A072605 -id:A072605 - cp's OEIS frontend

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

2, 5, 2, 9, 4, 7, 7, 4, 7, 2, 0, 7, 9, 1, 5, 2, 6, 4, 8, 1, 8, 0, 1, 1, 6, 1, 5, 4, 2, 5, 3, 9, 5, 4, 2, 4, 1, 1, 7, 8, 7, 0, 2, 3, 9, 4, 8, 4, 5, 9, 9, 7, 3, 3, 7, 5, 8, 4, 9, 3, 4, 9, 8, 2, 5, 0, 0, 2, 1, 1, 8, 7, 8, 0, 0, 8, 6, 6, 9, 9, 1, 2, 1, 9, 9, 8, 8, 3, 6, 4, 6, 2, 5, 2, 6, 1, 4, 9, 5, 5, 1, 6, 4, 3, 2

Offset: 1

Vaclav Kotesovec, Sep 19 2014

			2.5294774720791526481801161542539542411787023948459973375849349825...

Steven R. Finch, Mathematical Constants II, Encyclopedia of Mathematics and Its Applications, Cambridge University Press, Cambridge, 2018, p. 269.
A. Knopfmacher, A. M. Odlyzko, B. Pittel, L. B. Richmond, D. Stark, G. Szekeres, and N. C. Wormald, The Asymptotic Number of Set Partitions with Unequal Block Sizes, The Electronic Journal of Combinatorics, Volume 6 (1999), Research Paper #R2.

Cf. A005651, A035341, A072605, A140585, A183239, A209668, A261600, A282529, A292713, A320566, A327827, A335643.

evalf(1/product(1-1/k!,k=2..infinity),100); # Vaclav Kotesovec, Sep 19 2014

digits = 105;
RealDigits[NProduct[1/(1-1/k!), {k, 2, Infinity}, WorkingPrecision -> digits+10, NProductFactors -> digits], 10, digits][[1]] (* Jean-François Alcover, Nov 02 2020 *)

default(realprecision,150); 1/prodinf(k=2,1 - 1/k!) \\ Vaclav Kotesovec, Sep 21 2014

Product_{k>=2} 1/(1-1/k!).
Equals lim_{n -> oo} A005651(n) / n!.
Equals 1/A282529. - Amiram Eldar, Sep 15 2023

A212359 Partition array for the number of representative necklaces (only cyclic symmetry) with n beads, each available in n colors. Only the color type (signature) matters.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 1, 2, 3, 6, 1, 1, 2, 4, 6, 12, 24, 1, 1, 3, 4, 5, 10, 16, 20, 30, 60, 120, 1, 1, 3, 5, 6, 15, 20, 30, 30, 60, 90, 120, 180, 360, 720, 1, 1, 4, 7, 10, 7, 21, 35, 54, 70, 42, 105, 140, 210, 318, 210, 420, 630, 840, 1260, 2520, 5040

