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%I #17 Apr 30 2014 01:53:47
%S 0,1,3,2,8,6,5,7,19,15,4,22,16,52,14,13,20,60,43,51,41,11,18,53,178,
%T 42,153,39,10,21,47,155,177,125,151,38,12,61,56,136,154,555,123,150,
%U 40,33,55,179,164,135,479,553,122,152,117,29,17,159,557,163,417,477,552,124
%N Inverse permutation to A071653.
%C A014137(n-1) = A071654(A072638(n)) for n>0 - _Antti Karttunen_, Jul 30 2012, based on _Paul D. Hanna_'s similar observation in A071653.
%H Antti Karttunen, <a href="/A071654/b071654.txt">Rows n=0..66 of triangle, flattened</a>
%H A. Karttunen, <a href="https://oeis.org/wiki/Alternative_Catalan_Orderings">Alternative Catalan Orderings</a>
%H <a href="/index/Per#IntegerPermutation">Index entries for sequences that are permutations of the natural numbers</a>
%o (Scheme functions below show the essential idea. For a complete source, see "Alternative Catalan Orderings" OEIS Wiki page.)
%o (define arithrankA001477->lexrank (arithrank->lexrank-bijection A025581 A002262))
%o (define (arithrank->lexrank-bijection pr1 pr2) (lambda (n) (CatalanRankGlobal (parenthesization->binexp (unrank-bintree n pr1 pr2)))))
%o (define (unrank-bintree rank pr1 pr2) (cond ((zero? rank) (list)) (else (cons (unrank-bintree (pr1 (-1+ rank)) pr1 pr2) (unrank-bintree (pr2 (-1+ rank)) pr1 pr2)))))
%o (define (A002262 n) (- n (binomial_n_2 (floor->exact (+ (/ 1 2) (sqrt (* 2 (1+ n))))))))
%o (define (A025581 n) (- (binomial_n_2 (1+ (floor->exact (+ (/ 1 2) (sqrt (* 2 (1+ n))))))) (1+ n)))
%Y Inverse permutation: A071653. A071672 gives the corresponding parenthesizations (from the term 1 onward) encoded as binary numbers, i.e. A071672(n) = A063171(A071654(n)) for n >= 1.
%Y a(n) = A057163(A071652(n)). Cf. also A014486, A002262, A025581, A071651, A071652.
%K nonn,tabl
%O 0,3
%A _Antti Karttunen_, May 30 2002
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