Cheers!
Here follows additions to three rows in "Permutation of natural numbers..."
index, a piece of Maple code for your transforms file,
another re-edit of A071673
and all-together 40 PRE_NUMBERED new sequences,
A072732-A072741, A072764-A072773, A072781-A072800.
-----------------------------------------------------------------------
Additions to the index entry
http://www.research.att.com/~njas/sequences/Sindx_Per.html#IntegerPermutation
(Filled the line 8 with 3 new pairs A072732-A072733 A072734-A072735 A072793-A072794)
permutations, of the integers, each paired with its inverse: ( 8)
A060736-A064788 A054068-A054069 A057028-A064789 A072732-A072733 A072734-A072735 A072793-A072794
(A072796 & A072797 added to the end of the following line)
permutations, of the integers, induced by Catalan automorphisms,
involutions: A057163 A057164 A057508 A069766 A069769 A069770 A069771
A069772 A069787 A069888 A069889 A072796 A072797
(A072787-A072788, A072764-A072765, A072766-A072767 added to the end of the following line)
permutations, of the integers, induced by Catalan rerankings, each paired
with its inverse: (2) A072646-A072647, A072787-A072788, A072764-A072765, A072766-A072767
Later when I have more time, I would like pick all the tabular/triangular
(with keyword tabl) permutations apart from those lists, and construct
a new sub-entry like:
permutations, of the integers, tabular, i.e. NxN->N bijections:
which would list A000027*, A001477*, A054238-A054239, Wythoff's array, etc.
i.e. all the permutations that are usable as NxN->N bijections.
-----------------------------------------------------------------------
If you wish, add the following Maple-code to
http://www.research.att.com/~njas/sequences/transforms.txt
# Transforms ADDONE, RIGHT0, SUMTABL, RAST and RASTxx contributed by
# Antti Karttunen 12. July 2002.
add1 := n -> n+1:
ADDONE := a -> map(add1,a): # Increment each term by one.
RIGHT0 := a -> [0,op(a)]: # Shift right, prepending 0.
# SUMTABL(A001477,A001477) gives A003056.
# SUMTABL(A000027,A000027) gives A003057.
# and SUMTABL(A000027,A001477) & SUMTABL(A001477,A000027) give A002024.
A025581 := n -> binomial(1+floor((1/2)+sqrt(2*(1+n))),2) - (n+1); # The X-projection &
A002262 := n -> n - binomial(floor((1/2)+sqrt(2*(1+n))),2); # Y-projection (of A001477)
SUMTABL := proc(a,b) local c,i,u;
u := binomial(min(nops(a),nops(b))+1,2);
c := [];
for i from 0 to u-1
do
c := [ op(c), a[A025581(i)+1]+b[A002262(i)+1] ];
od;
RETURN(c);
end:
# Theorem: The set of the sequences where each n >= 0 occurs A000108(n) times
# (i.e. the permutations of A072643) is closed under these two transformations.
# The fixed point of RASTxx is A071673.
RAST := (a,b) -> RIGHT0(ADDONE(SUMTABL(a,b)));
RASTxx := a -> RIGHT0(ADDONE(SUMTABL(a,a)));
-----------------------------------------------------------------------
(or otherwise I have to add it to %p lines of A072768 and/or A071673)
Here's a re-edition of A071673, again:
%I A071673
%S A071673 1,2,2,3,3,3,3,4,4,3,4,4,5,4,4,4,5,5,5,5,4,4,5,6,5,6,5,4,4,5,6,6,6,6,5,
%T A071673 4,5,5,6,6,7,6,6,5,5,5,6,6,6,7,7,6,6,6,5,4,6,7,6,7,7,7,6,7,6,4,5,5,7,7,
%U A071673 7,7,7,7,7,7,5,5,5,6,6,7,8,7,7,7,8,7,6,6,5,6,6,7,6,8,8,7,7,8,8,6,7,6,6
%N A071673 Sequence a(n) obtained by setting a(0) = 0; then reading the table T(x,y)=a(x)+a(y)+1 in antidiagonal fashion.
%C A071673 The fixed point of RASTxx transformation. The repeated applications of RASTxx starting from A072643 seem to converge toward this sequence. Cf. A072768 from which this differs first time at the position n=37, where A072768(37) = 4, while A071673(37) = 5.
%C A071673 Each value v occurs A000108(v) times. (The term a(0)=0 is not eplicitly listed here as to get a better looking triangle).
%C A071673 The size of each Catalan structure encoded by the corresponding terms in triangles A071671 & A071672 (i.e. the number of digits / 2), as obtained with the global ranking/unranking scheme presented in A071651-A071654.
%H A071673 A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
%H A071673 N. J. A. Sloane, Transforms (Maple code for RASTxx transform)
%F A071673 a(0)=0, a(n)=1+a(A025581(n-1))+a(A002262(n-1))
%e A071673 E.g. we have a(1) = T(0,0) = a(0)+a(0)+1 = 1, a(2)=T(1,0)=a(1)+a(0)+1 = 2, a(3)=T(0,1)=a(0)+a(1)+1 = 2, a(4)=T(2,0)=a(2)+a(0)+1 = 3, etc.
%o A071673 (Scheme function:) (define (A071673 n) (cond ((zero? n) n) (else (+ 1 (A071673 (A025581 (-1+ n))) (A071673 (A002262 (-1+ n)))))))
%Y A071673 Same triangle computed modulo 2: A071674. Permutations: A072643, A072644, A072645, A072660, A072768, 072789. Max. position where value v occurs: A072638(v). Cf. also A025581, A002262.
%K A071673 nonn,tabl,eigen
%O A071673 1,2
%A A071673 Antti Karttunen (my_firstname.my_surname@iki.fi) May 30 2002. Self-referential definition added Jun 03 2002.
-----------------------------------------------------------------------
%I A072732
%S A072732 0,1,4,7,12,2,3,17,24,8,11,31,40,18,5,6,23,49,60,32,13,16,39,71,84,50,25,9,10,30,59,97,112,
%T A072732 72,41,19,22,48,83,127,144,98,61,33,14,15,38,70,111,161,180,128,85,51,26,29,58,96,143,199,220,
%U A072732 162,113,73,42,20,21,47,82,126,179,241,264,200,145,99,62,34,37,69,110,160,219,287,312,242,181
%N A072732 Simple triangle-stretching NxN->N bijection: push terms in the middle twice as far down to make space for the terms obtained by bisecting the edges, which are thus contracted by the same factor. Do this for all "triangle-shells" successively contained inside each other in A001477.
