A057502 -id:A057502 - cp's OEIS frontend

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

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

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

nonn

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

Antti Karttunen, Table of n, a(n) for n = 0..2055
A. Karttunen, Introductory Survey of Catalan Automorphisms and Bijections (an unfinished draft), pp. 56-57.
A. Karttunen et al, Combinatorial interpretations of Catalan numbers, OEIS Wiki.
Torsten Mütze, Proof of the middle levels conjecture, arXiv preprint arXiv:1404.4442 [math.CO], 2014 (p. 24).
R. P. Stanley, Exercises on Catalan and Related Numbers (This sequence is concerned with rotations of the interpretations (e) and (n) in the Exercise 19)
Index entries for signature-permutations of Catalan automorphisms

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.

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.

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.

A074679 Signature permutation of a Catalan automorphism: Rotate binary tree left if possible, otherwise swap its sides.

Original entry on oeis.org

0, 1, 3, 2, 6, 7, 8, 4, 5, 14, 15, 16, 17, 18, 19, 20, 21, 9, 10, 22, 11, 12, 13, 37, 38, 39, 40, 41, 42, 43, 44, 45, 46, 47, 48, 49, 50, 51, 52, 53, 54, 55, 56, 57, 58, 23, 24, 59, 25, 26, 27, 60, 61, 62, 28, 29, 63, 30, 31, 32, 64, 33, 34, 35, 36, 107, 108, 109, 110, 111

Offset: 0

Antti Karttunen, Sep 11 2002

nonn

This automorphism effects the following transformation on the unlabeled rooted plane binary trees (letters A, B, C refer to arbitrary subtrees located on those nodes and () stands for an implied terminal node.)
...B...C.......A...B
....\./.........\./
.A...x....-->....x...C.................A..().........()..A..
..\./.............\./...................\./....-->....\./...
...x...............x.....................x.............x....
(a . (b . c)) -> ((a . b) . c) ____ (a . ()) --> (() . a)
That is, we rotate the binary tree left, in case it is possible and otherwise (if the right hand side of a tree is a terminal node) swap the left and right subtree (so that the terminal node ends to the left hand side), i.e., apply the automorphism *A069770. Look at the example in A069770 to see how this will produce the given sequence of integers.
This is the first multiclause nonrecursive automorphism in table A089840 and the first one whose order is not finite, i.e., the maximum size of cycles in this permutation is not bounded (see A089842). The cycle counts in range [A014137(n-1)..A014138(n)] of this permutation is given by A001683(n+1), which is otherwise the same sequence as for Catalan automorphisms *A057161/*A057162, but shifted once right. For an explanation, please see the notes in OEIS Wiki.

A. Karttunen, Table of n, a(n) for n = 0..2055
A. Karttunen, Introductory Survey of Catalan Automorphisms and Bijections, (an unfinished draft)
A. Karttunen, Notes on the orbits of this permutation, OEIS Wiki.
A. Karttunen, Prolog-program which illustrates the construction of this and similar nonrecursive bijections of oriented binary trees
Index entries for signature-permutations of Catalan automorphisms

This automorphism has several variants, where the first clause is same (rotate binary tree to the left, if possible), but something else is done (than just swapping sides), in case the right hand side is empty: A082335, A082349, A123499, A123695. The following automorphisms can be derived recursively from this one: A057502, A074681, A074683, A074685, A074687, A074690, A089865, A120706, A122321, A122332. See also somewhat similar ones: A069773, A071660, A071656, A071658, A072091, A072095, A072093.
Inverse: A074680.
Row 12 of A089840.
Occurs also in A073200 as row 557243 because a(n) = A073283(A073280(A072796(n))). a(n) = A083927(A123498(A057123(n))).
Number of cycles: LEFT(A001683). Number of fixed points: LEFT(A019590). Max. cycle size & LCM of all cycle sizes: A089410 (in range [A014137(n-1)..A014138(n)] of this permutation).

Description clarified Oct 10 2006

A002995 Number of unlabeled planar trees (also called plane trees) with n nodes.