Offset: 1

Wolfdieter Lang, Jun 25 2012

The row lengths sequence is A000041(n), n>=1.
The partitions are ordered like in Abramowitz-Stegun (A-St order). For the reference see A036036, where also a link to a work by C. F. Hindenburg from 1779 is found where this order has been used.
A necklace with n beads (n-necklace) has here only the cyclic C_n symmetry group. This is in contrast to, e.g., the Harary-Palmer reference, p. 44, where a n-necklace has the symmetry group D_n, the dihedral group of degree n (order 2n), which allows, in addition to C_n operations, also a turnover (in 3-space) or a reflection (in 2-space).
The necklace number a(n,k) gives the number of n-necklaces, with up to n colors for each bead, belonging to the k-th partition of n in A-St order in the following way. Write this partition with nonincreasing parts (this is the reverse of the partition as given by A-St), e.g., [3,1^2], not [1^2,3], is written as [3,1,1], a partition of n=5. In general [p[1],p[2],...,p[m]], with p[1]>=p[2]>=...>=p[m]>=1, with m the number of parts. To each such partition of n corresponds an n-multiset obtained by 'exponentiation'. For the given example the 5-multiset is {1^3,2^1,3^1}={1,1,1,2,3}. In general {1^p[1],2^p[2],...,m^p[m]}. Such an n-multiset representative (of a repetition class defined by the exponents, sometimes called signature) encodes the n-necklace color monomial by c[1]^p[1]*c[2]^p[2]*...*c[m]^p[m]. For the example one has c[1]^3*c[2]*c[3]. The number of 5-necklaces with this color assignment is a(5,4) because [3,1,1] is the 4th partition of 5 in A-St order. The a(5,4)=4 non-equivalent 5-necklaces with this color assignment are cyclic(c[1]c[1]c[1]c[2]c[3]), cyclic(c[1]c[1]c[1]c[3]c[2]), cyclic(c[1]c[1]c[2]c[1]c[3]) and cyclic(c[1]c[1]c[3]c[1]c[2]).
Such a set of a(n,k) n-necklaces for the given color assignment stands for other sets of the same order where different colors from the repertoire {c[1],...,c[n]} are chosen. In the example, the partition [3,1,1]  with the representative multiset [1^3,2,3] stands for all-together 5*binomial(4,2) = 30 such sets, each leading to 4 possible non-equivalent 5-necklace arrangements. Thus one has, in total, 30*4=120 5-necklaces with color signature determined from the partition [3,1,1]. See the partition array A212360 for these numbers.
For the example n=4, k=1..5, see the Stanley reference, last line, where the numbers a(4,k) are, in A-St order, 1, 1, 2, 3, 6, summing to A072605(4).
a(n,k) is computed from the cycle index Z(C_n) for the cyclic group (see A212357 and the link given there) after the variables x_j have been replaced by the j-th power sum sum(c[i]^j,i=1..n), abbreviated as Z(C_n,c_n) with c_n:=sum(c[i],i=1..n), n>=1. The coefficient of the color assignment representative determined by the k-th partition of n in A-St order, as explained above, is a(n,k). See the Harary-Palmer reference, p. 36, Theorem (PET) with A=C_n and p. 36 eq. (2.2.10) for the cycle index polynomial Z(C_n). See the W. Lang link for more details.
The corresponding triangle with summed entries of row n which belong to partitions of n with the same number of parts is A213934. [Wolfdieter Lang, Jul 12 2012]

			n\k  1 2 3 4 5  6  7  8  9 10  11  12  13  14  15
1    1
2    1 1
3    1 1 2
4    1 1 2 3 6
5    1 1 2 4 6 12 24
6    1 1 3 4 5 10 16 20 30 60 120
7    1 1 3 5 6 15 20 30 30 60  90 120 180 360 720
...
See the link for the rows n=1..15 and the corresponding color polynomials for n=1..10.
a(4,5)=6 because the partition in question is 1^4, the corresponding color type representative multinomial is c[1]*c[2]*c[3]*c[4] (all four colors are involved), and there are the 6 C_4 non-equivalent 4-necklaces (we use here j for color c[j]): 1234, 1243, 1324, 1342, 1423 and 1432 (all taken as cyclically). For this partition there is only one color choice.
a(4,4)=3 because the partition is [2,1^2]=[2,1,1], the color representative monomial is c[1]^2*c[2]*c[3], and the arrangements are 1123, 1132  and  1213 (all taken cyclically). There are, in total, 4*binomial(3,2)=12 color multinomials of this signature (color type) in Z(C_4,c_4).

F. Harary and E. M. Palmer, Graphical Enumeration, Academic Press, NY, 1973, p. 36, (2.2.10).
R. Stanley, Enumerative combinatorics, Vol. 2, Cambridge University Press, Cambridge, 1999, p. 392, 7.24.3 Example.

Álvar Ibeas, First 25 rows, flattened
Wolfdieter Lang, Rows n=1..15 and color polynomials n=1..10.

Cf. A212357 for Z(C_n), A072605 for the row sums.

