A061417 -id:A061417 - cp's OEIS frontend

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

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

A192332 For n >= 3, draw a regular n-sided polygon and its n(n-3)/2 diagonals, so there are n(n-1)/2 lines; a(n) is the number of ways to choose a subset of these lines (subsets differing by a rotation are regarded as identical). a(1)=1, a(2)=2 by convention.

Original entry on oeis.org

1, 2, 4, 22, 208, 5560, 299600, 33562696, 7635498336, 3518440564544, 3275345183542208, 6148914696963883712, 23248573454127484129024, 176848577040808821410837120, 2704321280486889389864215362560, 83076749736557243209409446411255936, 5124252113632955685095523500148980125696, 634332307869315502692705867068871886072665600

Offset: 1

N. J. A. Sloane, Jun 28 2011

nonn

Suggested by A192314.

Also the number of graphical necklaces with n vertices. We define a graphical necklace to be a simple graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of simple graphs under rotation of the vertices. These are a kind of partially labeled graphs. - Gus Wiseman, Mar 04 2019

			From _Gus Wiseman_, Mar 04 2019: (Start)
Inequivalent representatives of the a(1) = 1 through a(4) = 22 graphical necklace edge-sets:
  {}  {}      {}              {}
      {{12}}  {{12}}          {{12}}
              {{12}{13}}      {{13}}
              {{12}{13}{23}}  {{12}{13}}
                              {{12}{14}}
                              {{12}{24}}
                              {{12}{34}}
                              {{13}{24}}
                              {{12}{13}{14}}
                              {{12}{13}{23}}
                              {{12}{13}{24}}
                              {{12}{13}{34}}
                              {{12}{14}{23}}
                              {{12}{24}{34}}
                              {{12}{13}{14}{23}}
                              {{12}{13}{14}{24}}
                              {{12}{13}{14}{34}}
                              {{12}{13}{24}{34}}
                              {{12}{14}{23}{34}}
                              {{12}{13}{14}{23}{24}}
                              {{12}{13}{14}{23}{34}}
                              {{12}{13}{14}{23}{24}{34}}
(End)

Vincenzo Librandi, Table of n, a(n) for n = 1..80
Gus Wiseman, Inequivalent representatives of the a(5) = 22 graphical necklaces.
Gus Wiseman, Inequivalent representatives of the a(5) = 208 graphical necklaces.

Cf. A192314, A191563 (orbits under dihedral group).

Cf. A000031, A000939 (cycle necklaces), A008965, A059966, A060223, A061417, A086675 (digraph version), A184271, A275527, A323858, A324461, A324463, A324464.

with(numtheory);
f:=proc(n) local t0, t1, d; t0:=0; t1:=divisors(n);
for d in t1 do
if d mod 2 = 0 then t0:=t0+phi(d)*2^(n^2/(2*d))
else t0:=t0+phi(d)*2^(n*(n-1)/(2*d)); fi; od; t0/n; end;
[seq(f(n), n=1..30)];

Table[ 1/n* Plus @@ Map[Function[d, EulerPhi[d]*2^((n*(n - Mod[d, 2])/2)/d)], Divisors[n]], {n, 1, 20}]  (* Olivier Gérard, Aug 27 2011 *)
rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Subsets[Subsets[Range[n],{2}]],#=={}||#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&]],{n,0,5}] (* Gus Wiseman, Mar 04 2019 *)

a(n) = sumdiv(n, d, if (d%2, eulerphi(d)*2^(n*(n-1)/(2*d)), eulerphi(d)*2^(n^2/(2*d))))/n; \\ Michel Marcus, Mar 08 2019

a(n) = (1/n)*(Sum_{d|n, d odd} phi(d)*2^(n*(n-1)/(2*d)) + Sum_{d|n, d even} phi(d)*2^(n^2/(2*d))).

A006841 Permutation arrays of period n.

Original entry on oeis.org

1, 1, 1, 2, 4, 10, 28, 127, 686, 4975, 42529, 420948, 4622509, 55670332, 726738971, 10217376792, 153848448652, 2470073249960, 42120966152815, 760282326662191, 14481561464994821, 290289454462745374, 6108699653117045614

Offset: 1

N. J. A. Sloane

Also superpositions of cycles of order n of the groups S_2 and D_n. - Sean A. Irvine, Oct 25 2017

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 171.
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
A. P. Street and R. Day, Sequential binary arrays II: Further results on the square grid, pp. 392-418 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982.

