A214609 -id:A214609 - cp's OEIS frontend

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

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.

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)).

cp's OEIS Frontend

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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