A056311 -id:A056311 - cp's OEIS frontend

A305621 Triangle read by rows: T(n,k) is the number of rows of n colors with exactly k different colors counting chiral pairs as equivalent, i.e., the rows are reversible.

1, 1, 1, 1, 4, 3, 1, 8, 18, 12, 1, 18, 78, 120, 60, 1, 34, 273, 780, 900, 360, 1, 70, 921, 4212, 8400, 7560, 2520, 1, 134, 2916, 20424, 63000, 95760, 70560, 20160, 1, 270, 9150, 93360, 417120, 952560, 1164240, 725760, 181440, 1, 526, 28065, 409380, 2551560, 8217720, 14817600, 15120000, 8164800, 1814400, 1, 1054, 85773, 1749780, 14804700, 64615680, 161247240, 239500800, 209563200, 99792000, 19958400

Offset: 1

Robert A. Russell, Jun 06 2018

			The triangle begins:
  1;
  1,   1;
  1,   4,    3;
  1,   8,   18,      12;
  1,  18,   78,     120,      60;
  1,  34,  273,     780,     900,     360;
  1,  70,  921,    4212,    8400,    7560,     2520;
  1, 134, 2916,   20424,   63000,   95760,    70560,    20160;
  1, 270, 9150,   93360,  417120,  952560,  1164240,   725760,  181440;
  ...
For T(3,2)=4, the achiral color rows are ABA and BAB, while the chiral pairs are AAB-BAA and ABB-BBA. For T(3,3)=3, the color rows are all chiral pairs: ABC-CBA, ACB-BCA, and BAC-CAB.

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

Columns 1-6 are A057427, A056309, A056310, A056311, A056312, and A056313.
Row sums are A326963.
A019538 counts chiral pairs as two, i.e., the rows are not reversible.
Cf. A277504, A305622.

Table[(k!/2) (StirlingS2[n, k] + StirlingS2[Ceiling[n/2], k]), {n, 1, 15}, {k, 1, n}] // Flatten

T(n,k) = {k! * (stirling(n,k,2) + stirling((n+1)\2,k,2)) / 2} \\ Andrew Howroyd, Sep 13 2019

T(n,k) = (k!/2) * (S2(n,k) + S2(ceiling(n/2),k)) where S2(n,k) is the Stirling subset number A008277.
T(n,k) = (A019538(n,k) + A019538(ceiling(n/2),k)) / 2.
G.f. for column k: k! x^k / (2*Product_{i=1..k}(1-ix)) + k! (x^(2k-1)+x^(2k)) / (2*Product{i=1..k}(1-i x^2)). - Robert A. Russell, Sep 25 2018
T(n, k) = Sum_{i=0..k} (-1)^(k-i)*binomial(k,i)*A277504(n, i). - Andrew Howroyd, Sep 13 2019

A056328 Number of reversible string structures with n beads using exactly four different colors.

Original entry on oeis.org

0, 0, 0, 1, 6, 37, 183, 877, 3930, 17185, 73095, 306361, 1267266, 5198557, 21182343, 85910917, 347187210, 1399451545, 5629911015, 22616256721, 90754855026, 363890126677, 1458172596903, 5840531635357, 23385650196090

Offset: 1

Marks R. Nester

nonn

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 four different elements. An unoriented row is equivalent to its reverse. - Robert A. Russell, Oct 14 2018

			For a(5)=6, the color patterns are ABCDA, ABCBD, AABCD, ABACD, ABCAD, and ABBCD. The first two are achiral. - _Robert A. Russell_, Oct 14 2018

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]

Index entries for linear recurrences with constant coefficients, signature (8, -12, -44, 121, 12, -228, 144).

