composition of the permutations A073290 & A073292.
%F A073296 a(n) = A069767^5(n) = A073290(A073292(n)).
%R A073296
%H A073296 Index entries for sequences that are permutations of the natural numbers
%O A073296 0,3
%K A073296 nonn,new
%A A073296 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073296 Inverse permutation: A073297. Cf. A073290-A073299. Occurs in A073200 as the row 36169536789029067.
%D A073296
%p A073296
> format(A0073297,73297);
%I A073297
%S A073297 0,1,3,2,8,7,6,4,5,21,22,20,18,17,19,16,14,10,9,15,11,13,12,59,58,62,64,63,57,61,55,50,49,54,
%T A073297 48,45,46,56,60,53,47,44,51,42,38,27,26,37,25,23,24,52,43,39,29,28,41,33,36,35,40,30,34,31,
%U A073297 32,176,175,174,170,171,185,184,193,196,195,188,194,189,190,173,169,183,192,187,167,181,162
%N A073297 Permutation A069768 applied five times or composition of the permutations A073291 & A073293.
%F A073297 a(n) = A069768^5(n) = A073291(A073293(n)).
%R A073297
%H A073297 Index entries for sequences that are permutations of the natural numbers
%O A073297 0,3
%K A073297 nonn,new
%A A073297 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073297 Inverse permutation: A073296. Cf. A073290-A073299.
%D A073297
%p A073297
%I A073298
%S A073298 0,1,2,3,5,4,6,8,7,13,12,11,9,10,15,14,19,22,21,16,20,17,18,35,36,34,32,31,33,30,28,24,23,29,
%T A073298 25,27,26,41,40,39,37,38,52,51,60,63,64,56,62,59,58,43,42,53,61,57,44,54,46,45,47,55,48,50,
%U A073298 49,100,101,104,105,106,99,103,97,91,92,96,90,88,87,98,102,95,89,86,93,84,80,68,69,79,67,66
%N A073298 Permutation A069767 applied six times or permutation A073290 "cubed" or permutation A073292 "squared".
%F A073298 a(n) = A069767^6(n) = A073290(A073290(A073290(n))) = A073292(A073292(n))
%R A073298
%H A073298 Index entries for sequences that are permutations of the natural numbers
%O A073298 0,3
%K A073298 nonn,new
%A A073298 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073298 Inverse permutation: A073299. Cf. A073290-A073297. Occurs in A073200 as the row 2596188043348680205834878944583883.
%D A073298
%p A073298
%I A073299
%S A073299 0,1,2,3,5,4,6,8,7,12,13,11,10,9,15,14,19,21,22,16,20,18,17,32,31,34,36,35,30,33,29,27,26,28,
%T A073299 25,23,24,40,41,39,38,37,52,51,56,59,58,60,62,64,63,43,42,53,57,61,47,55,50,49,44,54,48,45,
%U A073299 46,92,91,90,87,88,97,96,103,106,105,99,104,100,101,89,86,95,102,98,85,94,83,78,77,82,76,73
%N A073299 Permutation A069768 applied six times or permutation A073291 "cubed" or permutation A073293 "squared".
%F A073299 a(n) = A069768^6(n) = A073291(A073291(A073291(n))) = A073293(A073293(n))
%R A073299
%H A073299 Index entries for sequences that are permutations of the natural numbers
%O A073299 0,3
%K A073299 nonn,new
%A A073299 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 25 2002
%Y A073299 Inverse permutation: A073298. Cf. A073290-A073297.
%D A073299
%p A073299
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Yours,
Antti Karttunen