A056330 Number of reversible string structures with n beads using exactly six different colors.

0, 0, 0, 0, 0, 1, 12, 142, 1346, 11511, 89974, 662674, 4662574, 31724735, 210361046, 1367510326, 8752976610, 55343947975, 346541488998, 2153041587538, 13292844257198, 81652683550119, 499484958151630

Offset

1,7

Comments

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure.

Number of set partitions for an unoriented row of n elements using exactly six different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

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

Table of n, a(n) for n=1..23.

Index entries for linear recurrences with constant coefficients, signature (21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600).

Formula

a(n) = A056325(n) - A056324(n).

From Robert A. Russell, Oct 14 2018: (Start)

a(n) = (S2(n,k) + A(n,k))/2, where k=6 is the number of colors (sets), S2 is the Stirling subset number A008277 and A(n,k) = [n>1] * (k*A(n-2,k) + A(n-2,k-1) + A(n-2,k-2)) + [n<2 & n==k & n>=0].

G.f.: (x^6 / Product_{k=1..6} (1 - k*x) + x^6 (1+x) (1-4x^2) (1+2x-x^2-4x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.

a(n) = (A000770(n) + A304976(n)) / 2 = A000770(n) - A320529(n) = A320529(n) + A304976(n). (End)

Example

For a(7)=12, the color patterns are ABCDEFA, ABCDEBF, ABCDCEF, AABCDEF, ABACDEF, ABCADEF, ABCDAEF, ABBCDEF, ABCBDEF, ABCDBEF, and ABCCDEF. The first three are achiral. - Robert A. Russell, Oct 14 2018

Mathematica

k=6; Table[(StirlingS2[n, k] + If[EvenQ[n], StirlingS2[n/2+3, 6] - 3StirlingS2[n/2+2, 6] - 8StirlingS2[n/2+1, 6] + 16StirlingS2[n/2, 6], 3StirlingS2[(n+5)/2, 6] - 17StirlingS2[(n+3)/2, 6] + 20StirlingS2[(n+1)/2, 6]])/2, {n, 30}] (* Robert A. Russell, Oct 14 2018 *)
Ach[n_, k_] := Ach[n, k] = If[n < 2, Boole[n == k && n >= 0], k Ach[n-2, k] + Ach[n-2, k-1] + Ach[n-2, k-2]]
k = 6; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 1, 12, 142, 1346, 11511, 89974, 662674, 4662574, 31724735}, 40] (* Robert A. Russell, Oct 14 2018 *)

Crossrefs

Column 6 of A284949.

Cf. A056313.

Cf. A000770 (oriented), A320529 (chiral), A304976 (achiral).

Sequence in context: A266177 A135326 A056340 * A158516 A163448 A219307

Adjacent sequences: A056327 A056328 A056329 * A056331 A056332 A056333

Keyword

nonn,easy

Author

Marks R. Nester

Status

approved