Original entry on oeis.org

1, 1, 1, 1, 2, 3, 6, 14, 34, 95, 280, 854, 2694, 8714, 28640, 95640, 323396, 1105335, 3813798, 13269146, 46509358, 164107650, 582538732, 2079165208, 7457847082, 26873059986, 97239032056, 353218528324, 1287658723550, 4709785569184

Offset: 0

N. J. A. Sloane

Noncrossing handshakes of 2(n-1) people (each using only one hand) on round table, up to rotations - Antti Karttunen, Sep 03 2000
Equivalently, the number of noncrossing partitions up to rotation composed of n-1 blocks of size 2. - Andrew Howroyd, May 04 2018
a(n), n>2, is also the number of oriented cacti on n-1 unlabeled nodes with all cutpoints of separation degree 2, i.e. ones shared only by two (cyclic) blocks. These are digraphs (without loops) that have a unique Eulerian tour. Such digraphs with labeled nodes are enumerated by A102693. - Valery A. Liskovets, Oct 19 2005
Labeled plane trees are counted by A006963. - David Callan, Aug 19 2014
This sequence is similar to A000055 but those trees are not embedded in a plane. - Michael Somos, Aug 19 2014

			G.f. = 1 + x + x^2 + x^3 + 2*x^4 + 3*x^5 + 6*x^6 + 14*x^7 + 34*x^8 + 95*x^9 + ...
a(7) = 14 = 11 + 3 because there are 11 trees with 7 nodes but three of them can be embedded in a plane in two ways. These three trees have degree sequences 4221111, 3321111, 3222111, where there are two trees with each degree sequence but in the first, the two nodes of degree two are adjacent, in the second, the two nodes of degree three are adjacent, and in the third, the node of degree three is adjacent to two nodes of degree two. - _Michael Somos_, Aug 19 2014

Miklos Bona, editor, Handbook of Enumerative Combinatorics, CRC Press, 2015, page 304.
A. Errera, De quelques problèmes d'analysis situs, Comptes Rend. Congr. Nat. Sci. Bruxelles, (1930), 106-110.
F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 67, (3.3.26).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).

T. D. Noe, Table of n, a(n) for n=0..200
F. Bergeron, G. Labelle and P. Leroux, Combinatorial Species and Tree-Like Structures, Cambridge, 1998, pp. 285 (4.1.26), 291 (4.1.48)
CombOS - Combinatorial Object Server, Generate free plane trees
R. Cori and M. Marcus, Counting non-isomorphic chord diagrams, Theor. Comp. Sci. 204 (1998) 55-75, Corollary 5.2.
Paul Drube and Puttipong Pongtanapaisan, Annular Non-Crossing Matchings, Journal of Integer Sequences, Vol. 19 (2016), #16.2.4.
A. Errera, Reviews of two articles on Analysis Situs, from Fortschritte [Annotated scanned copy]
D. Feldman, Counting plane trees, Unpublished manuscript, 1992. (Annotated scanned copy)
Bernold Fiedler, Scalar polynomial vector fields in real and complex time, arXiv:2410.05043 [math.DS], 2024. See pp. 21, 50.
Bernold Fiedler, Real-time blow-up and connection graphs of rational vector fields on the Riemann sphere, arXiv:2504.20503 [math.DS], 2025. See p. 23.
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335.
F. Harary and R. W. Robinson, The number of achiral trees, J. Reine Angew. Math., 278 (1975), 322-335. (Annotated scanned copy)
G. Labelle, Sur la symétrie et l'asymétrie des structures combinatoires, Theor. Comput. Sci. 117, No. 1-2, 3-22 (1993).
P. Leroux and B. Miloudi, Généralisations de la formule d'Otter, Ann. Sci. Math. Québec, Vol. 16, No. 1, pp. 53-80, 1992. (Annotated scanned copy)
Torsten Mütze, Proof of the middle levels conjecture, arXiv preprint arXiv:1404.4442 [math.CO], 2014.
Torsten Mütze, A book proof of the middle levels theorem, arXiv:2306.13019 [math.CO], 2023.
Torsten Mütze and Franziska Weber, Construction of 2-factors in the middle layer of the discrete cube, arXiv preprint arXiv:1111.2413 [math.CO], 2011.
Torsten Mütze and F. Weber, Construction of 2-factors in the middle layer of the discrete cube, Journal of Combinatorial Theory, Series A, 119(8) (2012), 1832-1855.
J. Sawada, Generating rooted and free plane trees, ACM Transactions on Algorithms, Vol. 2 No. 1 (2006), 1-13.
Seunghyun Seo and Heesung Shin, Two involutions on vertices of ordered trees, FPSAC'02 (2002). (see p_n).
Alexander Stoimenow, On the number of chord diagrams, Discr. Math. 218 (2000), 209-233. See Table 1.
D. W. Walkup, The number of plane trees, Mathematika, vol. 19, No. 2 (1972), 200-204.
Index entries for sequences related to trees

Column k=2 of A303694 and A303864.

