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 10 results.

A126315 Signature-permutation of a Catalan automorphism: composition of A069771 and A125976.

Original entry on oeis.org

0, 1, 3, 2, 8, 6, 5, 4, 7, 22, 19, 15, 16, 10, 13, 21, 12, 11, 20, 14, 18, 17, 9, 64, 60, 52, 56, 43, 41, 32, 38, 47, 29, 55, 27, 24, 46, 36, 63, 53, 59, 44, 35, 62, 34, 33, 61, 51, 58, 57, 42, 40, 31, 39, 50, 30, 37, 49, 48, 28, 54, 26, 25, 23, 45, 196, 191, 178, 186, 164
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126313/A126314, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126316. a(n) = A069771(A125976(n)) = A126290(A069771(n)) = A126313(A057164(n)). The number of cycles, number of fixed points and maximum cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127281, A127282 and A127283. See also the comment at A127280.

A126316 Signature-permutation of a Catalan automorphism: composition of A125976 and A069771.

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 5, 8, 4, 22, 13, 17, 16, 14, 19, 11, 12, 21, 20, 10, 18, 15, 9, 63, 35, 62, 61, 34, 59, 32, 55, 52, 29, 45, 44, 42, 37, 56, 30, 53, 51, 28, 50, 27, 41, 64, 36, 31, 58, 57, 54, 47, 25, 39, 60, 33, 26, 49, 48, 40, 24, 46, 43, 38, 23, 196, 120, 106, 148, 78
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126313/A126314, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126315. a(n) = A125976(A069771(n)) = A069771(A126290(n)) = A057164(A126314(n)).

A069766 Self-inverse permutation of natural numbers induced by the automorphism RotateHandshakes180 (A069771) or xReflectHandshakes (A069772) acting on the symmetric parenthesizations encoded by A061855.

Original entry on oeis.org

0, 1, 2, 3, 5, 4, 6, 7, 11, 9, 10, 8, 12, 17, 14, 21, 19, 13, 20, 16, 18, 15, 22, 23, 37, 29, 33, 27, 41, 25, 39, 31, 32, 26, 40, 35, 36, 24, 38, 30, 34, 28, 42, 57, 47, 72, 64, 44, 68, 54, 61, 51, 76, 59, 49, 74, 66, 43, 67, 53, 60, 50, 75, 70, 46, 71, 56, 58, 48, 73, 63, 65
Offset: 0

Views

Author

Antti Karttunen, Apr 16 2002

Keywords

Links

Crossrefs

Cf. A061855, A069771, A069772.

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.

A057502 Permutation of natural numbers: rotations of non-crossing handshakes encoded by A014486 (to opposite direction of A057501).

Original entry on oeis.org

0, 1, 3, 2, 7, 6, 8, 4, 5, 17, 16, 18, 14, 15, 20, 19, 21, 9, 10, 22, 11, 12, 13, 45, 44, 46, 42, 43, 48, 47, 49, 37, 38, 50, 39, 40, 41, 54, 53, 55, 51, 52, 57, 56, 58, 23, 24, 59, 25, 26, 27, 61, 60, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 129, 128, 130, 126, 127
Offset: 0

Views

Author

Antti Karttunen, Sep 03 2000

Keywords

Comments

In A057501 and A057502, the cycles between (A014138(n-1)+1)-th and (A014138(n))-th term partition A000108(n) objects encoded by the corresponding terms of A014486 into A002995(n+1) equivalence classes of planar trees, thus the latter sequence can be produced also with Maple procedure RotHandshakesPermutationCycleCounts given below.

Links

Crossrefs

Inverse of A057501 and the car/cdr-flipped conjugate of A069774, i.e. A057502(n) = A057163(A069774(A057163(n))). Cf. also A057507, A057510, A057513, A069771, A069772.

Programs

  • Maple
    map(CatalanRankGlobal,map(RotateHandshakesR, A014486));
RotateHandshakesR := n -> pars2binexp(deepreverse(RotateHandshakesP(deepreverse(binexp2pars(n)))));
deepreverse := proc(a) if 0 = nops(a) or list <> whattype(a) then (a) else [op(deepreverse(cdr(a))), deepreverse(a[1])]; fi; end;
with(group); CountCycles := b -> (nops(convert(b,'disjcyc')) + (nops(b)-convert(map(nops,convert(b,'disjcyc')),`+`)));
RotHandshakesPermutationCycleCounts := proc(upto_n) local u,n,a,r,b; a := []; for n from 0 to upto_n do b := []; u := (binomial(2*n,n)/(n+1)); for r from 0 to u-1 do b := [op(b),1+CatalanRank(n,RotateHandshakes(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
# For other procedures, follow A057501.

A069772 Self-inverse permutation of natural numbers induced by the automorphism xReflectHandshakes acting on the parenthesizations encoded by A014486.

Original entry on oeis.org

0, 1, 2, 3, 7, 6, 5, 4, 8, 9, 10, 21, 20, 19, 14, 15, 18, 17, 16, 13, 12, 11, 22, 45, 46, 44, 42, 43, 31, 32, 30, 28, 29, 63, 62, 61, 60, 54, 55, 53, 51, 52, 26, 27, 25, 23, 24, 59, 58, 57, 56, 40, 41, 39, 37, 38, 50, 49, 48, 47, 36, 35, 34, 33, 64, 65, 67, 66, 68, 69, 170
Offset: 0

Views

Author

Antti Karttunen, Apr 16 2002

Keywords

Comments

This automorphism reflects over the x-axis the interpretation n (the non-crossing handshakes) of Stanley's exercise 19.
Note that DeepRev (A057164) reflects over y-axis.
This transformation keeps palindromic parenthesizations/Dyck paths/rooted planar trees palindromic, but not necessarily same, meaning that this induces a permutation on the sequence A061855 (= A069766).