%R A072732
%H A072732 Index entries for sequences that are permutations of the natural numbers
%O A072732 0,3
%K A072732 nonn,tabl,new
%A A072732 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072732 Inverse: A072733, projections: A072736 & A072737, variant of the same theme: A072734. Cf. also A001477 and its projections A025581 & A002262.
%D A072732
%o A072732 (Scheme) (define (A072732 n) (packA072732 (A025581 n) (A002262 n)))
%o A072732 (define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
%o A072732 (define (packA072732 x y) (let ((x-y (- x y))) (cond ((<= x-y 0) (packA001477 (+ (* 2 x) (modulo x-y 2)) (+ (* 2 x) (floor->exact (/ (1+ (- x-y)) 2))))) (else (packA001477 (+ (* 2 (1+ y)) (floor->exact (/ (- x-y 2) 2))) (+ (* 2 y) (modulo (1+ x-y) 2)))))))
%I A072733
%S A072733 0,1,5,6,2,14,15,3,9,27,28,10,4,20,44,45,21,7,13,35,65,66,36,16,8,26,54,90,91,55,29,11,19,43,
%T A072733 77,119,120,78,46,22,12,34,64,104,152,153,105,67,37,17,25,53,89,135,189,190,136,92,56,30,18,
%U A072733 42,76,118,170,230,231,171,121,79,47,23,33,63,103,151,209,275,276,210,154,106,68,38,24,52,88
%N A072733 Simple triangle-stretching NxN->N bijection: Inverse of A072732.
%R A072733
%H A072733 Index entries for sequences that are permutations of the natural numbers
%O A072733 0,3
%K A072733 nonn,tabl,new
%A A072733 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072733 Inverse: A072732, projections: A072738 & A072739, variant of the same theme: A072735. Cf. also A001477 and its projections A025581 & A002262.
%D A072733
%o A072733 (Scheme) (define (A072733 n) (packA072733 (A025581 n) (A002262 n)))
%o A072733 (define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
%o A072733 (define (packA072733 x y) (cond ((<= x y) (let ((half-x (floor->exact (/ x 2)))) (packA001477 half-x (+ half-x (* 2 (- y (* 2 half-x) (modulo x 2))) (modulo x 2))))) (else (let ((half-y (floor->exact (/ y 2)))) (packA001477 (+ 1 half-y (* 2 (- (-1+ x) (* 2 half-y) (modulo y 2))) (modulo y 2)) half-y)))))
%I A072734
%S A072734 0,1,2,3,12,4,7,17,18,5,6,23,40,24,8,11,31,49,50,25,9,10,30,59,84,60,32,13,16,39,71,97,98,61,
%T A072734 33,14,15,38,70,111,144,112,72,41,19,22,48,83,127,161,162,113,73,42,20,21,47,82,126,179,220,
%U A072734 180,128,85,51,26,29,58,96,143,199,241,242,181,129,86,52,27,28,57,95,142,198,263,312,264,200
%N A072734 Simple triangle-stretching NxN->N bijection, variant of A072732.
%R A072734
%H A072734 Index entries for sequences that are permutations of the natural numbers
%O A072734 0,3
%K A072734 nonn,tabl,new
%A A072734 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072734 Inverse: A072735, projections: A072740 & A072741, variant of the same theme: A072732. Used to construct the global arithmetic ranking scheme of plane binary trees presented in A072787/A072788. Cf. also A001477 and its projections A025581 & A002262.
%D A072734
%o A072734 (Scheme) (define (A072734 n) (packA072734 (A025581 n) (A002262 n)))
%o A072734 (define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
%o A072734 (define (packA072734 x y) (let ((x-y (- x y))) (cond ((negative? x-y) (packA001477 (+ (* 2 x) (modulo (1+ x-y) 2)) (+ (* 2 x) (floor->exact (/ (+ (- x-y) (modulo x-y 2)) 2))))) ((< x-y 3) (packA001477 (+ (* 2 y) x-y) (* 2 y))) (else (packA001477 (+ (* 2 y) (floor->exact (/ (1+ x-y) 2)) (modulo (1+ x-y) 2)) (+ (* 2 y) (modulo x-y 2)))))))
%I A072735
%S A072735 0,1,2,3,5,9,10,6,14,20,21,15,4,27,35,36,28,7,8,44,54,55,45,11,13,19,65,77,78,66,22,16,26,34,
%T A072735 90,104,105,91,37,29,12,43,53,119,135,136,120,56,46,17,18,64,76,152,170,171,153,79,67,23,25,
%U A072735 33,89,103,189,209,210,190,106,92,38,30,42,52,118,134,230,252,253,231,137,121,57,47,24,63,75
%N A072735 Simple triangle-stretching NxN->N bijection: Inverse of A072734, variant of A072733.
%R A072735
%H A072735 Index entries for sequences that are permutations of the natural numbers
%O A072735 0,3
%K A072735 nonn,tabl,new
%A A072735 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072735 Inverse: A072734, projections: A072781 & A072782, variant of the same theme: A072733. Cf. also A001477 and its projections A025581 & A002262.
%D A072735
%o A072735 (Scheme) (define (A072735 n) (packA072735 (A025581 n) (A002262 n)))
%o A072735 (define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
%o A072735 (define (packA072735 x y) (cond ((<= x y) (let ((half-x (floor->exact (/ x 2)))) (packA001477 half-x (+ half-x (* 2 (- y (* 2 half-x))) (modulo x 2) (if (and (eq? x y) (even? x)) 0 -1))))) (else (let ((half-y (floor->exact (/ y 2)))) (packA001477 (+ half-y (* 2 (- (-1+ x) (* 2 half-y))) (modulo y 2) (if (and (eq? x (1+ y)) (even? y)) 1 0)) half-y)))))
%I A072736
%S A072736 0,1,0,3,0,0,5,2,0,0,7,4,1,0,0,9,6,2,1,0,0,11,8,4,1,1,0,0,13,10,6,3,1,1,0,0,15,12,8,5,
%T A072736 2,1,1,0,0,17,14,10,7,3,2,1,1,0,0,19,16,12,9,5,2,2,1,1,0,0,21,18,14,11,7,4,2,2,1,1,0,0,23,20,
%U A072736 16,13,9,6,3,2,2,1,1,0,0,25,22,18,15,11,8,4,3,2,2,1,1,0,0,27,24,20,17,13,10,6,3,3,2,2,1,1,0
%N A072736 X-projection of the tabular NxN->N bijection A072732.
