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

Author

Wolfdieter Lang, Jul 20 2012

Keywords

• nonn

Comments

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

Examples

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

Links

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

Crossrefs

Row sums of A213940.

Row sums of A214609.

Cf. A072605 (representative necklaces).

Programs

PARI

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

Formula

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.