Links

Crossrefs

Composition of A057164 and A069771 in either order, i.e. A069772(n) = A057164(A069771(n)) = A069771(A057164(n)). Cf. also A061855, A069766, A057501, A069888, A069889.

A126313 Signature-permutation of a Catalan automorphism: composition of A069772 and A125976.

Original entry on oeis.org

0, 1, 3, 2, 8, 5, 6, 4, 7, 22, 13, 15, 12, 14, 19, 21, 16, 11, 18, 10, 20, 17, 9, 64, 36, 41, 35, 40, 52, 53, 38, 34, 39, 55, 51, 37, 54, 60, 63, 32, 62, 31, 56, 59, 47, 33, 50, 27, 58, 49, 26, 43, 44, 29, 61, 30, 24, 57, 48, 25, 46, 42, 28, 23, 45, 196, 106, 120, 105, 119
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126315/A126316, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126314. a(n) = A069772(A125976(n)) = A126290(A069772(n)) = A126315(A057164(n)). The number of cycles, number of fixed points, maximum cycle sizes and LCM's of all cycle sizes in range [A014137(n-1)..A014138(n-1)] of this permutation are given by A127277, A127278, A127279 and A127280. The fixed points are given by A127306. Note the curiosity: this automorphism partitions the A000108(8) = 1430 Catalan structures of size eight (e.g. Dyck paths of length 16) into 79 equivalence classes, of which the largest contains 79 members.

A126314 Signature-permutation of a Catalan automorphism: composition of A125976 and A069772.

Original entry on oeis.org

0, 1, 3, 2, 7, 5, 6, 8, 4, 22, 19, 17, 12, 10, 13, 11, 16, 21, 18, 14, 20, 15, 9, 63, 56, 59, 50, 47, 62, 53, 55, 41, 39, 45, 31, 26, 24, 35, 30, 32, 27, 25, 61, 51, 52, 64, 60, 44, 58, 49, 46, 34, 28, 29, 36, 33, 42, 57, 48, 43, 37, 54, 40, 38, 23, 196, 178, 191, 192, 177
Offset: 0

Views

Author

Antti Karttunen, Jan 16 2007

Keywords

Comments

Like A069771, A069772, A125976 and A126315/A126316, this automorphism keeps symmetric Dyck paths symmetric, but not necessarily same.

Links

Crossrefs

Inverse: A126313. a(n) = A125976(A069772(n)) = A069772(A126290(n)) = A057164(A126316(n)).

A126290 Signature-permutation of a Catalan automorphism: A069772-conjugate of A125976.

Original entry on oeis.org

0, 1, 3, 2, 4, 6, 5, 8, 7, 22, 13, 17, 20, 10, 19, 21, 18, 11, 16, 14, 12, 15, 9, 33, 50, 47, 56, 59, 34, 39, 38, 52, 53, 23, 28, 42, 46, 61, 30, 29, 43, 44, 35, 40, 41, 64, 36, 25, 48, 57, 24, 62, 31, 32, 60, 63, 26, 49, 58, 27, 54, 37, 51, 55, 45, 196, 120, 106, 105, 119
Offset: 0

Views

Author

Antti Karttunen, Jan 02 2007

Keywords

Comments

This is a self-inverse permutation (involution) of the nonnegative integers.

Links

Crossrefs

a(n) = A069772(A125976(A069772(n))) = A069771(A125976(A069771(n))).

A129599 Prime-factorization encoded partition code for the Łukasiewicz-word, variant of A129593.

Original entry on oeis.org

1, 3, 25, 25, 343, 35, 35, 343, 35, 14641, 847, 847, 847, 55, 847, 55, 847, 14641, 847, 55, 847, 847, 55, 371293, 24167, 24167, 1573, 1183, 24167, 1183, 1573, 24167, 1183, 1183, 1183, 1183, 65, 24167, 1183, 1183, 1183, 65, 1573, 1183, 24167
Offset: 0

Views

Author

Antti Karttunen, May 01 2007

Keywords

Comments

In addition to all the automorphisms whose signature permutation satisfies the more restricted condition A127301(SP(n)) = A127301(n) for all n, there are also general tree-rotating automorphisms like *A057501, *A057502, *A069771 and *A069772 that satisfy also the condition A129599(SP(n)) = A129599(n) for all n. However, in contrast to A129593 this is not invariant under the automorphism *A072797. A000041(n) distinct values (seem to) occur in each range [A014137(n)..A014138(n)].

Examples

			The terms A079436(5), A079436(6) and A079436(8) are 2010, 2100 and 1110. After adding one to each number except the first one we get 2121, 2211 and 1221, each one which produces partition 1+1+2+2. Converting it to prime-exponents like explained in A129595, we get 2^0 * 3^0 * 5^1 * 7^1 = 35, thus a(5) = a(6) = a(8) = 35.

Links

Crossrefs

Variant: A129593.

Formula

Construction: add one to each number of the Łukasiewicz-word of a general plane tree encoded by A014486(n) (i.e. A079436(n)) except the first number, sort the numbers into ascending order and interpreting it as a partition of a natural number, encode it in the manner explained in A129595.
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