%R A072736
%O A072736 0,4
%K A072736 nonn,tabl,new
%A A072736 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072736 The Y-projection is A072737. Composition of A025581 & A072733. A072783(n) = A072740(n)-A072736(n)
%D A072736
%o A072736 (Scheme) (define (A072736 n) (A025581 (A072733 n)))
%I A072737
%S A072737 0,0,2,0,1,4,0,0,3,6,0,0,1,5,8,0,0,1,3,7,10,0,0,1,2,5,9,12,0,0,1,1,4,7,11,14,0,0,1,1,
%T A072737 2,6,9,13,16,0,0,1,1,2,4,8,11,15,18,0,0,1,1,2,3,6,10,13,17,20,0,0,1,1,2,2,5,8,12,15,19,22,0,
%U A072737 0,1,1,2,2,3,7,10,14,17,21,24,0,0,1,1,2,2,3,5,9,12,16,19,23,26,0,0,1,1,2,2,3,4,7,11,14,18,21
%N A072737 Y-projection of the tabular NxN->N bijection A072732.
%R A072737
%O A072737 0,3
%K A072737 nonn,tabl,new
%A A072737 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072737 The X-projection is A072736. Composition of A002262 & A072733. A072784(n) = A072741(n)-A072737(n)
%D A072737
%o A072737 (Scheme) (define (A072737 n) (A002262 (A072733 n)))
%I A072738
%S A072738 0,1,1,2,2,0,2,3,3,1,3,4,4,2,0,3,4,5,5,3,1,4,5,6,6,4,2,0,4,5,6,7,7,5,3,1,5,6,7,8,
%T A072738 8,6,4,2,0,5,6,7,8,9,9,7,5,3,1,6,7,8,9,10,10,8,6,4,2,0,6,7,8,9,10,11,11,9,7,5,3,1,7,8,
%U A072738 9,10,11,12,12,10,8,6,4,2,0,7,8,9,10,11,12,13,13,11,9,7,5,3,1,8,9,10,11,12,13,14,14,12,10,8
%N A072738 X-projection of the tabular NxN->N bijection A072733.
%R A072738
%O A072738 0,4
%K A072738 nonn,tabl,new
%A A072738 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072738 The Y-projection is A072739. Composition of A025581 & A072732. A072785(n) = A072781(n)-A072738(n)
%o A072738 (Scheme) (define (A072738 n) (A025581 (A072732 n)))
%I A072739
%S A072739 0,0,1,1,2,1,0,2,3,2,1,3,4,3,2,0,2,4,5,4,3,1,3,5,6,5,4,3,0,2,4,6,7,6,5,4,1,3,5,7,
%T A072739 8,7,6,5,4,0,2,4,6,8,9,8,7,6,5,1,3,5,7,9,10,9,8,7,6,5,0,2,4,6,8,10,11,10,9,8,7,6,1,3,
%U A072739 5,7,9,11,12,11,10,9,8,7,6,0,2,4,6,8,10,12,13,12,11,10,9,8,7,1,3,5,7,9,11,13,14,13,12,11,10
%N A072739 Y-projection of the tabular NxN->N bijection A072733.
%R A072739
%O A072739 0,5
%K A072739 nonn,tabl,new
%A A072739 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072739 The X-projection is A072738. Composition of A002262 & A072732. A072786(n) = A072782(n)-A072739(n)
%D A072739
%o A072739 (Scheme) (define (A072739 n) (A002262 (A072732 n)))
%I A072740
%S A072740 0,1,0,2,0,0,4,3,0,0,6,5,1,0,0,8,7,2,1,0,0,10,9,3,1,1,0,0,12,11,5,4,1,1,0,0,14,13,7,6,
%T A072740 2,1,1,0,0,16,15,9,8,3,2,1,1,0,0,18,17,11,10,4,2,2,1,1,0,0,20,19,13,12,6,5,2,2,1,1,0,0,22,21,
%U A072740 15,14,8,7,3,2,2,1,1,0,0,24,23,17,16,10,9,4,3,2,2,1,1,0,0,26,25,19,18,12,11,5,3,3,2,2,1,1,0
%N A072740 X-projection of the tabular NxN->N bijection A072734.
%R A072740
%O A072740 0,4
%K A072740 nonn,tabl,new
%A A072740 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072740 The Y-projection is A072741. Composition of A025581 & A072735. A072783(n) = A072740(n)-A072736(n)
%D A072740
%o A072740 (Scheme) (define (A072740 n) (A025581 (A072735 n)))
%I A072741
%S A072741 0,0,1,0,2,3,0,0,4,5,0,0,1,6,7,0,0,1,2,8,9,0,0,1,3,4,10,11,0,0,1,1,5,6,12,13,0,0,1,1,
%T A072741 2,7,8,14,15,0,0,1,1,2,3,9,10,16,17,0,0,1,1,2,4,5,11,12,18,19,0,0,1,1,2,2,6,7,13,14,20,21,0,
%U A072741 0,1,1,2,2,3,8,9,15,16,22,23,0,0,1,1,2,2,3,4,10,11,17,18,24,25,0,0,1,1,2,2,3,5,6,12,13,19,20
%N A072741 Y-projection of the tabular NxN->N bijection A072734.
%R A072741
%O A072741 0,5
%K A072741 nonn,tabl,new
%A A072741 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072741 The X-projection is A072740. Composition of A002262 & A072735. A072784(n) = A072741(n)-A072737(n)
%D A072741
%o A072741 (Scheme) (define (A072741 n) (A002262 (A072735 n)))
%I A072781
%S A072781 0,1,0,2,2,1,2,3,2,0,3,4,4,3,1,3,4,5,4,2,0,4,5,6,6,5,3,1,4,5,6,7,6,4,2,0,5,6,7,8,
%T A072781 8,7,5,3,1,5,6,7,8,9,8,6,4,2,0,6,7,8,9,10,10,9,7,5,3,1,6,7,8,9,10,11,10,8,6,4,2,0,7,8,
%U A072781 9,10,11,12,12,11,9,7,5,3,1,7,8,9,10,11,12,13,12,10,8,6,4,2,0,8,9,10,11,12,13,14,14,13,11,9
%N A072781 X-projection of the tabular NxN->N bijection A072735.