M. Engelhardt, Java program
V. Meally, Letter to N. J. A. Sloane, Feb 1975
E. M. Palmer and R. W. Robinson, Enumeration under two representations of the wreath product, Acta Math., 131 (1973), 123-143. (See Table 2, but note errors.)
A. P. Street and R. Day, Sequential binary arrays II: Further results on the square grid, pp. 392-418 of Combinatorial Mathematics IX. Proc. Ninth Australian Conference (Brisbane, August 1981). Ed. E. J. Billington, S. Oates-Williams and A. P. Street. Lecture Notes Math., 952. Springer-Verlag, 1982. (Annotated scanned copy)
Venta Terauds, J. Sumner, Circular genome rearrangement models: applying representation theory to evolutionary distance calculations, arXiv preprint arXiv:1712.00858 [q-bio.PE], 2017.

Cf. A061417.

Asymptotic behavior: The n-th term T(n) is always larger than n! / (8*n^2) = (n-1)! / 8n; for large n, it is approximated by that value. Stated as formula: T(n) > (n-1)! / 8n; lim 8n * T(n) / (n-1) = 1 as n tends to infinity.

Terms for n=1..8 from A. P. Street and R. Day; other terms computed by Matthias Engelhardt. For n=9..12, he used a program which shifts, rotates and mirrors permutations. Terms for n=13..29 computed with a Java program implementing the formulas.

A060495 Each permutation in the list A060117 converted to Site Swap notation, with "zero throws" (fixed elements) replaced with n, the length of siteswap.

Original entry on oeis.org

1, 11, 312, 111, 231, 222, 4413, 1313, 4112, 1111, 2411, 2312, 4242, 1241, 4233, 1223, 2222, 2231, 3441, 3342, 3131, 3122, 3423, 3333, 55514, 14514, 51414, 11314, 25314, 24414, 55113, 14113, 51112, 11111, 25111, 24112, 52512, 12511, 52413

Offset: 0

Antti Karttunen, Mar 21 2001

This sequence is not well-defined for n >= 3628800 because the Site Swap notation can contain values exceeding 9, for example, the Site Swap notation for a(3628800) is [0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 10]. - Sean A. Irvine, Nov 25 2022

Site Swap notation explained.

Cf. factorial base representation A007623 and A060496, A006694.

See also A060498, A060499, A061417. Average of digits gives number of balls: A060501.

Perm2SiteSwap1 := proc(p) local ip,n,i,a; n := nops(p); ip := convert(invperm(convert(p,'disjcyc')),'permlist',n); a := []; for i from 1 to n do a := [op(a),((ip[i]-i) mod n)]; od; RETURN(a); end;
SiteSwap1ToDec := proc(s) local i,z,n; n := nops(s); z := 0; for i from 1 to n do z := 10*z; if(0 = s[i]) then z := z+n; else z := z+s[i]; fi; od; RETURN(z); end;

a(n) = SiteSwap1ToDec(Perm2SiteSwap1(PermUnrank3R(n))).

A064640 Positions of non-crossing fixed-point-free involutions (encoded by A014486) in A055089, sorted to ascending order.

Original entry on oeis.org

0, 1, 7, 23, 127, 143, 415, 659, 719, 5167, 5183, 5455, 5699, 5759, 16687, 16703, 26815, 28495, 36899, 36959, 38579, 40031, 40319, 368047, 368063, 368335, 368579, 368639, 379567, 379583, 389695, 391375, 399779, 399839, 401459, 402911, 403199

Offset: 0

Antti Karttunen, Oct 02 2001

nonn

These permutations belong to the interpretation (kk) of the exercise 19 in the sixth chapter "Exercises on Catalan and Related Numbers" of Enumerative Combinatorics, Vol. 2, 1999 by R. P. Stanley, Wadsworth, Vol. 1, 1986: Fixed-point-free involutions w of [2n] such that if i < j < k < l and w(i) = k, then w(j) <> l.

From this, it follows that when they are subjected to the same automorphism as used in A061417 and A064636, one gets A002995.

