0,2

The binary Dyck-Language (A063171) in decimal representation.

These encode width 2n mountain ranges, rooted planar trees of n+1 vertices and n edges, planar planted trees with n nodes, rooted plane binary trees with n+1 leaves (2n edges, 2n+1 vertices, n internal nodes, the root included), Dyck words, binary bracketings, parenthesizations, non-crossing handshakes and partitions and many other combinatorial structures in Catalan family, enumerated by A000108.

Is sum(k=1,n,a(k)) / n^(5/2) bounded? - Benoit Cloitre (benoit7848c(AT)orange.fr), Aug 18 2002

N. G. De Bruijn and B. J. M. Morselt, A note on plane trees, J. Combinatorial Theory 2 (1967), 27-34

R. K. Guy, The second strong law of small numbers. Math. Mag. 63 (1990), no. 1, 3-20.

Franklin T. Adams-Watters, Table of n, a(n) for n = 0..2500

A. Karttunen, Illustration of 626 initial terms (up to size n=7) with various structures and natural isomorphisms between them

A. Karttunen, gatomorf.c - C program for computing this sequence and many of the related automorphisms

A. Karttunen, Some notes on Catalan's Triangle

D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998.

F. Ruskey, Algorithmic Solution of Two Combinatorial Problems

R. P. Stanley, Hipparchus, Plutarch, Schroeder and Hough, Am. Math. Monthly, Vol. 104, No. 4, p. 344, 1997.

R. P. Stanley, Exercises on Catalan and Related Numbers

Index entries for encodings of plane rooted trees (various subsets of this sequence).

Index entries for sequences related to parenthesizing

Index entries for signature-permutations induced by Catalan automorphisms (permutations of natural numbers induced by various bijective operations acting on these structures)

Index entries for the sequences induced by list functions of Lisp (sequences induced by various other operations on these codes or the corresponding structures).

a(19) = 226[dec] = 11100010[bin] = A063171(19) as bracket expression: ( ( ( ) ) )( ) and as a binary tree, proceeding from left to right in depth-first fashion, with 1's in binary expansion standing for internal (branching) nodes and 0's for leaves:

..0...0

...\./

....1...0.0..(0)

.....\./...\./

......1.....1

.......\.../

.........1

Note that in this coding scheme the last leaf of the binary trees (here in parentheses) is implicit. This tree can be also converted to a particular S-expression in languages like Lisp, Scheme and Prolog, if we interpret its internal nodes (1's) as cons cells with each leftward leaning branch being the "car" and the rightward leaning branch the "cdr" part of the pair, with the terminal nodes (0's) being ()'s (NILs). Thus we have (cons (cons (cons () ()) ()) (cons () ())) = '( ( ( () . () ) . () ) . ( () . () ) ) = (((())) ()) i.e. the same bracket epression as above, but surrounded by extra parentheses. This mapping is effected by the Scheme function A014486->parenthesization given below.

Maple procedure CatalanUnrank is adapted from the algorithm 3.24 of the CAGES book and the Scheme function CatalanUnrank from Ruskey's thesis. See the gatomorf.c program for the corresponding C-procedures.

CatalanSequences := proc(upto_n) local n, a, r; a := []; for n from 0 to upto_n do for r from 0 to (binomial(2*n, n)/(n+1))-1 do a := [op(a), CatalanUnrank(n, r)]; od; od; RETURN(a); end;

CatalanUnrank := proc(n, rr) local r, x, y, lo, m, a; r := (binomial(2*n, n)/(n+1))-(rr+1); y := 0; lo := 0; a := 0; for x from 1 to 2*n do m := Mn(n, x, y+1); if(r <= lo+m-1) then y := y+1; a := 2*a + 1; else lo := lo+m; y := y-1; a := 2*a; fi; od; RETURN(a); end;

Mn := (n, x, y) -> binomial(2*n-x, n-((x+y)/2)) - binomial(2*n-x, n-1-((x+y)/2));

cat[ n_ ] := (2 n)!/n!/(n+1)!; b2d[li_List] := Fold[2#1+#2&, 0, li]; d2b[n_Integer] := IntegerDigits[n, 2] tree[n_] := Join[Table[1, {i, 1, n}], Table[0, {i, 1, n}]]

nexttree[t_] := Flatten[Reverse[t]/. {a___, 0, 0, 1, b___}:> Reverse[{Sort[{a, 0}]//Reverse, 1, 0, b}]]

wood[ n_ /; n<8 ] := NestList[ nexttree, tree[ n ], cat[ n ]-1 ]

Table[ Reverse[ b2d/@wood[ j ] ], {j, 0, 6} ]//Flatten

(MIT Scheme) (define (A014486 n) (let ((w/2 (A072643 n))) (CatalanUnrank w/2 (if (zero? n) 0 (- n (A014137 (-1+ w/2)))))))

(Here 'm' is the row on A009766 and 'y' is the position on row 'm' of A009766, both >= 0. The resulting totally balanced binary string is computed into variable 'a'): (define (CatalanUnrank size rank) (let loop ((a 0) (m (-1+ size)) (y size) (rank rank) (c (A009766 (-1+ size) size))) (if (negative? m) a (if (>= rank c) (loop (1+ (* 2 a)) m (-1+ y) (- rank c) (A009766 m (-1+ y))) (loop (* 2 a) (-1+ m) y rank (A009766 (-1+ m) y))))))

(This converts the totally balanced binary string 'n' into the corresponding S-expression:) (define (A014486->parenthesization n) (let loop ((n n) (stack (list (list)))) (cond ((zero? n) (car stack)) ((zero? (modulo n 2)) (loop (floor->exact (/ n 2)) (cons (list) stack))) (else (loop (floor->exact (/ n 2)) (cons2top! stack))))))

(define (cons2top! stack) (let ((ex-cdr (cdr stack))) (set-cdr! stack (car ex-cdr)) (set-car! ex-cdr stack) ex-cdr))

Characteristic function: A080116. Inverse function: A080300.

The terms of binary width 2n are counted by A000108[n]. Subset of A036990. Number of peaks in each mountain (number of leaves in rooted plane general trees): A057514. Number of trailing zeros in the binary expansion: A080237. First differences: A085192.

Cf. also A009766, A014137, A071156, A072643, A079436, A085184.

Sequence in context: A055701 A154391 A035928 this_sequence A166751 A071162 A075165

Adjacent sequences: A014483 A014484 A014485 this_sequence A014487 A014488 A014489

nonn,nice,easy,base

Wouter Meeussen (wouter.meeussen(AT)pandora.be)

Additional comments from Antti Karttunen, Aug 11 2000 and May 25 2004.