Cf. A000055, A000108, A002996, A003239, A005354, A057502, A061417, A064640.

with (powseries): with (combstruct): n := 27: Order := n+2: sys := {C = Cycle(B), B = Union(Z,Prod(B,B))}: G003239 := (convert(gfseries(sys,unlabeled,x) [C(x)], polynom)) / x: G000108 := convert(taylor((1-sqrt(1-4*x)) / (2*x),x),polynom): G002995 := 1 + G003239 + (eval(G000108,x=x^2) - G000108^2)/2: A002995 := 1,1,1,seq(coeff(G002995,x^i),i=1..n); # Ulrich Schimke, Apr 05 2002
with(combinat): with(numtheory): m := 2: for p from 2 to 28 do s1 := 0: s2 := 0: for d from 1 to p do if p mod d = 0 then s1 := s1+phi(p/d)*binomial(m*d, d) fi: od: for d from 1 to p-1 do if gcd(m, p-1) mod d = 0 then s2 := s2+phi(d)*binomial((p*m)/d, (p-1)/d) fi: od: printf(`%d, `, (s1+s2)/(m*p)-binomial(m*p, p)/(p*(m-1)+1)) od : # Zerinvary Lajos, Dec 01 2006

a[0] = a[1] = 1; a[n_] := (1/(2*(n-1)))*Sum[ EulerPhi[(n-1)/d]*Binomial[2*d, d], {d, Divisors[n-1]}] - CatalanNumber[n-1]/2 + If[ EvenQ[n], CatalanNumber[n/2-1]/2, 0]; Table[ a[n], {n, 0, 29}] (* Jean-François Alcover, Mar 07 2012, from formula *)

catalan(n) = binomial(2*n, n)/(n+1);
a(n) = if (n<2, 1, n--; sumdiv(n, d, eulerphi(n/d)*binomial(2*d, d))/(2*n) - catalan(n)/2 + if ((n-1) % 2, 0, catalan((n-1)/2)/2)); \\ Michel Marcus, Jan 23 2016

G.f.: 1+B(x)+(C(x^2)-C(x)^2)/2 where B is g.f. of A003239 and C is g.f. of A000108(n-1).

a(n) = 1/(2*(n-1))*sum{d|(n-1)}(phi((n-1)/d)*binomial(2d, d)) - A000108(n-1)/2 + (if n is even) A000108(n/2-1)/2.

More terms, formula from Christian G. Bower, Dec 15 1999

Name corrected ("labeled" --> "unlabeled") by David Callan, Aug 19 2014

A130402 Signature permutations of ENIPS-transformations of A057163-conjugates of Catalan automorphisms in table A122203.

Original entry on oeis.org

0, 1, 0, 2, 1, 0, 3, 3, 1, 0, 4, 2, 2, 1, 0, 5, 7, 3, 2, 1, 0, 6, 8, 4, 3, 2, 1, 0, 7, 6, 5, 4, 3, 2, 1, 0, 8, 4, 7, 5, 4, 3, 2, 1, 0, 9, 5, 6, 6, 5, 4, 3, 2, 1, 0, 10, 17, 8, 8, 8, 5, 4, 3, 2, 1, 0, 11, 18, 9, 7, 6, 8, 5, 5, 3, 2, 1, 0, 12, 20, 10, 9, 7, 7, 7, 4, 4, 3, 2, 1, 0, 13, 22, 12, 10, 9, 6

Offset: 0

Antti Karttunen, Jun 11 2007

Row n is the signature permutation of the Catalan automorphism which is obtained from A057163-conjugate of the n-th automorphism in the table A122203 with the recursion scheme "ENIPS", i.e. row n is obtained as ENIPS(A057163 o SPINE(A089840[n]) o A057163). See A122203 and A122204 for the description of SPINE and ENIPS. Each row occurs only once in this table. Inverses of these permutations can be found in table A130403. This table contains also all the rows of A122204 and A089840.

Cf. The first 22 rows of this table: row 0 (identity permutation): A001477, 1: A082346, 2: A130935, 3: A073289, 4: A130937, 5: A130939, 6: A130941, 7: A130943, 8: A130945, 9: A130947, 10: A130949, 11: A130951, 12: A074687, 13: A130953, 14: A130955, 15: A130957, 16: A130959, 17: A057162, 18: A130961, 19: A130963, 20: A130965, 21: A069768. Other rows: 251: A069770, 3613: A082340, 65352: A057502.
Cf. also tables A089840, A122201-A122204, A122285-A122286, A130400-A130401.
Cf. As a sequence differs from A130403 for the first time at n=92, where a(n)=22, while A130403(n)=21.