%R A072781
%O A072781 0,4
%K A072781 nonn,tabl,new
%A A072781 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072781 The Y-projection is A072782. Composition of A025581 & A072734. A072785(n) = A072781(n)-A072738(n)
%D A072781
%o A072781 (Scheme) (define (A072781 n) (A025581 (A072734 n)))
%I A072782
%S A072782 0,0,1,0,2,1,1,2,3,2,0,2,4,3,2,1,3,4,5,4,3,0,2,4,6,5,4,3,1,3,5,6,7,6,5,4,0,2,4,6,
%T A072782 8,7,6,5,4,1,3,5,7,8,9,8,7,6,5,0,2,4,6,8,10,9,8,7,6,5,1,3,5,7,9,10,11,10,9,8,7,6,0,2,
%U A072782 4,6,8,10,12,11,10,9,8,7,6,1,3,5,7,9,11,12,13,12,11,10,9,8,7,0,2,4,6,8,10,12,14,13,12,11,10
%N A072782 Y-projection of the tabular NxN->N bijection A072735.
%R A072782
%O A072782 0,5
%K A072782 nonn,tabl,new
%A A072782 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072782 The X-projection is A072781. Composition of A002262 & A072734. A072786(n) = A072782(n)-A072739(n)
%D A072782
%o A072782 (Scheme) (define (A072782 n) (A002262 (A072734 n)))
%I A072783
%S A072783 0,0,0,1,0,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,
%T A072783 0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,
%U A072783 1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0
%V A072783 0,0,0,-1,0,0,-1,1,0,0,-1,1,0,0,0,-1,1,0,0,0,0,-1,1,-1,0,0,0,0,-1,1,-1,1,0,0,0,0,-1,1,-1,1,
%W A072783 0,0,0,0,0,-1,1,-1,1,0,0,0,0,0,0,-1,1,-1,1,-1,0,0,0,0,0,0,-1,1,-1,1,-1,1,0,0,0,0,0,0,-1,1,
%X A072783 -1,1,-1,1,0,0,0,0,0,0,0,-1,1,-1,1,-1,1,0,0,0,0,0,0,0,0,-1,1,-1,1,-1,1,-1,0,0,0,0,0,0,0,0
%N A072783 Differences between A072740 and A072736.
%R A072783
%O A072783 0,1
%K A072783 sign,done,tabl,new
%A A072783 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072783
%o A072783 (Scheme) (define (A072783 n) (- (A072740 n) (A072736 n)))
%I A072784
%S A072784 0,0,1,0,1,1,0,0,1,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,
%T A072784 0,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,0,0,
%U A072784 0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1
%V A072784 0,0,-1,0,1,-1,0,0,1,-1,0,0,0,1,-1,0,0,0,-1,1,-1,0,0,0,1,-1,1,-1,0,0,0,0,1,-1,1,-1,0,0,0,0,
%W A072784 0,1,-1,1,-1,0,0,0,0,0,-1,1,-1,1,-1,0,0,0,0,0,1,-1,1,-1,1,-1,0,0,0,0,0,0,1,-1,1,-1,1,-1,0,0,
%X A072784 0,0,0,0,0,1,-1,1,-1,1,-1,0,0,0,0,0,0,0,-1,1,-1,1,-1,1,-1,0,0,0,0,0,0,0,1,-1,1,-1,1,-1,1,-1
%N A072784 Differences between A072741 and A072737.
%R A072784
%O A072784 0,1
%K A072784 sign,done,tabl,new
%A A072784 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072784
%o A072784 (Scheme) (define (A072784 n) (- (A072741 n) (A072737 n)))
%I A072785
%S A072785 0,0,1,0,0,1,0,0,1,1,0,0,0,1,1,0,0,0,1,1,1,0,0,0,0,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0,
%T A072785 0,1,1,1,1,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,1,0,0,
%U A072785 0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1
%V A072785 0,0,-1,0,0,1,0,0,-1,-1,0,0,0,1,1,0,0,0,-1,-1,-1,0,0,0,0,1,1,1,0,0,0,0,-1,-1,-1,-1,0,0,0,0,
%W A072785 0,1,1,1,1,0,0,0,0,0,-1,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,0,0,
%X A072785 0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1
%N A072785 Differences between A072781 and A072738.
%R A072785
%O A072785 0,1
%K A072785 sign,done,tabl,new
%A A072785 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072785
%o A072785 (Scheme) (define (A072785 n) (- (A072781 n) (A072738 n)))
%I A072786
%S A072786 0,0,0,1,0,0,1,0,0,0,1,1,0,0,0,1,1,0,0,0,0,1,1,1,0,0,0,0,1,1,1,0,0,0,0,0,1,1,1,1,
%T A072786 0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,1,1,
%U A072786 1,1,1,1,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0
%V A072786 0,0,0,-1,0,0,1,0,0,0,-1,-1,0,0,0,1,1,0,0,0,0,-1,-1,-1,0,0,0,0,1,1,1,0,0,0,0,0,-1,-1,-1,-1,
%W A072786 0,0,0,0,0,1,1,1,1,0,0,0,0,0,0,-1,-1,-1,-1,-1,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,0,0,0,-1,-1,
%X A072786 -1,-1,-1,-1,0,0,0,0,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0,0,0,-1,-1,-1,-1,-1,-1,-1,0,0,0,0,0,0,0,0
%N A072786 Differences between A072782 and A072739.
%R A072786
%O A072786 0,1
%K A072786 sign,done,tabl,new
%A A072786 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072786
%o A072786 (Scheme) (define (A072786 n) (- (A072782 n) (A072739 n)))
%I A072787
%S A072787 0,1,3,2,6,5,13,8,4,14,10,36,20,9,25,19,24,11,12,18,38,16,7,44,27,209,77,21,105,66,104,28,35,
%T A072787 65,230,54,15,34,33,75,43,26,85,50,40,37,22,31,191,67,23,51,41,69,107,68,49,92,30,29,32,56,
%U A072787 211,46,17,299,120,5671,820,78,1485,629,1484,135,190,595,6902,434,45,189,171,780,276,119,990
%N A072787 Permutation of natural numbers induced by reranking plane binary trees given in the standard lexicographic order (A014486) with an "arithmetic global ranking algorithm", using A072734 as the packing bijection NxN -> N.