			The first eight such permutations (after the identity) are in positions 1, 7, 23, 127, 143, 415, 659, 719 of A055089: 21, 2143, 4321, 214365, 432165, 216543, 632541, 654321 which written as disjoint cycles are (1 2), (1 2)(3 4), (1 4)(2 3), (1 2)(3 4)(5 6), (1 4)(2 3)(5 6), (1 2)(3 6)(4 5), (1 6)(2 3)(4 5), (1 6)(2 5)(3 4).

R. P. Stanley, Exercises on Catalan and Related Numbers

For the needed Maple procedures see A064638. Cf. also A064639, A060112.

Maple
```
sort(A064638); or sort(A064639);
```

A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows.

Original entry on oeis.org

1, 2, 10, 176, 16456, 6710912, 11453291200, 80421421917440, 2305843009750581376, 268650182136584290872320, 126765060022823052739661424640, 241677817415439249618874010960064512, 1858395433210885261795036719974526548094976

Offset: 0

Yuval Dekel (dekelyuval(AT)hotmail.com), Jul 27 2003

Also the number of digraphical necklaces with n vertices. A digraphical necklace is defined to be a directed graph that is minimal among all n rotations of the vertices. Alternatively, it is an equivalence class of directed graphs under rotation of the vertices. These are a kind of partially labeled digraphs. - Gus Wiseman, Mar 04 2019

			From _Gus Wiseman_, Mar 04 2019: (Start)
Inequivalent representatives of the a(2) = 10 digraphical necklace edge-sets:
  {}
  {(1,1)}
  {(1,2)}
  {(1,1),(1,2)}
  {(1,1),(2,1)}
  {(1,1),(2,2)}
  {(1,2),(2,1)}
  {(1,1),(1,2),(2,1)}
  {(1,1),(1,2),(2,2)}
  {(1,1),(1,2),(2,1),(2,2)}
(End)

Alois P. Heinz, Table of n, a(n) for n = 0..57
Peter Kagey and William Keehn, Counting tilings of the n X m grid, cylinder, and torus, arXiv:2311.13072 [math.CO], 2023. See p. 3.

Cf. A000031 (binary necklaces), A000939 (cycle necklaces), A008965, A060690, A061417 (permutation necklaces), A184271, A192332 (graphical necklaces), A275527 (path necklaces), A323858 (toroidal necklaces), A323870.

Cf. A324463, A324464, A324513, A324514.

Table[Fold[ #1+EulerPhi[ #2] 2^(n^2 /#2)&, 0, Divisors[n]]/n, {n, 16}]
(* second program *)
rotdigra[g_,m_]:=Sort[g/.k_Integer:>If[k==m,1,k+1]];
Table[Length[Select[Subsets[Tuples[Range[n],2]],#=={}||#==First[Sort[Table[Nest[rotdigra[#,n]&,#,j],{j,n}]]]&]],{n,0,4}] (* Gus Wiseman, Mar 04 2019 *)

a(n) = (1/n)*Sum_{ d divides n } phi(d)*2^(n^2/d) for n > 0, a(0) = 1.

More terms from Wouter Meeussen, Jul 29 2003

a(0)=1 prepended by Gus Wiseman, Mar 04 2019

A324513 Number of aperiodic cycle necklaces with n vertices.

Original entry on oeis.org

1, 0, 0, 0, 2, 7, 51, 300, 2238, 18028, 164945, 1662067, 18423138, 222380433, 2905942904, 40864642560, 615376173176, 9880203467184, 168483518571789, 3041127459127222, 57926238289894992, 1161157775616335125, 24434798429947993043, 538583682037962702384

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

We define an aperiodic cycle necklace to be an equivalence class of (labeled, undirected) Hamiltonian cycles under rotation of the vertices such that all n of these rotations are distinct.

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Inequivalent representatives of the a(6) = 7 aperiodic cycle necklaces.
Gus Wiseman, Inequivalent representatives of the a(7) = 51 aperiodic cycle necklaces.

Cf. A000740, A000939, A001037 (binary Lyndon words), A008965, A059966 (Lyndon compositions), A060223 (normal Lyndon words), A061417, A064852 (if cycle is oriented), A086675, A192332, A275527, A323866 (aperiodic toroidal arrays), A323871.