A057510 Permutation of natural numbers: rotations of the bottom branches of the rooted plane trees encoded by A014486. (to opposite direction of A057509).

Original entry on oeis.org

0, 1, 2, 3, 4, 6, 5, 7, 8, 9, 14, 10, 16, 19, 11, 15, 12, 17, 18, 13, 20, 21, 22, 23, 37, 24, 42, 51, 25, 38, 26, 44, 47, 27, 53, 56, 60, 28, 39, 29, 43, 52, 30, 40, 31, 45, 46, 32, 48, 49, 50, 33, 41, 34, 54, 55, 35, 57, 58, 59, 36, 61, 62, 63, 64, 65, 107, 66, 121, 149, 67

Offset: 0

Antti Karttunen, Sep 03 2000

nonn

Index entries for sequences that are permutations of the natural numbers
A. Karttunen, Gatomorphisms (Includes the complete Scheme program for computing this sequence)

Inverse of A057509 and the car/cdr-flipped conjugate of A069776 and also composition of A057502 & A069770, i.e. A057510(n) = A057163(A069776(A057163(n))) = A069770(A057502(n)).

Cycle counts given by A003239. Cf. also A057512, A057513.

# reverse given in A057508, for CountCycles, see A057502, for other procedures, follow A057501.
map(CatalanRankGlobal,map(RotateBottomBranchesR, A014486));
RotateBottomBranchesR := n -> pars2binexp(rotateR(binexp2pars(n)));
rotateR := a -> reverse(rotateL(reverse(a)));
RotBBPermutationCycleCounts := 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,RotateBottomBranchesL(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
A003239 := RotBBPermutationCycleCounts(some_value); (e.g. 9. Cf. A057502, A057162)

A061417 Number of permutations up to cyclic rotations; permutation siteswap necklaces.

Original entry on oeis.org

1, 2, 4, 10, 28, 136, 726, 5100, 40362, 363288, 3628810, 39921044, 479001612, 6227066928, 87178295296, 1307675013928, 20922789888016, 355687438476444, 6402373705728018, 121645100594641896, 2432902008177690360, 51090942175425331320, 1124000727777607680022

Offset: 1

Antti Karttunen, May 02 2001

If permutations are converted to (i,p(i)) permutation arrays, then this automorphism is obtained by their "SW-NE diagonal toroidal shifts" (see Matthias Engelhardt's Java program in A006841), while the Maple procedure below converts each permutation to a siteswap pattern (used in juggling), rotates it by one digit and converts the resulting new (or same) siteswap pattern back to a permutation.
When the subset of permutations listed by A064640 are subjected to the same automorphism one gets A002995.
The number of conjugacy classes of the symmetric group of degree n when conjugating only with the cyclic permutation group of degree n. - Attila Egri-Nagy, Aug 15 2014
Also the number of equivalence classes of permutations of {1...n} under the action of rotation of vertices in the cycle decomposition. The corresponding action on words applies m -> m + 1 for m < n and n -> 1, and rotates once to the right. For example, (24531) first becomes (35142) under the application of cyclic rotation, and then is rotated right to give (23514). - Gus Wiseman, Mar 04 2019

			If I have a five-element permutation like 25431, in cycle notation (1 2 5)(3 4), I mark the numbers 1-5 clockwise onto a circle and draw directed edges from 1 to 2, from 2 to 5, from 5 to 1 and a double-way edge between 3 and 4. All the 5-element permutations that produce some rotation (discarding the labels of the nodes) of that chord diagram belong to the same equivalence class with 25431. The sequence gives the count of such equivalence classes.

Cf. A006841, A060495. For other Maple procedures, see A060501 (Perm2SiteSwap2), A057502 (CountCycles), A057509 (rotateL), A060125 (PermRank3R and permul).
A061417[p] = A061860[p] = (p-1)!+(p-1) for all prime p's.
A064636 (derangements-the same automorphism).
A061417[n] = A064649[n]/n.
Cf. A000031, A000939, A002995, A008965, A060223, A064640, A086675 (digraphical necklaces), A179043, A192332, A275527 (path necklaces), A323858, A323859, A323870, A324513, A324514 (aperiodic permutations).