%H A072787 A. Karttunen, Gatomorphisms
%H A072787 Index entries for sequences that are permutations of the natural numbers
%C A072787 This ranking scheme condenses the structures of the same size (cf. A072789) somewhat better than scheme presented in A072656 (which uses the NxN->N bijection A072793). Compare the sequences A072790 and A072640 giving the max positions where the last structure with size n will occur in these orderings, and the respective binary widths A072791 & A072642. However, by using the second or third power of the bijection A072734 one gets even better results in a certain range.
%R A072787
%O A072787 0,3
%K A072787 nonn,new
%A A072787 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072787 Inverse permutation: A072788. C.f. also A014486, A072734, A072789.
%D A072787
%o A072787 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.)
%o A072787 (define A072787 (lexrank->arithrank-bijection packA072734))
%o A072787 (define (lexrank->arithrank-bijection packfun) (lambda (n) (rank-bintree (binexp->parenthesization (A014486 n)) packfun)))
%o A072787 (define (rank-bintree bt packfun) (cond ((not (pair? bt)) 0) (else (1+ (packfun (rank-bintree (car bt) packfun) (rank-bintree (cdr bt) packfun))))))
%I A072788
%S A072788 0,1,3,2,8,5,4,22,7,13,10,17,18,6,9,36,21,64,19,15,12,27,46,50,16,14,41,24,31,59,58,47,60,38,
%T A072788 37,32,11,45,20,191,44,52,120,40,23,78,63,148,144,56,43,51,111,108,35,106,61,196,168,131,178,
%U A072788 153,150,115,116,33,29,49,55,53,172,127,136,149,474,39,107,26,69,171,134,455,128,164,600,42
%N A072788 Inverse permutation of A072787.
%R A072788
%H A072788 A. Karttunen, Gatomorphisms
%H A072788 Index entries for sequences that are permutations of the natural numbers
%O A072788 0,3
%K A072788 nonn,tabl,new
%A A072788 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072788 Inverse permutation: A072787. Cf. also A072740, A072741, A072789.
%D A072788
%o A072788 (Scheme functions below show the essential idea. For a complete source, follow the "Gatomorphisms" link.)
%o A072788 (define A072788 (arithrank->lexrank-bijection A072740 A072741))
%o A072788 (define (arithrank->lexrank-bijection pr1 pr2) (lambda (n) (CatalanRankGlobal (parenthesization->binexp (unrank-bintree n pr1 pr2)))))
%o A072788 (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)))))
%I A072789
%S A072789 1,2,2,3,3,3,4,3,4,4,4,4,3,4,5,4,5,4,4,4,5,5,5,4,4,5,5,5,5,5,5,5,5,5,5,4,5,4,6,5,
%T A072789 5,6,5,5,6,5,6,6,5,5,5,6,6,5,6,5,6,6,6,6,6,6,6,6,5,5,5,5,5,6,6,6,6,7,5,6,5,6,6,6,
%U A072789 7,6,6,7,5,6,7,7,6,6,6,5,6,7,6,7,7,6,6,7,7,7,6,5,5,6,5,6,6,7,7,6,6,6,7,6,6,6,6,6
%N A072789 The size of the parenthesizations obtained with the global ranking/unranking scheme presented in A072787-A072788.
%C A072789 To get a cleaner looking table, the term a(0)=0 is not listed here.
%R A072789
%O A072789 1,2
%K A072789 nonn,tabl,new
%A A072789 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072789 A072790(n) gives the maximum position where the value n occurs. See the comment at A072787. Same triangle computed modulo 2: A072792.
%D A072789
%I A072790
%S A072790 0,1,3,13,38,230,6902,5963330,4445170540277,2469942641522759680444872,762577081551553466590743598512392989564579443140,
%T A072790 72690475663460578305076472056787461439583062421562715705151023011944336192599271053989785936377
%N A072790 Maximum position in A072789 where the value n occurs.
%R A072790
%O A072790 0,3
%K A072790 nonn,new
%A A072790 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072790 Binary widths given by A072791. See the comment at A072787.
%D A072790
%o A072790 (define A072790 (max-n-fun-with-arithrank-scheme packA072734))
%I A072791
%S A072791 0,1,2,4,6,8,13,23,43,82,160,316
%N A072791 Binary widths of A072790.
%R A072791
%O A072791 0,3
%K A072791 nonn,new
%A A072791 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072791 See the comment at A072787. Compare for example with A072641 and A072642.
%D A072791
%I A072792
%S A072792 1,0,0,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0,0,0,1,1,1,0,0,1,1,1,1,1,1,1,1,1,1,0,1,0,0,1,
%T A072792 1,0,1,1,0,1,0,0,1,1,1,0,0,1,0,1,0,0,0,0,0,0,0,0,1,1,1,1,1,0,0,0,0,1,1,0,1,0,0,0,
%U A072792 1,0,0,1,1,0,1,1,0,0,0,1,0,1,0,1,1,0,0,1,1,1,0,1,1,0,1,0,0,1,1,0,0,0,1,0,0,0,0,0
%N A072792 Triangle A072789 computed modulo 2.
%C A072792 To get a cleaner looking table, the term a(0)=0 is not listed here. This is just for the sake of abstract art.
%R A072792
%O A072792 0,1
%K A072792 nonn,tabl,new
%A A072792 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072792
%I A072793
%S A072793 0,1,2,3,4,5,6,7,9,10,8,12,11,17,13,14,15,19,20,22,23,16,25,18,33,24,38,26,27,28,30,31,40,41,
%T A072793 43,44,21,46,32,51,29,67,45,72,34,35,36,53,54,48,49,74,75,56,57,37,59,39,88,50,80,52,122,58,
%U A072793 93,60,61,62,64,65,82,83,85,86,95,96,98,99,42,101,66,106,47,135,87,140,63,156,100,161,68,69
%N A072793 Simple tabular NxN->N bijection: first interleave the bits as with A054238, then apply the bijection A048680.