Cf. A000041 (row lengths), A036036, A185974, A212360, A213934, A318810.

a(n,k) is the number of necklace arrangements with n beads (respecting the cyclic C_n symmetry) with color assignment given by the multiset representative obtained uniquely from the k-th partition of n in A-St order. See the comment for more details and the A-St reference.
From Álvar Ibeas, Dec 12 2020: (Start)
Let L be the k-th partition of n in A-St and d be the gcd of its parts. Abusing the notation, we write a(n, L) for a(n, k) and accordingly for other partition arrays.
a(n, L) = n^(-1) * Sum_{v|d} phi(v) * A036038(n/v, L/v), where L/v is the partwise division of L by v.
a(n, L) = Sum_{v|d} A339677(L/v).
(End)
a(n,k) = A318810(A185974(n,k)). - Andrew Howroyd, Jan 23 2025

A261600 Number of primitive (aperiodic, or Lyndon) necklaces with n beads such that beads of a largest subset have label 0, beads of a largest remaining subset have label 1, and so on.

Original entry on oeis.org

1, 1, 1, 3, 11, 49, 265, 1640, 11932, 96780, 887931, 8939050, 99298073, 1195617442, 15619180497, 219049941148, 3293800823995, 52746930894773, 897802366153076, 16167544246362566, 307372573010691195, 6148811682561388635, 129164845357775064609

Offset: 0

Alois P. Heinz, Aug 27 2015

nonn

			a(3) = 3: 001, 012, 021.
a(4) = 11: 0001, 0011, 0012, 0021, 0102, 0123, 0132, 0213, 0231, 0312, 0321.

Alois P. Heinz, Table of n, a(n) for n = 0..300
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Eric Weisstein's World of Mathematics, Necklace
Wikipedia, Lyndon word
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Cf. A072605, A247551, A261531, A261599.

with(numtheory):
b:= proc(n, i, g, d, j) option remember; `if`(g>0 and gn, 0, binomial(n/j, i/j)*b(n-i, i, igcd(i, g), d, j)))))
    end:
a:= n-> `if`(n=0, 1, add(add((f-> `if`(f=0, 0, f*b(n$2, 0, d, j)))(
                     mobius(j)), j=divisors(d)), d=divisors(n))/n):
seq(a(n), n=0..25);

b[n_, i_, g_, d_, j_] := b[n, i, g, d, j] = If[g>0 && gn, 0, Binomial[n/j, i/j]*b[n-i, i, GCD[i, g], d, j]]]]]; a[n_] := If[n==0, 1, Sum[Sum[ Function[f, If[f==0, 0, f*b[n, n, 0, d, j]]][MoebiusMu[j]], {j, Divisors[ d]}], {d, Divisors[n]}]/n]; Table[a[n], {n, 0, 25}] (* Jean-François Alcover, Feb 22 2017, translated from Maple *)

a(n) ~ c * (n-1)!, where c = Product_{k>=2} 1/(1-1/k!) = A247551 = 2.52947747207915264818011615... . - Vaclav Kotesovec, Aug 27 2015

A213943 Row sums of partition array A213939 and triangle A213940: number of representative bracelets with n beads and up to n colors.

Original entry on oeis.org

1, 2, 3, 9, 28, 144, 832, 6012, 48447, 444198, 4469834, 49650464, 597810739, 7809600123, 109524985564, 1646900490716, 26373465572350, 448901183773766, 8083772124339442, 153686286512223573, 3074405841292532560, 64582422678961767945

Offset: 1

Wolfdieter Lang, Jul 20 2012

nonn

See A213939 for representative bracelets of a color class defined by a signature, given by a partition.
If color c[j] is written as j, for j from {1, 2, ... ,n}, the representative multisets, corresponding to the bracelets in question, are the ones with the least sum of their members.
E.g., n=4, m=3: signature [2,1,1] (partition of n with 4 parts), representative multiset (written as an ordered list by convention) [1,1,2,3], with the two representative bracelets 1123 and 1213, both taken cyclically.
Number of bracelets with n beads over a n-ary alphabet {a1,a2,...,an} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 0, where #(w,x) counts the letters x in word w. - Andrew Howroyd, Dec 21 2017

			The a(4)= 9 representative bracelets are (j for c[j]):  1111, 1112, 1122, 1212, 1123, 1213, 1234, 1324 and 1243, all taken cyclically.