List([1..10],n->Size( OrbitsDomain( CyclicGroup(IsPermGroup,n), SymmetricGroup( IsPermGroup,n),\^))); # Attila Egri-Nagy, Aug 15 2014

a061417 = sum . a047917_row  -- Reinhard Zumkeller, Mar 19 2014

Algebraic formula: with(numtheory); SSRPCC := proc(n) local d,s; s := 0; for d in divisors(n) do s := s + phi(n/d)*((n/d)^d)*(d!); od; RETURN(s/n); end;
Empirically: with(group); SiteSwapRotationPermutationCycleCounts := proc(upto_n) local b,u,n,a,r; a := []; for n from 1 to upto_n do b := []; u := n!; for r from 0 to u-1 do b := [op(b),1+PermRank3R(SiteSwap2Perm1(rotateL(Perm2SiteSwap2(PermUnrank3Rfix(n,r)))))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;
PermUnrank3Rfixaux := proc(n,r,p) local s; if(0 = n) then RETURN(p); else s := floor(r/((n-1)!)); RETURN(PermUnrank3Rfixaux(n-1, r-(s*((n-1)!)), permul(p,[[n,n-s]]))); fi; end;
PermUnrank3Rfix := (n,r) -> convert(PermUnrank3Rfixaux(n,r,[]),'permlist',n);
SiteSwap2Perm1 := proc(s) local e,n,i,a; n := nops(s); a := []; for i from 1 to n do e := ((i+s[i]) mod n); if(0 = e) then e := n; fi; a := [op(a),e]; od; RETURN(convert(invperm(convert(a,'disjcyc')),'permlist',n)); end;

a[n_] := (1/n)*Sum[ EulerPhi[n/d]*(n/d)^d*d!, {d, Divisors[n]}]; Table[a[n], {n, 1, 21}] (* Jean-François Alcover, Oct 09 2012, from formula *)
Table[Length[Select[Permutations[Range[n]],#==First[Sort[NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]]]&]],{n,8}] (* Gus Wiseman, Mar 04 2019 *)

a(n) = (1/n)*sumdiv(n, d, eulerphi(n/d)*(n/d)^d*d!); \\ Indranil Ghosh, Apr 10 2017

from sympy import divisors, factorial, totient
def a(n):
    return sum(totient(n//d)*(n//d)**d*factorial(d) for d in divisors(n))//n
print([a(n) for n in range(1, 22)]) # Indranil Ghosh, Apr 10 2017

a(n) = (1/n)*Sum_{d|n} phi(n/d)*((n/d)^d)*(d!).

A057513 Number of separate orbits to which permutations given in A057511/A057512 (induced by deep rotation of general parenthesizations/plane trees) partition each A000108(n) objects encoded by A014486 between (A014138(n-1)+1)-th and (A014138(n))-th terms.

Original entry on oeis.org

1, 1, 2, 4, 9, 21, 56, 153, 451, 1357, 4212, 13308, 42898, 140276, 465324, 1561955, 5300285, 18156813, 62732842, 218405402, 765657940

Offset: 0

Antti Karttunen Sep 03 2000

It is much faster to compute this sequence empirically with the given C-program than to calculate the terms with the formula in its present form.

A. Karttunen, Gatomorphisms (with the complete Scheme source)
Index entries for sequences related to rooted trees
A. Karttunen, C-program for computing empirically the initial terms of this sequence

CountCycles given in A057502, for other procedures, follow A057511 and A057501.
Similarly generated sequences: A001683, A002995, A003239, A038775, A057507. Cf. also A000081.
Occurs for first time in A073201 as row 12. Cf. A057546 and also A000081.

A057513 := proc(n) local i; `if`((0=n),1,(1/A003418(n-1))*add(A079216bi(n,i),i=1..A003418(n-1))); end;
# Or empirically:
DeepRotatePermutationCycleCounts := 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,DeepRotateL(CatalanUnrank(n,r)))]; od; a := [op(a),CountCycles(b)]; od; RETURN(a); end;

a(0)=1, a(n) = (1/A003418(n-1))*Sum_{i=1..A003418(n-1)} A079216(n, i) [Needs improvement.] - Antti Karttunen, Jan 03 2003

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

Antti Karttunen, Jan 16 2007

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.

A. Karttunen, Table of n, a(n) for n = 0..2055
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.
Index entries for signature-permutations of Catalan automorphisms

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, ...

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

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