%R A072793
%H A072793 Index entries for sequences that are permutations of the natural numbers
%O A072793 0,3
%K A072793 nonn,tabl,new
%A A072793 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072793 Inverse: A072794. Composition of A048680 & A054238. The X-projection: A072661, The Y-projection: A072662. Used in global arithmetic ranking scheme presented in A072656.
%D A072793
%o A072793 (define (A072793 n) (A048680 (A054238 n)))
%I A072794
%S A072794 0,1,2,3,4,5,6,7,10,8,9,12,11,14,15,16,21,13,23,17,18,36,19,20,25,22,27,28,29,40,30,31,38,24,
%T A072794 44,45,46,55,26,57,32,33,78,34,35,42,37,82,49,50,59,39,61,47,48,136,53,54,63,56,65,66,67,86,
%U A072794 68,69,80,41,90,91,92,105,43,107,51,52,144,95,96,109,60,111,70,71,140,72,73,84,58,152,153,154
%N A072794 Inverse permutation of A072793.
%R A072794
%H A072794 Index entries for sequences that are permutations of the natural numbers
%O A072794 0,3
%K A072794 nonn,new
%A A072794 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072794 Composition of A054239 & A048679.
%D A072794
%o A072794 (define (A072794 n) (A054239 (A048679 n)))
%I A072795
%S A072795 1,2,4,5,9,10,11,12,13,23,24,25,26,27,28,29,30,31,32,33,34,35,36,65,66,67,68,69,70,71,72,73,
%T A072795 74,75,76,77,78,79,80,81,82,83,84,85,86,87,88,89,90,91,92,93,94,95,96,97,98,99,100,101,102,
%U A072795 103,104,105,106,197,198,199,200,201,202,203,204,205,206,207,208,209,210,211,212,213,214,215
%N A072795 The first column of A072764, the first row of A072766.
%F A072795 a(n) = n + A000108(A072643(n)).
%C A072795 Gives in A014486 the positions of the plane binary trees AND plane general trees whose left subtree is just a stick: \. thus corresponding to the parenthesizations whose first element (of the top-level list) is an empty parenthesization: (), i.e. in A063171 positions of the terms which begin with digits 10...
%Y A072795 Also composition of A069770 & A057548, i.e. a(n) = A069770(A057548(n)).
%R A072795
%O A072795 0,2
%K A072795 nonn,new
%A A072795 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072795
%p A072795
%I A072796
%S A072796 0,1,2,3,4,6,5,7,8,9,10,14,16,19,11,15,12,17,18,13,20,21,22,23,24,25,26,27,37,38,42,44,47,51,
%T A072796 53,56,60,28,29,39,43,52,30,40,31,45,46,32,48,49,50,33,41,34,54,55,35,57,58,59,36,61,62,63,
%U A072796 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 A072796 Self-inverse permutation of natural numbers induced by the automorphism exch2first-cdr! acting on the parenthesizations encoded by A014486.
%C A072796 This automorphism swaps the two leftmost branches of the general trees if the degree of the root > 1, and keeps the tree intact if it's planted (root's degree = 1).
%R A072796
%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 0,3
%K A072796 nonn,new
%A A072796 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072796 The car/cdr-flipped conjugate of A072797, i.e. A072796(n) = A057163(A072797(A057163(n))).
%o A072796 (Scheme function implementing this automorphism on list-structures, first the constructive variant:)
%o A072796 (define (exch2first-cdr 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 (exch2first-cdr! 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))
%I A072797
%S A072797 0,1,2,3,4,5,7,6,8,9,10,11,12,13,17,18,16,14,15,20,19,21,22,23,24,25,26,27,28,29,30,31,32,33,
%T A072797 34,35,36,45,46,48,49,50,44,47,42,37,38,43,39,40,41,54,55,53,51,52,57,56,58,59,61,60,62,63,
%U A072797 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 A072797 Self-inverse permutation of natural numbers induced by the automorphism exch2first-car! acting on the parenthesizations encoded by A014486.
%R A072797
%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 0,3
%K A072797 nonn,new
%A A072797 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072797 The car/cdr-flipped conjugate of A072796, i.e. A072797(n) = A057163(A072796(A057163(n))).
%D A072797
%o A072797 (Scheme function implementing this automorphism on list-structures:)
%o A072797 (define (exch2first-car! 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))
%I A072798
%S A072798 0,1,3,2,4,8,9,7,5,6,12,11,10,13,22,23,26,25,27,21,24,19,14,15,20,17,16,18,35,36,34,31,32,33,
%T A072798 30,28,29,40,39,38,37,41,64,65,68,67,69,77,78,76,73,74,82,81,80,83,63,66,75,72,79,61,70,56,
%U A072798 42,43,57,45,44,46,62,71,59,50,51,58,49,47,48,60,54,53,52,55,105,106,110,109,111,104,108,101
%N A072798 Self-inverse permutation of non-negative integers obtained by contracting the permutation A069787 induced by the telescoping gatomorphism DeepRev1CarSide.
%R A072798
%H A072798 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A072798 Index entries for sequences that are permutations of the natural numbers
%O A072798 0,3
%K A072798 nonn,new
%A A072798 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072798 A072798(n) = A072799(n+1)-1
%D A072798
%p A072798
%I A072799
%S A072799 1,2,4,3,5,9,10,8,6,7,13,12,11,14,23,24,27,26,28,22,25,20,15,16,21,18,17,19,36,37,35,32,33,
%T A072799 34,31,29,30,41,40,39,38,42,65,66,69,68,70,78,79,77,74,75,83,82,81,84,64,67,76,73,80,62,71,
%U A072799 57,43,44,58,46,45,47,63,72,60,51,52,59,50,48,49,61,55,54,53,56,106,107,111,110,112,105,109
%N A072799 Self-inverse permutation of natural numbers: A072798(n-1)+1. (one-based).
%H A072799 Index entries for sequences that are permutations of the natural numbers
%R A072799
%O A072799 1,2
%K A072799 nonn,new
%A A072799 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072799
%p A072799
%I A072800
%S A072800 0,1,5,3,21,13,19,11,7,85,53,77,45,29,83,51,75,43,27,71,39,23,15,341,213,309,181,117,333,205,
%T A072800 301,173,109,285,157,93,61,339,211,307,179,115,331,203,299,171,107,283,155,91,59,327,199,295,
%U A072800 167,103,279,151,87,55,271,143,79,47,31,1365,853,1237,725,469,1333,821,1205,693,437,1141,629
%N A072800 Composition of A030101 and A014486. Binary encodings of parenthesizations & other Catalan structures reversed.
