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 1-7 of 7 results.

A127290 Signature-permutation of a Catalan automorphism: composition of A057164 and A127292.

Original entry on oeis.org

0, 1, 3, 2, 8, 7, 4, 6, 5, 22, 21, 17, 20, 18, 11, 9, 16, 19, 14, 12, 10, 13, 15, 64, 63, 58, 62, 59, 48, 45, 57, 61, 54, 49, 46, 50, 55, 33, 30, 23, 28, 25, 47, 44, 56, 60, 53, 37, 42, 39, 51, 34, 31, 24, 29, 26, 35, 32, 36, 27, 38, 43, 40, 52, 41, 196, 195, 189, 194, 190
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

Links

Crossrefs

Inverse: A127289. a(n) = A057164(A127292(n)) = A127300(A057164(n)).

A127293 Number of cycles in range [A014137(n-1)..A014138(n-1)] of permutation A127291/A127292.

Original entry on oeis.org

1, 1, 1, 2, 3, 6, 8, 8, 9, 10, 8, 14, 18, 10
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

A127294 Maximum cycle size in range [A014137(n-1)..A014138(n-1)] of permutation A127291/A127292.

Original entry on oeis.org

1, 1, 2, 3, 6, 21, 80, 255, 965, 3349, 9366, 30793, 80798, 396492
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

A127295 Least common multiple of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of permutation A127291/A127292.

Original entry on oeis.org

1, 1, 2, 6, 30, 1260, 1680, 19825740, 10028280, 182416547040, 271239404020200, 219240769050559711332360, 6467876945923041743744426827092900, 27867228909478820644943875389480
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

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

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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.

A127291 Signature-permutation of Elizalde's and Deutsch's 2003 bijection for Dyck paths.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 5, 4, 15, 18, 14, 16, 17, 20, 22, 19, 11, 12, 21, 13, 10, 9, 39, 47, 40, 48, 50, 41, 49, 38, 43, 46, 37, 42, 44, 45, 53, 60, 54, 61, 63, 55, 62, 52, 29, 32, 51, 28, 30, 31, 59, 64, 57, 34, 36, 56, 33, 25, 26, 58, 35, 27, 24, 23, 113, 136, 116, 139, 146
Offset: 0

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Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Deutsch and Elizalde show in their paper that this automorphism converts certain properties concerning "tunnels" of Dyck path to another set of properties concerning the number of hills, even and odd rises, as well as the number of returns (A057515), thus proving the equidistribution of the said parameters.
This automorphism is implemented with function "tau" (Scheme code given below) that takes as its arguments an S-expression and a Catalan automorphism that permutes only the top level of the list (i.e., the top-level branches of a general tree, or the whole arches of a Dyck path) and thus when the permuting automorphism is applied to a list (parenthesization) of length 2n it induces some permutation of [1..2n].
This automorphism is induced in that manner by the automorphism *A127287 and likewise, *A127289 is induced by *A127285, *A057164 by *A057508, *A057501 by *A057509 and *A057502 by *A057510.
Note that so far these examples seem to satisfy the homomorphism condition, e.g., as *A127287 = *A127285 o *A057508 so is *A127291 = *A127289 o *A057164. and likewise, as *A057510 = *A057508 o *A057509 o *A057508, so is *A057502 = *A057164 o *A057501 o *A057164.
However, it remains open what are the exact criteria of the "picking automorphism" and the corresponding permutation that this method would induce a bijection. For example, if we give *A127288 (the inverse of *A127287) to function "tau" it will not induce *A127292 and actually not a bijection at all.
Instead, we have to compute the inverse of this automorphism with another, more specific algorithm that implements Deutsch's and Elizalde's description and is given in A127300.

Links

Crossrefs

Inverse: A127292. a(n) = A127289(A057164(n)) = A057164(A127299(A057164(n))). A127291(A057548(n)) = A072795(A127291(n)), A127291(A072795(n)) = A127307(A127291(A057502(n))) for all n >= 1. The number of cycles, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127293, A127294 and A127295. Number of fixed points begins as 1, 1, 0, 0, 0, 1, 0, 0, 0, 0, 0, 1, 0, 0, ...

A127300 Signature-permutation of A057164-conjugate of the inverse of Elizalde's and Deutsch's 2003 bijection for Dyck paths.

Original entry on oeis.org

0, 1, 3, 2, 8, 4, 7, 6, 5, 22, 11, 17, 16, 12, 21, 9, 20, 19, 10, 18, 14, 13, 15, 64, 33, 48, 47, 34, 58, 23, 57, 56, 24, 49, 37, 35, 38, 63, 30, 45, 44, 31, 62, 28, 61, 60, 29, 46, 42, 32, 43, 59, 25, 54, 53, 26, 50, 39, 36, 40, 55, 51, 27, 52, 41, 196, 102, 145, 144, 103
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Used to construct the inverse for A127291.

References

  • Emeric Deutsch and Sergi Elizalde, A simple and unusual bijection for Dyck paths and its consequences, Annals of Combinatorics, 7 (2003), no. 3, 281-297.

Links

Crossrefs

Inverse: A127299. a(n) = A057164(A127292(A057164(n))) = A127290(A057164(n)). Cf. A014486.
Showing 1-7 of 7 results.

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