Andrew Howroyd, Table of n, a(n) for n = 1..200

Row sums of A213940.
Row sums of A214609.
Cf. A072605 (representative necklaces).

a(n)={ if(n==0, 1,
  my(p=serlaplace(prod(k=1, n, 1/(1-x^k/k!) + O(x*x^n))));
  my(c=sumdiv(n, d, eulerphi(n/d)*polcoeff(p, d))/n);
  my(r=if(n%2, sum(d=0, (n-1)/2, binomial((n-1)/2, d)*polcoeff(p, d)), polcoeff(p, n/2) + sum(d=0, n/2-1, binomial(n/2-1, d)*polcoeff(p, n/2-1-d)*(2^d + if(d%2, 0, binomial(d, d/2))))/2));
  ( (c + r)/2 ) )
} \\ Andrew Howroyd, Dec 21 2017

a(n) = sum(A213939(n,k),k=1..p(n)), with p(n)=A000041(n), n >= 1.

a(n) = sum(A213940(n,m),m=1..n), n >= 1.

A261531 Number of necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.

Original entry on oeis.org

1, 1, 1, 2, 2, 4, 15, 25, 69, 254, 1799, 4039, 16828, 61751, 349831, 3485031, 10391139, 49433136, 240065255, 1282012987, 9167583734, 131550812011, 459677216341, 2707382738559, 14318807603110, 94084166753927, 601900541251447, 5894253303715375

Offset: 0

Alois P. Heinz, Aug 23 2015

nonn

			a(4) = 2: 0000, 0001.
a(5) = 4: 00000, 00001, 00011, 00101.
a(6) = 15: 000000, 000001, 000011, 000101, 000112, 000121, 000122, 001001, 001012, 001021, 001022, 001102, 001201, 001202, 010102.

Alois P. Heinz, Table of n, a(n) for n = 0..260
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc.
F. Ruskey, Necklaces, Lyndon words, De Bruijn sequences, etc. [Cached copy, with permission, pdf format only]
Eric Weisstein's World of Mathematics, Necklace
Wikipedia, Necklace (combinatorics)
Index entries for sequences related to necklaces

Cf. A072605, A261599, A261600.

with(numtheory): with(combinat):
g:= l-> (n-> `if`(n=0, 1, add(phi(j)*multinomial(n/j,
        (l/j)[]), j=divisors(igcd(l[])))/n))(add(i, i=l)):
b:= proc(n, i, l) `if`(i*(i+1)/2n, 0, b(n-i, i-1, [l[], i]))))
    end:
a:= n-> b(n$2, []):
seq(a(n), n=0..35);

multinomial[n_, k_] := n!/Times @@ (k!);
g[l_] := Function[n, If[n==0, 1, Sum[EulerPhi[j]*multinomial[n/j, l/j], {j, Divisors[GCD @@ l]}]/n]][Total[l]];
b[n_, i_, l_] := If[i*(i+1)/2n, 0, b[n-i, i-1, Append[l, i]]]]];
a[n_] := b[n, n, {}];
Table[a[n], {n, 0, 35}] (* Jean-François Alcover, Mar 21 2017, translated from Maple *)

a(n)={if(n==0, 1, my(p=prod(k=1, n, (1+x^k/k!) + O(x*x^n))); sumdiv(n, d, eulerphi(n/d)*d!*polcoeff(p, d))/n)} \\ Andrew Howroyd, Dec 21 2017

a(n) = (1/n) * Sum_{d | n} phi(n/d) * A007837(d) for n>0. - Andrew Howroyd, Apr 02 2017

A213934 Triangle with entry a(n,m) giving the number of necklaces of n beads (C_N symmetry) with n colors available for each bead, but only m distinct fixed colors, say c[1],...,c[m], are present, with m from {1,...,n} and n>=1.