%F A072800 a(n) = A030101(A014486(n))
%R A072800
%O A072800 0,3
%K A072800 nonn,new
%A A072800 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072800
%p A072800
%I A072764
%S A072764 1,3,2,7,6,4,8,16,14,5,17,19,42,15,9,18,44,51,43,37,10,20,47,126,52,121,38,11,21,53,135,127,
%T A072764 149,122,39,12,22,56,154,136,385,150,123,40,13,45,60,163,155,413,386,151,124,41,23,46,128,177,
%U A072764 164,475,414,387,152,125,107,24,48,131,390,178,503,476,415,388,153,371,108,25,49,137,399,391
%N A072764 Tabular NxN->N bijection induced by Lisp/Scheme function 'cons' combining the two planar binary trees/general trees/parenthesizations encoded by A014486(X) and A014486(Y).
%R A072764
%H A072764 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A072764 Index entries for sequences that are permutations of the natural numbers
%O A072764 1,3
%K A072764 nonn,tabl,new
%A A072764 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072764 Inverse permutation: A072765. a(n) = A069770(A072766(n)). Also transpose of A072766, i.e. a(n) = A072766(A038722(n)). The upper triangular region: A072773. Projection functions are A072771 & A072772. The sizes of the corresponding Catalan structures: A072768. The first row: A057548, the first column: A072795. Cf. also A025581, A002262.
%D A072764
%o A072764 (Scheme function showing the general idea:) (define (A072764 n) (cond ((zero? n) n) (else (let ((x (A025581 (-1+ n))) (y (A002262 (-1+ n)))) (CatalanRankGlobal (parenthesization->binexp (cons (binexp->parenthesization (A014486 x)) (binexp->parenthesization (A014486 y)))))))))
%I A072765
%S A072765 0,1,3,2,6,10,5,4,7,15,21,28,36,45,9,14,8,11,16,12,22,29,37,55,66,78,91,105,120,136,153,171,
%T A072765 190,210,231,253,276,20,27,35,44,54,13,19,17,46,56,23,67,79,92,18,25,30,106,121,38,137,154,
%U A072765 172,47,191,211,232,254,300,325,351,378,406,435,465,496,528,561,595,630,666,703,741,780,820
%N A072765 Inverse permutation of A072764.
%R A072765
%H A072765 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A072765 Index entries for sequences that are permutations of the natural numbers
%O A072765 0,3
%K A072765 nonn,new
%A A072765 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072765 Cf. A072767, A001477, A072771, A072772.
%D A072765
%o A072765 (Scheme function showing the general idea:) (define (A072765 n) (cond ((zero? n) n) (else (1+ (packA001477 (A072771 n) (A072772 n))))))
%I A072766
%S A072766 1,2,3,4,6,7,5,14,16,8,9,15,42,19,17,10,37,43,51,44,18,11,38,121,52,126,47,20,12,39,122,149,
%T A072766 127,135,53,21,13,40,123,150,385,136,154,56,22,23,41,124,151,386,413,155,163,60,45,24,107,125,
%U A072766 152,387,414,475,164,177,128,46,25,108,371,153,388,415,476,503,178,390,131,48,26,109,372,461
%N A072766 Transpose of A072764. 'cons' with arguments swapped.
%R A072766
%H A072766 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A072766 Index entries for sequences that are permutations of the natural numbers
%O A072766 1,3
%K A072766 nonn,tabl,new
%A A072766 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072766 Inverse permutation: A072767. a(n) = A069770(A072764(n)). Also transpose of A072764, i.e. a(n) = A072764(A038722(n)). Projection functions are A072772 & A072771. The sizes of the corresponding Catalan structures: A072768. The first column: A057548, the first row: A072795. Cf. also A025581, A002262.
%D A072766
%o A072766 (Scheme function showing the general idea:) (define (A072766 n) (cond ((zero? n) n) (else (let ((y (A025581 (-1+ n))) (x (A002262 (-1+ n)))) (CatalanRankGlobal (parenthesization->binexp (cons (binexp->parenthesization (A014486 x)) (binexp->parenthesization (A014486 y)))))))))
%I A072767
%S A072767 0,1,2,3,4,7,5,6,10,11,16,22,29,37,8,12,9,15,21,14,28,36,45,46,56,67,79,92,106,121,137,154,
%T A072767 172,191,211,232,254,17,23,30,38,47,13,18,20,55,66,27,78,91,105,19,25,35,120,136,44,153,171,
%U A072767 190,54,210,231,253,276,277,301,326,352,379,407,436,466,497,529,562,596,631,667,704,742,781
%N A072767 Inverse permutation of A072766.
%R A072767
%H A072767 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%H A072767 Index entries for sequences that are permutations of the natural numbers
%O A072767 0,3
%K A072767 nonn,new
%A A072767 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072767 Cf. A072765, A001477, A072771, A072772.
%D A072767
%o A072767 (Scheme function showing the general idea:) (define (A072767 n) (cond ((zero? n) n) (else (1+ (packA001477 (A072772 n) (A072771 n))))))
%I A072768
%S A072768 1,2,2,3,3,3,3,4,4,3,4,4,5,4,4,4,5,5,5,5,4,4,5,6,5,6,5,4,4,5,6,6,6,6,5,4,4,5,6,6,
%T A072768 7,6,6,5,4,5,5,6,6,7,7,6,6,5,5,5,6,6,6,7,7,7,6,6,6,5,5,6,7,6,7,7,7,7,6,7,6,5,5,6,
%U A072768 7,7,7,7,7,7,7,7,7,6,5,5,6,7,7,8,7,7,7,7,8,7,7,6,5,5,6,7,7,8,8,7,7,7,8,8,7,7,6,5
%N A072768 The RASTxx transformation of the sequence A072643. The sizes of the parenthesizations produced by 'cons' combination A072764 & its transpose A072766.
%C A072768 Differs from A071673 first time at the position n=37, where A072768(37) = 4, while A071673(37) = 5. RASTxx(A072768) differs from A071673 first time at the position n=704, which leads to conjecture that the repeated applications of RASTxx starting from A072643 converge towards A071673, the fixed point of RASTxx transformation.
