A072788 -id:A072788 - cp's OEIS frontend

A072789 The size of the parenthesizations obtained with the global ranking/unranking scheme presented in A072787-A072788.

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, 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, 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

Offset: 1

Antti Karttunen, Jun 12 2002

To get a cleaner looking table, the term a(0)=0 is not listed here.

A072790(n) gives the maximum position where the value n occurs. See the comment at A072787. Same triangle computed modulo 2: A072792.

A072734 Simple triangle-stretching N X N -> N bijection, variant of A072732.

Original entry on oeis.org

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, 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, 180, 128, 85, 51, 26, 29, 58

Offset: 0

Antti Karttunen, Jun 12 2002

Index entries for sequences that are permutations of the natural numbers

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.

(define (A072734 n) (packA072734 (A025581 n) (A002262 n)))
(define (packA001477 x y) (/ (+ (expt (+ x y) 2) x (* 3 y)) 2))
(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)))))))

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 N X N -> N.

Original entry on oeis.org

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, 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, 211, 46, 17, 299, 120, 5671

Offset: 0

Antti Karttunen, Jun 12 2002

nonn

This ranking scheme condenses the structures of the same size (cf. A072789) somewhat better than scheme presented in A072656 (which uses the N X N -> 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.

A. Karttunen, Gatomorphisms
Index entries for sequences that are permutations of the natural numbers

Inverse permutation: A072788. Cf. also A014486, A072734, A072789.

A215406 A ranking algorithm for the lexicographic ordering of the Catalan families.

Original entry on oeis.org

0, 0, 1, 1, 1, 1, 1, 1, 1, 1, 2, 2, 3, 3, 3, 3, 1, 1, 1, 1, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 4, 4, 5, 5, 5, 5, 5, 5, 6, 6, 7, 7, 7, 7, 8, 8, 8, 8, 8, 8, 8, 8, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 2, 3, 3, 3, 3, 3, 3, 3, 3, 4, 4, 4

Offset: 0

Peter Luschny, Aug 09 2012

See Antti Karttunen's code in A057117. Karttunen writes: "Maple procedure CatalanRank is adapted from the algorithm 3.23 of the CAGES (Kreher and Stinson) book."

For all n>0, a(A014486(n)) = n = A080300(A014486(n)). The sequence A080300 differs from this one in that it gives 0 for those n which are not found in A014486. - Antti Karttunen, Aug 10 2012

G. C. Greubel, Table of n, a(n) for n = 0..1000
A. Karttunen, Catalan's Triangle and ranking of Mountain Ranges
D. L. Kreher and D. R. Stinson, Combinatorial Algorithms, Generation, Enumeration and Search, CRC Press, 1998.
F. Ruskey, Algorithmic Solution of Two Combinatorial Problems, Thesis, Department of Applied Physics and Information Science, University of Victoria, 1978.

Cf. A213704, A057117, A057164, A057505, A057506, A057501, A057502, A057511, A057512, A057123, A057117, A057509, A057510, A057161, A057162, A072766, A071654, A071652, A075161, A061856, A072635, A072788, A057518, A075168, A080119, A057120, A072773.

A215406 := proc(n) local m,a,y,t,x,u,v;
m := iquo(A070939(n), 2);
a := A030101(n);
y := 0; t := 1;
for x from 0 to 2*m-2 do
    if irem(a, 2) = 1 then y := y + 1
    else u := 2*m - x;
         v := m-1 - iquo(x+y,2);
         t := t + A037012(u,v);
         y := y - 1 fi;
    a := iquo(a, 2) od;
A014137(m) - t end:
seq(A215406(i),i=0..199); # Peter Luschny, Aug 10 2012

A215406[n_] := Module[{m, d, a, y, t, x, u, v}, m = Quotient[Length[d = IntegerDigits[n, 2]], 2]; a = FromDigits[Reverse[d], 2]; y = 0; t = 1; For[x = 0, x <= 2*m - 2, x++, If[Mod[a, 2] == 1, y++, u = 2*m - x; v = m - Quotient[x + y, 2] - 1; t = t - Binomial[u - 1, v - 1] + Binomial[u - 1, v]; y--]; a = Quotient[a, 2]]; (1 - I*Sqrt[3])/2 - 4^(m + 1)*Gamma[m + 3/2]*Hypergeometric2F1[1, m + 3/2, m + 3, 4]/(Sqrt[Pi]*Gamma[m + 3]) - t]; Table[A215406[n] // Simplify, {n, 0, 86}] (* Jean-François Alcover, Jul 25 2013, translated and adapted from Peter Luschny's Maple program *)

def A215406(n) : # CatalanRankGlobal(n)
    m = A070939(n)//2
    a = A030101(n)
    y = 0; t = 1
    for x in (1..2*m-1) :
        u = 2*m - x; v = m - (x+y+1)/2
        mn = binomial(u, v) - binomial(u, v-1)
        t += mn*(1 - a%2)
        y -= (-1)^a
        a = a//2
    return A014137(m) - t

cp's OEIS Frontend

A072789 The size of the parenthesizations obtained with the global ranking/unranking scheme presented in A072787-A072788.

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A072734 Simple triangle-stretching N X N -> N bijection, variant of A072732.

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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 N X N -> N.

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A215406 A ranking algorithm for the lexicographic ordering of the Catalan families.

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Maple

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Sage