Original entry on oeis.org

1, 1, 1, 1, 1, 2, 1, 3, 3, 6, 1, 3, 10, 12, 24, 1, 8, 31, 50, 60, 120, 1, 9, 71, 180, 300, 360, 720, 1, 22, 187, 815, 1260, 2100, 2520, 5040, 1, 29, 574, 2324, 6496, 10080, 16800, 20160, 40320, 1, 66, 1373, 9570, 32268, 58464, 90720, 151200, 181440, 362880

Offset: 1

Wolfdieter Lang, Jun 27 2012

This triangle is obtained from the partition array A212359 by summing in the row number n, for n>=1, all entries related to partitions of n with the same number of parts m.
a(n,m) is the number of necklaces of n beads (C_N symmetry) corresponding to the representative color multinomials obtained from all partitions of n with m parts by 'exponentiation', hence only m from the available n colors are present. As a representative multinomial of each of the p(n,m)=A008284(n,m) such m-color classes we take the one where the considered m part partition of n, [p[1],...,p[m]], written in a nonincreasing way, is distributed as exponents over c[1]^p[1]*...*c[m]^p[m]. That is only the first m colors from the n available ones are involved.
See the comments on A212359 for the Abramowitz-Stegun (A-St) order of partitions, and the 'exponentiation' to obtain multisets, used to encode color multinomials, from partitions.
The row sums of this triangle coincide with the ones of array A212359, and they are given by A072605.
Number of necklaces with n beads w over a k-ary alphabet {a1,a2,...,ak} such that #(w,a1) >= #(w,a2) >= ... >= #(w,ak) >= 1, where #(w,x) counts the letters x in word w (necklace analog of A226874). - Andrew Howroyd, Dec 20 2017

			n\m  1  2    3    4     5     6     7      8      9     10 ...
1    1
2    1  1
3    1  1    2
4    1  3    3    6
5    1  3   10   12    24
6    1  8   31   50    60   120
7    1  9   71  180   300   360   720
8    1 22  187  815  1260  2100  2520   5040
9    1 29  574 2324  6496 10080 16800  20160  40320
10   1 66 1373 9570 32268 58464 90720 151200 181440 362880
...
a(5,3) = 4 + 6 = 10, from A212359(5,4) + A212359(5,5), because k(5,3,1) = 4 and p(5,3) = 2.
a(2,1) = 1 because the partition [2] of n=2 with part number m=1 corresponds to the representative color multinomial (here monomial) c[1]^2=c[1]*c[1], and there is one such representative necklace. There is another necklace color monomial in this class of n=2 colors where only m=1 color is active: c[2]*c[2]. See the triangle entry A213935(2,1)=2.
a(3,1) = 1 from the color monomial representative c[1]^3. This class has 2 other members: c[2]^3 and c[3]^3. See A213935(3,1)=3.
In general a(n,1)=1 and A213935(n,1)=n from the partition [n] providing the color signature and a representative c[1]^n.
a(3,2)=1 from the representative color multinomial c[1]^2*c[2] (from the m=2 partition [2,1] of n=3) leading to just one representative necklace cyclic(112) (when one uses j for color c[j]). The whole class consists of A213935(3,2)=6 necklaces: cyclic(112), cyclic(113), cyclic(221), cyclic(223), cyclic(331) and  cyclic(332).
a(3,3)=2. The representative color multinomial is c[1]*c[2]*c[3] (from the m=3 partition [1,1,1]). There are the two non-equivalent representative necklaces cyclic(1,2,3) and cyclic(1,3,2) which constitute already the whole class (A213935(3,3)=2).
a(4,2) = 3 from two representative color multinomials c[1]^3*c[2] and c[1]^2*c[2]^2 (from the two m=2 partitions of n=4: [3,1] and [2,2]). The first one has one representative necklace, namely cyclic(1112), the second one originates from two representative necklaces: cyclic(1122) and cyclic(1212). Together these are the 3 necklaces counted by a(4,2). The class with the first representative consists of 4*3=12 necklaces, when all 4 colors are used. The class of the second representative consists of 2*6=12 necklaces. Together they sum up to the 24 necklaces counted by A213935(4,2).

Andrew Howroyd, Table of n, a(n) for n = 1..1275

Cf. A008284, A072605 (row sums), A212359, A213935.

