cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-6 of 6 results.

A057518 The global ranks of each term of A057517, i.e., tells that A057515(A057518(n)) = 2 for all n.

Original entry on oeis.org

2, 5, 6, 12, 13, 15, 16, 19, 31, 32, 34, 35, 36, 40, 41, 43, 44, 47, 52, 53, 56, 60, 87, 88, 90, 91, 92, 96, 97, 99, 100, 101, 103, 104, 105, 106, 115, 116, 118, 119, 120, 124, 125, 127, 128, 131, 136, 137, 140, 144, 152, 153, 155, 156, 159, 164, 165, 168, 172, 178
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Crossrefs

Cf. A057517, A057519.

Programs

  • Maple
    map(CatalanRankGlobal, A057517); # CatalanRankGlobal given in A057117.

A057519 The local ranks of each term of A057517.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 10, 8, 9, 11, 12, 13, 17, 18, 20, 21, 24, 29, 30, 33, 37, 22, 23, 25, 26, 27, 31, 32, 34, 35, 36, 38, 39, 40, 41, 50, 51, 53, 54, 55, 59, 60, 62, 63, 66, 71, 72, 75, 79, 87, 88, 90, 91, 94, 99, 100, 103, 107, 113, 114, 117, 121, 126, 64, 65, 67, 68, 69
Offset: 1

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Crossrefs

Cf. A057517, A057518.

Programs

A057501 Signature-permutation of a Catalan Automorphism: Rotate non-crossing chords (handshake) arrangements; rotate the root position of general trees as encoded by A014486.

Original entry on oeis.org

0, 1, 3, 2, 7, 8, 5, 4, 6, 17, 18, 20, 21, 22, 12, 13, 10, 9, 11, 15, 14, 16, 19, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 31, 32, 34, 35, 36, 26, 27, 24, 23, 25, 29, 28, 30, 33, 40, 41, 38, 37, 39, 43, 42, 44, 47, 52, 51, 53, 56, 60, 129, 130, 132, 133, 134
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000; entry revised Jun 06 2014

Keywords

Comments

This is a permutation of natural numbers induced when "noncrossing handshakes", i.e., Stanley's interpretation (n), "n nonintersecting chords joining 2n points on the circumference of a circle", are rotated.
The same permutation is induced when the root position of plane trees (Stanley's interpretation (e)) is successively changed around the vertices.
For a good illustration how the rotation of the root vertex works, please see the Figure 6, "Rotation of an ordered rooted tree" in Torsten Mütze's paper (on page 24 in 20 May 2014 revision).
For yet another application of this permutation, please see the attached notes for A085197.
By "recursivizing" either the left or right hand side argument of A085201 in the formula, one ends either with A057161 or A057503. By "recursivizing" the both sides, one ends with A057505. - Antti Karttunen, Jun 06 2014

Links

Crossrefs

Inverse: A057502.
Also, a "SPINE"-transform of A074680, and thus occurs as row 17 of A122203. (Also as row 65167 of A130403.)
Successive powers of this permutation, a^2(n) - a^6(n): A082315, A082317, A082319, A082321, A082323.
Other related permutations: A057161, A057163, A057503, A057505, A057508, A057509, A057511, A069770, A069771, A069772, A069773, A069888, A069889, A082313, A082314, A085173, A086427, A123501, A127291, A127292.
Cf. also A057548, A072771, A072772, A085201, A002995 (cycle counts), A057543 (max cycle lengths), A085197, A129599, A057517, A064638, A064640.

Programs

  • Maple
    map(CatalanRankGlobal,map(RotateHandshakes, A014486));
RotateHandshakes := n -> pars2binexp(RotateHandshakesP(binexp2pars(n)));
RotateHandshakesP := h -> `if`((0 = nops(h)),h,[op(car(h)),cdr(h)]); # This does the trick! In Lisp: (defun RotateHandshakesP (h) (append (car h) (list (cdr h))))
car := proc(a) if 0 = nops(a) then ([]) else (op(1,a)): fi: end: # The name is from Lisp, takes the first element (head) of the list.
cdr := proc(a) if 0 = nops(a) then ([]) else (a[2..nops(a)]): fi: end: # As well. Takes the rest (the tail) of the list.
PeelNextBalSubSeq := proc(nn) local n,z,c; if(0 = nn) then RETURN(0); fi; n := nn; c := 0; z := 0; while(1 = 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); if(c >= 0) then RETURN((z - 2^(floor_log_2(z)))/2); fi; od; end;
RestBalSubSeq := proc(nn) local n,z,c; n := nn; c := 0; while(1 = 1) do c := c + (-1)^n; n := floor(n/2); if(c >= 0) then break; fi; od; z := 0; c := -1; while(1 = 1) do z := 2*z + (n mod 2); c := c + (-1)^n; n := floor(n/2); if(c >= 0) then RETURN(z/2); fi; od; end;
pars2binexp := proc(p) local e,s,w,x; if(0 = nops(p)) then RETURN(0); fi; e := 0; for s in p do x := pars2binexp(s); w := floor_log_2(x); e := e * 2^(w+3) + 2^(w+2) + 2*x; od; RETURN(e); end;
binexp2pars := proc(n) option remember; `if`((0 = n),[],binexp2parsR(binrev(n))); end;
binexp2parsR := n -> [binexp2pars(PeelNextBalSubSeq(n)),op(binexp2pars(RestBalSubSeq(n)))];
# Procedure CatalanRankGlobal given in A057117, other missing ones in A038776.