%C A072768 Each value v occurs A000108(v) times. (The term a(0)=0 is not explicitly listed here as to get a better looking triangle).
%R A072768
%H A072768 A. Karttunen, Gatomorphisms (Includes the complete Scheme source for computing this sequence)
%H A072768 N. J. A. Sloane, Transforms (Maple code for RASTxx transform)
%O A072768 1,2
%K A072768 nonn,tabl,new
%A A072768 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072768 Same triangle computed modulo 2: A072770. Permutations: A072643, A071673, A072644, A072645, A072660, A072789. Cf. also A072769, A025581, A002262.
%D A072768
%o A072768 (Scheme) (define (A072768 n) (cond ((zero? n) n) (else (+ 1 (A072643 (A025581 (-1+ n))) (A072643 (A002262 (-1+ n)))))))
%I A072769
%S A072769 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%T A072769 0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,1,0,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0,0,0,1,0,1,0,0,0,
%U A072769 1,0,1,0,0,0,1,0,1,0,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0,0,1,0
%V A072769 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0,0,0,
%W A072769 0,0,0,0,1,0,1,0,0,0,0,0,0,1,0,-1,0,1,0,0,0,0,0,1,0,-1,0,-1,0,1,0,0,0,0,1,0,-1,0,0,0,
%X A072769 -1,0,1,0,0,0,1,0,-1,0,0,1,0,0,-1,0,1,0,0,1,0,-1,0,0,1,0,1,0,0,-1,0,1,0,1,0,-1,0,0,1,0
%N A072769 Differences between A071673 and A072768.
%C A072769 The term a(0) = A071673(0)-A072768(0) = 0-0 = 0 is not explicitly listed here as to get a better looking triangle.
%R A072769
%O A072769 1,1
%K A072769 sign,done,tabl,new
%A A072769 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072769
%o A072769 (Scheme) (define (A072769 n) (- (A071673 n) (A072768 n)))
%I A072770
%S A072770 1,0,0,1,1,1,1,0,0,1,0,0,1,0,0,0,1,1,1,1,0,0,1,0,1,0,1,0,0,1,0,0,0,0,1,0,0,1,0,0,
%T A072770 1,0,0,1,0,1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,1,0,1,1,1,1,0,1,0,1,1,0,
%U A072770 1,1,1,1,1,1,1,1,1,0,1,1,0,1,1,0,1,1,1,1,0,1,1,0,1,1,0,1,1,0,0,1,1,1,0,0,1,1,0,1
%N A072770 Triangle A072768 computed modulo 2.
%C A072770 Differs from A071674 first time at the position n=37, where A072770(37) = 0, while A071674(37) = 1.
%C A072770 The term a(0)=0 is not explicitly listed here as to get a better looking triangle.
%R A072770
%O A072770 1,1
%K A072770 nonn,tabl,new
%A A072770 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072770 a(n) = (A072769(n) + A071674(n)) mod 2.
%D A072770
%o A072770 (Scheme) (define (A072770 n) (modulo (A072768 n) 2))
%I A072771
%S A072771 0,0,1,0,0,1,2,3,0,0,0,0,0,1,1,2,4,5,3,6,7,8,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,
%T A072771 1,1,2,2,4,9,10,5,11,12,13,3,3,6,14,15,7,16,17,18,8,19,20,21,22,0,0,0,0,0,0,0,0,0,0,0,0,0,0,
%U A072771 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1
%N A072771 X-projection of the tabular NxN->N bijection A072764 and Y-projection of its transpose A072766.
%C A072771 This corresponds to Lisp/Scheme function 'car' computed with respect to the lexicographical ordering of parenthesizations/planar binary trees (A014486), i.e. with planar binary trees this is equal to extracting the left subtree (from the root), with general parenthesizations equal to taking the first sub-parenthesization of the top-level list, and with general plane trees equal to taking the leftmost branch of the tree (at the root).
%R A072771
%H A072771 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%O A072771 1,8
%K A072771 nonn,new
%A A072771 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072771
%p A072771
%I A072772
%S A072772 0,1,0,2,3,1,0,0,4,5,6,7,8,2,3,1,0,0,1,0,0,0,9,10,11,12,13,14,15,16,17,18,19,20,21,22,4,5,
%T A072772 6,7,8,2,3,1,0,0,1,0,0,0,2,3,1,0,0,1,0,0,0,1,0,0,0,0,23,24,25,26,27,28,29,30,31,32,33,34,35,
%U A072772 36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,52,53,54,55,56,57,58,59,60,61,62,63,64,9,10
%N A072772 Y-projection of the tabular NxN->N bijection A072764 and X-projection of its transpose A072766.
%C A072772 This corresponds to Lisp/Scheme function 'cdr' computed with respect to the lexicographical ordering of parenthesizations/planar binary trees (A014486), i.e. with planar binary trees this is equal to extracting the right subtree (from the root), with general parenthesizations equal to discarding the first sub-parenthesization of the top-level list, and with general plane trees equal to discarding the leftmost branch from the root.
%R A072772
%H A072772 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%O A072772 1,5
%K A072772 nonn,new
%A A072772 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%D A072772
%p A072772
%I A072773
%S A072773 1,3,6,7,16,42,8,19,51,52,17,44,126,127,385,18,47,135,136,413,414,20,53,154,155,475,476,477,
%T A072773 21,56,163,164,503,504,505,506,22,60,177,178,551,552,553,554,555,45,128,390,391,1243,1244,1245,
%U A072773 1246,1247,4089,46,131,399,400,1271,1272,1273,1274,1275,4179,4180,48,137,418,419,1333,1334
%N A072773 The upper triangular region of A072764.
%R A072773
%H A072773 A. Karttunen, Gatomorphisms (With the complete Scheme source)
%O A072773 0,3
%K A072773 nonn,tabl,new
%A A072773 Antti Karttunen (my_firstname.my_surname@iki.fi) Jun 12 2002
%Y A072773 Cf. also A014486, A003056, A002262.
%D A072773
%o A072773 (Scheme function showing the general idea:) (define (A072773 n) (CatalanRankGlobal (parenthesization->binexp (cons (binexp->parenthesization (A014486 (A003056 n))) (binexp->parenthesization (A014486 (A002262 n)))))))
----------------------------------------------------------------------------
Yours,
Antti Karttunen