Cf. A213940 (bracelets), A226874 (words).

b[n_, i_, t_] := b[n, i, t] = If[t == 1, 1/n!, Sum[b[n - j, j, t - 1]/j!, {j, i, n/t}]];
a226874[n_, k_] := If[n k == 0, If[n == k, 1, 0], n! b[n, 1, k]];
T[n_, k_] := (1/n) Sum[EulerPhi[n/d] a226874[d, k], {d, Divisors[n]}];
Table[T[n, k], {n, 1, 10}, {k, 1, n}] // Flatten (* Jean-François Alcover, Jul 06 2018, after Alois P. Heinz and Andrew Howroyd *)

\\ here U is A226874 as vector of polynomials.
U(n)={Vec(serlaplace(prod(k=1, n, 1/(1-y*x^k/k!) + O(x*x^n))))}
C(n)={my(t=U(n)); vector(n, n, vector(n, k, (1/n)*sumdiv(n, d, eulerphi(n/d) * polcoeff(t[d+1], k))))}
{ my(t=C(10)); for(n=1, #t, print(t[n])) } \\ Andrew Howroyd, Dec 20 2017

a(n,m) = Sum_{j=1..p(n,m)}A212359(n,k(n,m,1)+j-1), with k(n,m,1) the position where in the list of partitions of n in A-St order the first with m parts appears, and p(n,m) the number of partitions of n with m parts shown in the array A008284. E.g., n=5, m=3: k(5,3,1)=4, p(5,3)=2.

T(n,k) = (1/n)*Sum_{d|n} phi(n/d)*A226874(d, k). - Andrew Howroyd, Dec 20 2017

A339677 Partition array: T(n, k) is the number of aperiodic necklaces (Lyndon words) on a multiset of colored beads (of size n) whose color multiplicities form the k-th partition of n in Abramowitz-Stegun order.

Original entry on oeis.org

1, 0, 1, 0, 1, 2, 0, 1, 1, 3, 6, 0, 1, 2, 4, 6, 12, 24, 0, 1, 2, 3, 5, 10, 14, 20, 30, 60, 120, 0, 1, 3, 5, 6, 15, 20, 30, 30, 60, 90, 120, 180, 360, 720, 0, 1, 3, 7, 8, 7, 21, 35, 51, 70, 42, 105, 140, 210, 312, 210, 420, 630, 840, 1260, 2520, 5040, 0, 1, 4, 9, 14, 8, 28, 56, 70, 84, 140

Offset: 1

Álvar Ibeas, Dec 12 2020

As in A212359, A072605, and A261600, for each partition, the base set of beads is fixed.

Abuse of notation: we write T(n, L) for T(n, k), where L is the k-th partition of n in A-St order. We do accordingly for A036038 and A212359.

			Array begins:
  k:  1 2 3 4 5  6  7  8  9 10  11  12  13  14  15
      --------------------------------------------
n=1:  1
n=2:  0 1
n=3:  0 1 2
n=4:  0 1 1 3 6
n=5:  0 1 2 4 6 12 24
n=6:  0 1 2 3 5 10 14 20 30 60 120
n=7:  0 1 3 5 6 15 20 30 30 60  90 120 180 360 720
Consider partition L = (4, 2). There are 3 = A212359(6, L) necklaces on the bead set {a^4, b^2}: (aaaabb), (aaabab), and (aabaab). The latter has a period smaller than its size (3 < 6), whereas the other two are aperiodic. Hence, T(6, L) = 2.
T(n, (1,...,1)) = A212359(n, (1,...,1)) = (n-1)!, counting necklaces with n beads, each in a different color.

Álvar Ibeas, First 25 rows, flattened

Cf. A036035, A036038, A072605, A185974, A212359, A261600 (row sums), A298941, A318808.