Cf. A306669, A324461, A324462, A324512, A324514.

rotgra[g_,m_]:=Sort[Sort/@(g/.k_Integer:>If[k==m,1,k+1])];
Table[Length[Select[Union[Sort[Sort/@Partition[#,2,1,1]]&/@Permutations[Range[n]]],#==First[Sort[Table[Nest[rotgra[#,n]&,#,j],{j,n}]]]&&UnsameQ@@Table[Nest[rotgra[#,n]&,#,j],{j,n}]&]],{n,8}]

a(n)={if(n<3, n==0||n==1, (if(n%2, 0, -(n/2-1)!*2^(n/2-2)) + sumdiv(n, d, moebius(n/d)*eulerphi(n/d)*(n/d)^d*d!/n^2))/2)} \\ Andrew Howroyd, Aug 19 2019

a(n) = A324512(n)/n.

a(2*n+1) = A064852(2*n+1)/2 for n > 0; a(2*n) = (A064852(2*n) - A002866(n-1))/2 for n > 1. - Andrew Howroyd, Aug 16 2019

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

A324514 Number of aperiodic permutations of {1..n}.

Original entry on oeis.org

1, 0, 3, 16, 115, 660, 5033, 39936, 362718, 3624920, 39916789, 478953648, 6227020787, 87177645996, 1307674338105, 20922779566080, 355687428095983, 6402373519409856, 121645100408831981, 2432902004460734000, 51090942171698415483, 1124000727695858073380

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

A permutation is defined to be aperiodic if every cyclic rotation of {1..n} acts on the cycle decomposition to produce a different digraph.

			The a(4) = 16 aperiodic permutations:
  (1243) (1324) (1342) (1423)
  (2134) (2314) (2413) (2431)
  (3124) (3142) (3241) (3421)
  (4132) (4213) (4231) (4312)

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Cycle decompositions of the a(4) = 16 aperiodic permutations.

Cf. A000740, A000939, A027375, A061417, A079128, A086675, A192332, A323860, A323867, A323869.

Cf. A306669, A324461, A324462, A324512, A324513, A324514.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]],{n,6}]

a(n) = sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

a(n) = A306669(n) * n.

a(n) = Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

A306669 Number of aperiodic permutation necklaces of weight n.

Original entry on oeis.org

1, 0, 1, 4, 23, 110, 719, 4992, 40302, 362492, 3628799, 39912804, 479001599, 6226974714, 87178289207, 1307673722880, 20922789887999, 355687417744992, 6402373705727999, 121645100223036700, 2432902008176115023, 51090942167993548790, 1124000727777607679999

Offset: 1

Gus Wiseman, Mar 04 2019

nonn

A permutation is aperiodic if every rotation of {1...n} acts on the vertices of the cycle decomposition to produce a different digraph. A permutation necklace is an equivalence class of permutations 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).

Andrew Howroyd, Table of n, a(n) for n = 1..200
Gus Wiseman, Inequivalent representatives of the a(5) = 23 aperiodic necklace permutations.

Cf. A000031, A000740, A000939, A001037, A059966, A060223, A061417, A086675, A323861, A323865, A323866, A323871.

Cf. A324461, A324512, A324513, A324514.

Table[Length[Select[Permutations[Range[n]],UnsameQ@@NestList[RotateRight[#/.k_Integer:>If[k==n,1,k+1]]&,#,n-1]&]]/n,{n,6}]

a(n) = (1/n)*sumdiv(n, d, moebius(n/d)*(n/d)^d*d!); \\ Andrew Howroyd, Aug 19 2019

a(n) = A324514(n)/n.

a(n) = (1/n)*Sum_{d|n} mu(n/d)*(n/d)^d*d!. - Andrew Howroyd, Aug 19 2019

Terms a(10) and beyond from Andrew Howroyd, Aug 19 2019

cp's OEIS Frontend

A061860 Variant of A061417.

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

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A192332 For n >= 3, draw a regular n-sided polygon and its n(n-3)/2 diagonals, so there are n(n-1)/2 lines; a(n) is the number of ways to choose a subset of these lines (subsets differing by a rotation are regarded as identical). a(1)=1, a(2)=2 by convention.

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Maple

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PARI

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A006841 Permutation arrays of period n.

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A060495 Each permutation in the list A060117 converted to Site Swap notation, with "zero throws" (fixed elements) replaced with n, the length of siteswap.

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A064640 Positions of non-crossing fixed-point-free involutions (encoded by A014486) in A055089, sorted to ascending order.

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Maple

A086675 Number of n X n (0,1)-matrices modulo cyclic permutations of the rows.

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