A064649 -id:A064649 - cp's OEIS frontend

A002619 Number of 2-colored patterns on an n X n board.

1, 1, 2, 3, 8, 24, 108, 640, 4492, 36336, 329900, 3326788, 36846288, 444790512, 5811886656, 81729688428, 1230752346368, 19760413251956, 336967037143596, 6082255029733168, 115852476579940152, 2322315553428424200, 48869596859895986108

Offset: 1

N. J. A. Sloane, Colin Mallows

Also number of orbits in the set of circular permutations (up to rotation) under cyclic permutation of the elements. - Michael Steyer, Oct 06 2001

Moser shows that (1/n^2)*Sum_{d|n} k^d*phi(n/d)^2*(n/d)^d*d! is an integer. Here we have k=1. - Michel Marcus, Nov 02 2012

			n=6: {(123456)}, {(135462), (246513), (351624)} and {(124635), (235146), (346251), (451362), (562413), (613524)} are 3 of the 24 orbits, consisting of 1, 3 and 6 permutations, respectively.

N. J. A. Sloane, A Handbook of Integer Sequences, Academic Press, 1973 (includes this sequence).
N. J. A. Sloane and Simon Plouffe, The Encyclopedia of Integer Sequences, Academic Press, 1995 (includes this sequence).
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
K. Yordzhev, On the cardinality of a factor set in the symmetric group. Asian-European Journal of Mathematics, Vol. 7, No. 2 (2014) 1450027, doi: 10.1142/S1793557114500272, ISSN:1793-5571, E-ISSN:1793-7183, Zbl 1298.05035.

T. D. Noe, Table of n, a(n) for n = 1..100
M. Kazarian, E. Krasilnikov, S. Lando, and M. Shapiro, Generalized chord diagrams and weight systems, arXiv:2505.24491 [math.CO], 2025. See Theorem 5.3 p. 24.
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
W. O. J. Moser, A (modest) generalization of the theorems of Wilson and Fermat, Canad. Math. Bull. 33(1990), pp. 253-256.
Ludovic Schwob, On the enumeration of double cosets and self-inverse double cosets, arXiv:2506.04007 [math.CO], 2025. See p. 9.
N. J. A. Sloane, Notes on A002618, A002619, etc.
András Szilárd, A combinatorial generalization of Wilson's theorem, Australasian Journal of Combinatorics, Volume 49 (2011), Pages 265-272. See Theorem 3.c p. 269.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]
A. Vella, Pattern avoidance in permutations: linear and cyclic orders, Electron. J. Combin. 9 (2002/03), no. 2, #R18, 43 pp.
K. Yordzhev, On the cardinality of a factor set in the symmetric group, arXiv:1410.8408 [math.CO], 2014. See Table 2 p. 14.
Saeed Zakeri, Cyclic Permutations: Degrees and Combinatorial Types, arXiv:1909.03300 [math.DS], 2019. See Table 2 p. 10.

Cf. A002618, A047916, A064852, A064649.
Cf. A000010.
Cf. A000939, A000940, A089066, A262480, A275527 (other classes of permutations under various symmetries).

with(numtheory): a:=proc(n) local div: div:=divisors(n): sum(phi(div[j])^2*(n/div[j])!*div[j]^(n/div[j]),j=1..tau(n))/n^2 end: seq(a(n),n=1..23); # Emeric Deutsch, Aug 23 2005

a[n_] := EulerPhi[#]^2*(n/#)!*#^(n/#)/n^2 & /@ Divisors[n] // Total; a /@ Range[23] (* Jean-François Alcover, Jul 11 2011, after Emeric Deutsch *)

a(n)={sumdiv(n, d, eulerphi(n/d)^2*d!*(n/d)^d)/n^2} \\ Andrew Howroyd, Sep 09 2018

from sympy import totient, factorial, divisors
def A002619(n): return sum(totient(m:=n//d)**2*factorial(d)*m**d for d in divisors(n,generator=True))//n**2 # Chai Wah Wu, Nov 07 2022

