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 1-10 of 11 results. Next

A085197 Positions of ones in A007001. Repeating part in each sub-permutation A082315[A014137(n-1)..A014138(n-1)] normalized to begin from 1.

Original entry on oeis.org

1, 3, 6, 8, 11, 15, 17, 20, 22, 25, 29, 31, 34, 38, 43, 45, 48, 50, 53, 57, 59, 62, 64, 67, 71, 73, 76, 80, 85, 87, 90, 92, 95, 99, 101, 104, 108, 113, 115, 118, 122, 127, 133, 135, 138, 140, 143, 147, 149, 152, 154, 157, 161, 163, 166, 170, 175, 177, 180, 182, 185, 189
Offset: 1

Views

Author

Antti Karttunen, Jun 14 2003. Proposed by Wouter Meeussen Mar 15 2003

Keywords

Comments

From the second term 3 onward also one more than the partial sums of A076050.

Links

Crossrefs

Cf. A085196. First column of A085180.

Programs

  • Mathematica
    PositionIndex[Nest[Flatten[Map[Range[#+1] &, #]] &, {1}, 6]][[1]] (* Paolo Xausa, Mar 04 2024 *)

Formula

a(n) = A080336(n-1) + n = A082854(A082315(A072795(A081291(n-1)))).
a(n) = n if n < 2, otherwise a(n-1)+A076050(n-1).

A082319 Permutation A057501 applied four times ("^4"), permutation A082315 squared.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 5, 7, 6, 9, 14, 21, 18, 13, 10, 15, 20, 17, 12, 19, 16, 11, 22, 23, 28, 37, 42, 51, 58, 62, 49, 46, 55, 35, 32, 27, 41, 24, 29, 38, 43, 52, 57, 61, 48, 45, 54, 34, 31, 26, 40, 56, 60, 47, 44, 53, 33, 30, 25, 39, 64, 63, 59, 50, 36, 65, 70, 79, 84, 93
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082320. Occurs in A073200. Cf. also A082321-A082324.

Formula

a(n) = A082315(A082315(n))

A082323 Permutation A057501 applied six times ("^6"), permutation A082317 squared, permutation A082315 cubed.

Original entry on oeis.org

0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 21, 10, 20, 19, 11, 22, 12, 17, 16, 13, 18, 14, 15, 23, 37, 58, 49, 35, 24, 38, 57, 48, 34, 56, 47, 33, 64, 25, 39, 59, 50, 36, 26, 40, 54, 45, 31, 53, 44, 30, 63, 27, 41, 55, 46, 32, 51, 42, 28, 62, 52, 43, 29, 61, 60, 65, 79, 107, 121, 149
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082324. a(n) = A057501(A082321(n)). Occurs in A073200.

Formula

a(n) = A082317(A082317(n)) = A082315(A082315(A082315(n)))

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.

A085159 Permutation of natural numbers induced by the Catalan bijection gma085159 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 7, 5, 8, 9, 11, 14, 16, 19, 17, 20, 12, 10, 13, 21, 15, 18, 22, 23, 25, 28, 30, 33, 37, 39, 42, 44, 47, 51, 53, 56, 60, 45, 48, 54, 57, 61, 31, 34, 26, 24, 27, 35, 29, 32, 36, 58, 62, 40, 38, 41, 49, 43, 46, 50, 63, 52, 55, 59, 64, 65, 67, 70, 72, 75
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "rising slope" mapping illustrated in A085161.

Links

Crossrefs

Inverse: A085160. a(n) = A085161(A085160(A085161(n))) = A085169(A082315(A085170(n))) = A074684(A082315(A074683(n))) = A085173(A085173(n)). Occurs in A073200. Cf. also A085165-A085168, A086429. Scheme-function app-to-xrt given in A085203.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082316 Permutation A057502 applied twice ("squared").

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 5, 7, 6, 9, 21, 10, 20, 19, 11, 22, 12, 17, 16, 13, 18, 14, 15, 23, 58, 24, 57, 56, 25, 59, 26, 54, 53, 27, 55, 51, 52, 28, 62, 29, 61, 60, 30, 63, 31, 45, 44, 32, 46, 42, 43, 33, 64, 34, 48, 47, 35, 49, 37, 38, 36, 50, 39, 40, 41, 65, 170, 66, 169, 168
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003, Proposed by Wouter Meeussen

Keywords

Links

Crossrefs

Inverse of A082315. a(n) = A057164(A082313(n)). Occurs in A073200 as row 549755978251. Cf. also A082317-A082324.
Number of cycles: A054357. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057502(A057502(n))

A086429 Permutation of natural numbers induced by the Catalan bijection gma086429 acting on symbolless S-expressions encoded by A014486/A063171.