C(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, moebius(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
Row(n)=[C(Vec(p)) | p<-partitions(n)]
for(n=1, 7, print(Row(n))) \\ Andrew Howroyd, Dec 14 2020

Let L be a partition of n and d be the gcd of its parts. Then,
T(n, L) = n^(-1) * Sum_{v|d} mu(v) * A036038(n/v, L/v), where L/v is the partition obtained from L after dividing each part by v.
T(n, L) = Sum_{v|d} mu(v) * A212359(n/v, L/v).
T(n, L) = n^(-1) * A036038(n, L) - Sum_{1T(n,k) = A298941(A036035(n,k)) = A318808(A185974(n,k)). - Andrew Howroyd, Dec 14 2020

A380401 Triangle read by rows: T(n,k) is the number of necklace permutations of a multiset whose multiplicities are given by the k-th partition of n in graded reflected lexicographic order.

Original entry on oeis.org

1, 1, 1, 2, 1, 1, 6, 3, 2, 1, 1, 24, 12, 6, 4, 2, 1, 1, 120, 60, 30, 16, 20, 10, 4, 5, 3, 1, 1, 720, 360, 180, 90, 120, 60, 30, 20, 30, 15, 5, 6, 3, 1, 1, 5040, 2520, 1260, 630, 318, 840, 420, 210, 140, 70, 210, 105, 54, 35, 10, 42, 21, 7, 7, 4, 1, 1, 40320, 20160, 10080, 5040, 2520, 6720, 3360, 1680, 840, 1120, 560, 188, 1680, 840, 420, 280, 140, 70, 336, 168, 84, 56, 14, 56, 28, 10, 8, 4, 1, 1

Offset: 1

Marko Riedel, Jan 23 2025

See A318810 for a definition of necklace permutation.

			The ordering of the partitions used here is graded reflected lexicographic illustrated below with n=5:
  1,1,1,1,1 => 24
  1,1,1,2 => 12
  1,2,2 => 6
  1,1,3 => 4
  2,3 => 2
  1,4 => 1
  5 => 1
Table begins:
  1
  1,1
  2,1,1
  6,3,2,1,1
  24,12,6,4,2,1,1

F. Harary and E. Palmer, Graphical Enumeration, Academic Press, 1973, pages 36-37, 42-43.

Math StackExchange, Marko Riedel et. al, Free circular permutations
Marko Riedel, Maple code for sequence by PET and closed form.

Cf. A000041 (row lengths), A072605 (row sums), A080576 (graded reflected lexicographic order), A212359 (similar triangle for Abramowitz-Stegun order), A318810, A334434, A214609 (up to rotations and reflections).

C(sig)={my(n=vecsum(sig)); sumdiv(gcd(sig), d, eulerphi(d)*(n/d)!/prod(i=1, #sig, (sig[i]/d)!))/n}
Row(n)={apply(C, vecsort([Vecrev(p) | p<-partitions(n)]))} \\ Andrew Howroyd, Jan 23 2025

For a distribution of colors n1+n2+...+nm = n the number of necklaces is (1/n)*Sum_{d|gcd(n1,n2,...,nm)} phi(d) (n/d)!/Prod_{q=1..m} (nq/d)!

T(n,k) = A318810(A334434(n,k)).

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cp's OEIS Frontend

A247551 Decimal expansion of Product_{k>=2} 1/(1-1/k!).

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A212359 Partition array for the number of representative necklaces (only cyclic symmetry) with n beads, each available in n colors. Only the color type (signature) matters.

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A261600 Number of primitive (aperiodic, or Lyndon) necklaces with n beads such that beads of a largest subset have label 0, beads of a largest remaining subset have label 1, and so on.

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A213943 Row sums of partition array A213939 and triangle A213940: number of representative bracelets with n beads and up to n colors.

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A261531 Number of necklaces with n beads of unlabeled colors such that the numbers of beads per color are distinct.

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A213934 Triangle with entry a(n,m) giving the number of necklaces of n beads (C_N symmetry) with n colors available for each bead, but only m distinct fixed colors, say c[1],...,c[m], are present, with m from {1,...,n} and n>=1.

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A339677 Partition array: T(n, k) is the number of aperiodic necklaces (Lyndon words) on a multiset of colored beads (of size n) whose color multiplicities form the k-th partition of n in Abramowitz-Stegun order.

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