Formula

a(0) = 0, and for n>=1, a(n) = A085201(A072771(n), A057548(A072772(n))). [This formula reflects directly the given non-destructive Lisp/Scheme function: A085201 is a 2-ary function corresponding to 'append', A072771 and A072772 correspond to 'car' and 'cdr' (known also as first/rest or head/tail in some dialects), and A057548 corresponds to unary form of function 'list'].
As a composition of related permutations:
a(n) = A057509(A069770(n)).
a(n) = A057163(A069773(A057163(n))).
Invariance-identities:
A129599(a(n)) = A129599(n) holds for all n.

A057548 A014486-indices of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

Original entry on oeis.org

1, 3, 7, 8, 17, 18, 20, 21, 22, 45, 46, 48, 49, 50, 54, 55, 57, 58, 59, 61, 62, 63, 64, 129, 130, 132, 133, 134, 138, 139, 141, 142, 143, 145, 146, 147, 148, 157, 158, 160, 161, 162, 166, 167, 169, 170, 171, 173, 174, 175, 176, 180, 181, 183, 184, 185, 187, 188
Offset: 0

Views

Author

Antti Karttunen, Sep 07 2000

Keywords

Comments

This sequence is induced by the unary form of function 'list' (present in Lisp and Scheme) when it acts on symbolless S-expressions encoded by A014486/A063171.

Links

Crossrefs

We have A057515(A057548(n)) = 1 for all n. Row 0 of A072764. Column 1 of A085203. Cf. A057517, A057549, A057551.

Formula

a(n) = A080300(A057547(n)) = A069770(A072795(n)).

A057122 The binary encoding (as a rooted planar tree) of each rooted planar binary tree. See A057123 for illustration.

Original entry on oeis.org

0, 10, 180, 210, 2920, 2980, 3380, 3490, 3730, 46800, 46920, 47720, 47940, 48420, 54120, 54180, 55860, 56130, 56610, 59700, 59810, 60690, 62610, 748960, 749200, 750800, 751240, 752200, 763600, 763720, 767080, 767620, 768580, 774760, 774980
Offset: 0

Views

Author

Antti Karttunen, Aug 11 2000

Keywords

Comments

bintree_depth_first2tree given in A057119.

Crossrefs

Cf. A057119, A057123, A057124. Subset of A057517.

Programs

  • Maple
    map(bintree_depth_first2tree, A014486);

A057547 A014486-encodings of Catalan mountain ranges with no sea-level valleys, i.e., the rooted plane general trees with root degree = 1.

Original entry on oeis.org

2, 12, 52, 56, 212, 216, 228, 232, 240, 852, 856, 868, 872, 880, 916, 920, 932, 936, 944, 964, 968, 976, 992, 3412, 3416, 3428, 3432, 3440, 3476, 3480, 3492, 3496, 3504, 3524, 3528, 3536, 3552, 3668, 3672, 3684, 3688, 3696, 3732, 3736, 3748, 3752, 3760
Offset: 0

Views

Author

Antti Karttunen Sep 07 2000

Keywords

Comments

This one-to-one correspondence between all rooted plane trees and one node larger, root degree = 1 trees illustrates the fact that INVERT(A000108) = LEFT(A000108). (Catalan numbers shift left under Cameron's A transformation.)
From Ruud H.G. van Tol, May 13 2024: (Start)
Sequence on a lattice:
Tree Paths Decimal Count
|_ 10 2 1
|. 1100 12 1
||._ 110100 -111000 52,56 2
|||_._ 11010100 -11110000 212-240 5
|||_|. 1101010100-1111100000 852-992 14
... (End)

Links

Crossrefs

Double-trunked trees: A057517. Cf. also A057548, A057549.

Programs

  • Maple
    alltrees2singletrunked := n -> pars2binexp([binexp2pars(n)]); # Just surround with extra parentheses.
  • PARI
    a_rows(N) = my(a=Vec([[2]], N)); for(r=1, N-1, my(b=a[r], c=List()); foreach(b, t, for(i=1, valuation(t, 2), listput(~c, (t<<2)+(2<Ruud H.G. van Tol, May 25 2024

Formula

a(n) = A014486(A057548(n)) and also from n > 0 onward = A079946(A014486(n)).
a(n) = alltrees2singletrunked(A014486[n]) (see Maple code below and in A057501).
Showing 1-6 of 6 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.