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A320747 Array read by antidiagonals: T(n,k) is the number of color patterns (set partitions) in an oriented cycle of length n using k or fewer colors (subsets). 3
1, 1, 1, 1, 2, 1, 1, 2, 2, 1, 1, 2, 3, 4, 1, 1, 2, 3, 6, 4, 1, 1, 2, 3, 7, 9, 8, 1, 1, 2, 3, 7, 11, 26, 10, 1, 1, 2, 3, 7, 12, 39, 53, 20, 1, 1, 2, 3, 7, 12, 42, 103, 146, 30, 1, 1, 2, 3, 7, 12, 43, 123, 367, 369, 56, 1, 1, 2, 3, 7, 12, 43, 126, 503, 1235, 1002, 94, 1, 1, 2, 3, 7, 12, 43, 127, 539, 2008, 4439, 2685, 180, 1, 1, 2, 3, 7, 12, 43, 127, 543, 2304, 8720, 15935, 7434, 316, 1, 1, 2, 3, 7, 12, 43, 127, 544, 2356, 11023, 38365, 58509, 20441, 596, 1 (list; table; graph; refs; listen; history; text; internal format)
OFFSET

1,5

COMMENTS

Two color patterns are equivalent if the colors are permuted. An oriented cycle counts each chiral pair as two.

Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.

In other words, the number of n-bead necklace structures using a maximum of k different colored beads. - Andrew Howroyd, Oct 30 2019

REFERENCES

M. R. Nester (1999). Mathematical investigations of some plant interaction designs. PhD Thesis. University of Queensland, Brisbane, Australia. [See A056391 for pdf file of Chap. 2]

LINKS

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

E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

FORMULA

T(n,k) = (1/n)*Sum_{j=1..k} Sum_{d|n} phi(d)*A(d,n/d,j), where A(d,n,k) = [n==0 & k==0] + [n>0 & k>0]*(k*A(d,n-1,k) + Sum_{j|d} A(d,n-1,k-j)).

T(n,k) = A320748(n,k) + A320742(n,k) = 2*A320748(n,k) - A305749(n,k) = A305749(n,k) + 2*A320742(n,k).

EXAMPLE

Array begins with T(1,1):

1   1    1     1      1      1      1      1      1      1      1      1 ...

1   2    2     2      2      2      2      2      2      2      2      2 ...

1   2    3     3      3      3      3      3      3      3      3      3 ...

1   4    6     7      7      7      7      7      7      7      7      7 ...

1   4    9    11     12     12     12     12     12     12     12     12 ...

1   8   26    39     42     43     43     43     43     43     43     43 ...

1  10   53   103    123    126    127    127    127    127    127    127 ...

1  20  146   367    503    539    543    544    544    544    544    544 ...

1  30  369  1235   2008   2304   2356   2360   2361   2361   2361   2361 ...

1  56 1002  4439   8720  11023  11619  11697  11702  11703  11703  11703 ...

1  94 2685 15935  38365  54682  60499  61579  61684  61689  61690  61690 ...

1 180 7434 58509 173609 284071 336447 349746 351619 351766 351772 351773 ...

For T(4,2)=4, the patterns are AAAA, AAAB, AABB, and ABAB.

For T(4,3)=6, the patterns are the above four, AABC and ABAC.

MATHEMATICA

Adnk[d_, n_, k_] := Adnk[d, n, k] = If[n>0 && k>0, Adnk[d, n-1, k]k + DivisorSum[d, Adnk[d, n-1, k-#] &], Boole[n==0 && k==0]]

Table[Sum[DivisorSum[n, EulerPhi[#] Adnk[#, n/#, j] &], {j, k-n+1}]/n, {k, 15}, {n, k}] // Flatten

PROG

(PARI) \\ R is A152175 as square matrix

R(n) = {Mat(Col([Vecrev(p/y, n) | p<-Vec(intformal(sum(m=1, n, eulerphi(m) * subst(serlaplace(-1 + exp(sumdiv(m, d, y^d*(exp(d*x + O(x*x^(n\m)))-1)/d))), x, x^m))/x))]))}

T(n)={my(M=R(n)); for(i=2, n, M[, i] += M[, i-1]); M}

{ my(A=T(12)); for(n=1, #A, print(A[n, ])) } \\ Andrew Howroyd, Nov 03 2019

CROSSREFS

Partial row sums of A152175.

Columns 1-6 are A057427, A000013, A002076, A056292, A056293, A056294.

For increasing k, columns converge to A084423.

Cf. A320748 (unoriented), A320742 (chiral), A305749 (achiral).

Sequence in context: A159936 A305749 A320748 * A238392 A144464 A138015

Adjacent sequences:  A320744 A320745 A320746 * A320748 A320749 A320750

KEYWORD

nonn,tabl,easy

AUTHOR

Robert A. Russell, Oct 21 2018

STATUS

approved

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Last modified July 24 14:21 EDT 2021. Contains 346273 sequences. (Running on oeis4.)