Original entry on oeis.org

0, 1, 2, 3, 4, 7, 5, 6, 8, 9, 17, 10, 16, 21, 11, 18, 12, 14, 15, 13, 19, 20, 22, 23, 45, 24, 44, 58, 25, 46, 26, 42, 43, 27, 56, 57, 63, 28, 48, 29, 47, 59, 30, 49, 31, 37, 38, 32, 39, 40, 41, 33, 50, 34, 51, 52, 35, 53, 54, 55, 36, 60, 61, 62, 64, 65, 129, 66, 128, 170
Offset: 0

Views

Author

Antti Karttunen, Jun 23 2003

Keywords

Comments

This Catalan bijection rotates the interpretations (pp)-(rr) of Stanley, using the "descending slope" mapping illustrated in A086431.

Links

Crossrefs

Inverse: A086430. a(n) = A086427(A086427(n)) = A086431(A086430(A086431(n))) = A057164(A085159(A057164(n))) = A086425(A082315(A086426(n))). Occurs in A073200.
Number of cycles: A054357. Number of fixed points: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

A082313 Involution of natural numbers: A057501-conjugate of A057164.

Original entry on oeis.org

0, 1, 2, 3, 4, 8, 6, 7, 5, 9, 21, 14, 18, 13, 11, 22, 16, 17, 12, 19, 20, 10, 15, 23, 58, 37, 49, 35, 28, 62, 42, 46, 32, 51, 55, 27, 41, 25, 59, 39, 50, 36, 30, 63, 44, 45, 31, 53, 54, 26, 40, 33, 64, 47, 48, 34, 56, 57, 24, 38, 60, 61, 29, 43, 52, 65, 170, 107, 142, 100
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003. Proposed by Wouter Meeussen in Dec 15 2001

Keywords

Comments

Note: This is isomorphic with Meeussen's "skewcatacycleft" operation acting on the interpretation (gg) of the exercise 19 by Stanley.

Links

Crossrefs

a(n) = A069888(A057502(n)). Occurs in A073200 as row 604463486276865131809167. Cf. also A082314, A082315, A082333, A082334.
Number of cycles: A007123. Number of fixed-points: A001405. Max. cycle size: A046698. LCM of cycle sizes: A046698. (In range [A014137(n-1)..A014138(n-1)] of this permutation, possibly shifted one term left or right).

Formula

a(n) = A057501(A057164(A057502(n)))

A082317 Permutation A057501 applied thrice ("cubed").

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 6, 4, 8, 17, 20, 12, 10, 15, 16, 19, 11, 9, 14, 22, 21, 18, 13, 45, 48, 54, 57, 61, 31, 34, 26, 24, 29, 40, 38, 43, 52, 44, 47, 53, 56, 60, 30, 33, 25, 23, 28, 39, 37, 42, 51, 63, 64, 59, 58, 62, 50, 49, 46, 55, 36, 35, 32, 27, 41, 129, 132, 138, 141, 145
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082318. Occurs in A073200 as row 11150372599265311570767859136464952601119119. Cf. also A082319-A082324.

Formula

a(n) = A057501(A082315(n))

A082321 Permutation A057501 applied five times ("^5").

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 4, 5, 17, 12, 16, 11, 22, 18, 13, 14, 9, 21, 15, 10, 20, 19, 45, 54, 31, 26, 40, 44, 53, 30, 25, 39, 63, 59, 50, 36, 46, 55, 32, 27, 41, 42, 51, 28, 23, 37, 62, 58, 49, 35, 43, 52, 29, 24, 38, 61, 57, 48, 34, 60, 56, 47, 33, 64, 129, 138, 157, 166, 180
Offset: 0

Views

Author

Antti Karttunen, Apr 17 2003

Keywords

Links

Crossrefs

Inverse of A082322. Occurs in A073200. Cf. also A082323-A082324.

Formula

a(n) = A082315(A082317(n))
Showing 1-10 of 11 results. Next

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