a(n) = Sum_{k|n} u(n, k)/(nk), where u(n, k) = A047918(n, k).
a(n) = (1/n^2)*Sum_{d|n} phi(d)^2*(n/d)!*d^(n/d), where phi is Euler's totient function (A000010). - Emeric Deutsch, Aug 23 2005
From Richard L. Ollerton, May 09 2021: (Start)
Let A(n,k) = (1/n^2)*Sum_{d|n} k^d*phi(n/d)^2*(n/d)^d*d!, then:
A(n,k) = (1/n^2)*Sum_{i=1..n} k^gcd(n,i)*phi(n/gcd(n,i))*(n/gcd(n,i))^gcd(n,i)*gcd(n,i)!.
A(n,k) = (1/n^2)*Sum_{i=1..n} k^(n/gcd(n,i))*phi(gcd(n,i))^2*(gcd(n,i))^(n/gcd(n,i))*(n/gcd(n,i))!.
a(n) = A(n,1). (End)

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!).

A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).

Original entry on oeis.org

1, 2, 2, 6, 0, 6, 8, 8, 0, 24, 20, 0, 0, 0, 120, 12, 36, 48, 0, 0, 720, 42, 0, 0, 0, 0, 0, 5040, 32, 64, 0, 384, 0, 0, 0, 40320, 54, 0, 324, 0, 0, 0, 0, 0, 362880, 40, 200, 0, 0, 3840, 0, 0, 0, 0, 3628800, 110, 0, 0, 0, 0, 0, 0, 0, 0, 0, 39916800, 48, 144

Offset: 1

N. J. A. Sloane

T(n,k) = A054523(n,k) * A010766(n,k)^A002260(n,k) * A166350(n,k). - Reinhard Zumkeller, Jan 20 2014

			1; 2,2; 6,0,6; 8,8,0,24; 20,0,0,0,120; 12,36,48,0,0,720; ...

J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.

Reinhard Zumkeller, Rows n = 1..125 of triangle, flattened
C. L. Mallows and N. J. A. Sloane, Notes on A002618, A002619, etc.
N. J. A. Sloane, Notes on A002618, A002619, etc.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61.
J. E. A. Steggall, On the numbers of patterns which can be derived from certain elements, Mess. Math., 37 (1907), 56-61. [Annotated scanned copy. Note that the scanned pages are out of order]

A064649 gives the row sums.
Cf. A002618 (left edge), A000142 (right edge), A049820 (zeros per row), A000005 (nonzeros per row).
See also A247917, A047918, A047919.

import Data.List (zipWith4)
a047916 n k = a047916_tabl !! (n-1) !! (k-1)
a047916_row n = a047916_tabl !! (n-1)
a047916_tabl = zipWith4 (zipWith4 (\x u v w -> x * v ^ u * w))
               a054523_tabl a002260_tabl a010766_tabl a166350_tabl
-- Reinhard Zumkeller, Jan 20 2014

a[n_, k_] := If[Divisible[n, k], EulerPhi[n/k]*(n/k)^k*k!, 0]; Flatten[ Table[ a[n, k], {n, 1, 12}, {k, 1, n}]] (* Jean-François Alcover, May 04 2012 *)

a(n,k)=if(n%k, 0, eulerphi(n/k)*(n/k)^k*k!) \\ Charles R Greathouse IV, Feb 09 2017

cp's OEIS Frontend

A002619 Number of 2-colored patterns on an n X n board.

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

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A047916 Triangular array read by rows: a(n,k) = phi(n/k)*(n/k)^k*k! if k|n else 0 (1<=k<=n).

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Author

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Comments

Examples

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Crossrefs

Programs

Haskell

Mathematica

PARI

A047916 Triangular array read by rows: a(n,k) = phi(n/k)(n/k)^kk! if k|n else 0 (1<=k<=n).