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A056325 Number of reversible string structures with n beads using a maximum of six different colors. 8
1, 1, 2, 4, 11, 32, 117, 467, 2135, 10480, 55091, 301633, 1704115, 9819216, 57365191, 338134521, 2005134639, 11937364184, 71254895955, 426063226937, 2550552314219, 15280103807200, 91588104196415, 549159428968825 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,3

COMMENTS

A string and its reverse are considered to be equivalent. Permuting the colors will not change the structure. Thus aabc, cbaa and bbac are all considered to be identical.

Number of set partitions of an unoriented row of n elements with six or fewer nonempty subsets. - Robert A. Russell, Oct 28 2018

There are nonrecursive formulas, generating functions, and computer programs for A056273 and A305752, which can be used in conjunction with the first formula. - Robert A. Russell, Oct 28 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=0..23.

FORMULA

Use de Bruijn's generalization of Polya's enumeration theorem as discussed in reference.

From Robert A. Russell, Oct 28 2018: (Start)

a(n) = (A056273(n) + A305752(n)) / 2.

a(n) = A056273(n) - A320936(n) = A320936(n) + A305752(n).

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

a(n) = A000007(n) + A057427(n) + A056326(n) + A056327(n) + A056328(n) + A056329(n) + A056330(n). (End)

EXAMPLE

For a(4)=11, the 7 achiral patterns are AAAA, AABB, ABAB, ABBA, ABCA, ABBC, and ABCD.  The 4 chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

MATHEMATICA

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]] (* A304972 *)

k=6; Table[Sum[StirlingS2[n, j]+Ach[n, j], {j, 0, k}]/2, {n, 0, 40}] (* Robert A. Russell, Oct 28 2018 *)

LinearRecurrence[{16, -84, 84, 685, -2140, 180, 7200, -8244, -4176, 11664, -5184}, {1, 1, 2, 4, 11, 32, 117, 467, 2135, 10480, 55091, 301633}, 40] (* Robert A. Russell, Oct 28 2018 *)

CROSSREFS

Cf. A056308.

Column 6 of A320750.

Cf. A056273 (oriented), A320936 (chiral), A305752 (achiral).

Sequence in context: A113774 A124504 A056324 * A103293 A123418 A123412

Adjacent sequences:  A056322 A056323 A056324 * A056326 A056327 A056328

KEYWORD

nonn

AUTHOR

Marks R. Nester

EXTENSIONS

Another term from Robert A. Russell, Oct 29 2018

a(0)=1 prepended by Robert A. Russell, Nov 09 2018

STATUS

approved

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Last modified August 18 06:42 EDT 2019. Contains 326072 sequences. (Running on oeis4.)