Column 4 of A284949.
Cf. A056311.
Cf. A000453 (oriented), A320527 (chiral), A304974 (achiral).

k=4; Table[(StirlingS2[n,k] + If[EvenQ[n], StirlingS2[n/2+2,4] - StirlingS2[n/2+1,4] - 2StirlingS2[n/2,4], 2StirlingS2[(n+3)/2,4] - 4StirlingS2[(n+1)/2,4]])/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 = 4; Table[(StirlingS2[n, k] + Ach[n, k])/2, {n, 1, 30}] (* Robert A. Russell, Oct 14 2018 *)
LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 1, 6, 37, 183}, 30] (* Robert A. Russell, Oct 14 2018 *)

a(n) = A056323(n) - A001998(n-1).
Empirical g.f.: -x^4*(3*x^3 + x^2 - 2*x + 1) / ((x-1)*(2*x-1)*(2*x+1)*(3*x-1)*(4*x-1)*(3*x^2-1)). - Colin Barker, Nov 25 2012
From Robert A. Russell, Oct 14 2018: (Start)
a(n) = (S2(n,k) + A(n,k))/2, where k=4 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].
a(n) = (A000453(n) + A304974(n)) / 2 = A000453(n) - A320527(n) = A320527(n) + A304974(n). (End)

A305624 Number of chiral pairs of rows of n colors with exactly 4 different colors.

Original entry on oeis.org

0, 0, 0, 12, 120, 780, 4188, 20400, 93120, 409140, 1748220, 7337232, 30386160, 124696740, 508250988, 2061566400, 8331954240, 33585590580, 135115594140, 542784981552, 2178107091600, 8733341736900, 34996103558988, 140172672276000, 561255446475360, 2246716252964820, 8991948337723260, 35983044114659472, 143977928423467440, 576048972752188260, 2304607666801990188, 9219666007300387200, 36882370043723748480

Offset: 1

Robert A. Russell, Jun 06 2018

If the row is achiral, i.e., the same as its reverse, we ignore it. If different from its reverse, we count it and its reverse as a chiral pair.

			For a(4) = 12, the chiral pairs are the 4! = 24 permutations of ABCD, each paired with its reverse.

Fourth column of A305622.
A056455(n) is number of achiral rows of n colors with exactly 4 different colors.
Cf. A000919, A056311.

k=4; Table[(k!/2) (StirlingS2[n,k] - StirlingS2[Ceiling[n/2],k]), {n, 1, 40}]

a(n) = my(k=4); (k!/2) * (stirling(n,k,2) - stirling(ceil(n/2),k,2)); \\ Michel Marcus, Jun 07 2018

a(n) = (k!/2) * (S2(n,k) - S2(ceiling(n/2),k)), with k=4 colors used and where S2(n,k) is the Stirling subset number A008277.
a(n) = (A000919(n) - A056455(n)) / 2.
a(n) = A000919(n) - A056311(n) = A056311(n) - A056455(n).
G.f.: -(k!/2) * (x^(2k-1) + x^(2k)) / Product_{j=1..k} (1 - j*x^2) + (k!/2) * x^k / Product_{j=1..k} (1 - j*x) with k=4 colors used.

A056320 Number of primitive (aperiodic) reversible strings with n beads using exactly four different colors.

Original entry on oeis.org

0, 0, 0, 12, 120, 780, 4212, 20412, 93360, 409260, 1749780, 7338000, 30394560, 124700928, 508291692, 2061586800, 8332140720, 33585682920, 135116412660, 542785390680, 2178110585388, 8733343485120

Offset: 1

Marks R. Nester

nonn

A string and its reverse are considered to be equivalent.

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]

Cf. A056315.

sum mu(d)*A056311(n/d) where d|n.

cp's OEIS Frontend

A305621 Triangle read by rows: T(n,k) is the number of rows of n colors with exactly k different colors counting chiral pairs as equivalent, i.e., the rows are reversible.

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A056328 Number of reversible string structures with n beads using exactly four different colors.

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A305624 Number of chiral pairs of rows of n colors with exactly 4 different colors.

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A056320 Number of primitive (aperiodic) reversible strings with n beads using exactly four different colors.

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