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.

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

Author

Álvar Ibeas, Dec 12 2020

Keywords

• nonn
• tabf

Comments

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.

Examples

			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.
		

Links

• Álvar Ibeas, First 25 rows, flattened

Crossrefs

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

Programs

PARI

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

Formula

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_{1

T(n,k) = A298941(A036035(n,k)) = A318808(A185974(n,k)). - Andrew Howroyd, Dec 14 2020

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.