A304972 -id:A304972 - cp's OEIS frontend

A305752 Number of achiral color patterns (set partitions) in a row or cycle of length n with 6 or fewer colors (subsets).

1, 1, 2, 3, 7, 12, 31, 58, 159, 312, 883, 1774, 5103, 10368, 30067, 61414, 178815, 366168, 1068259, 2190190, 6395919, 13120944, 38335123, 78665590, 229890591, 471814344, 1378985155, 2830350526, 8272839855, 16980500640, 49633834099, 101878204486

Offset: 0

Robert A. Russell, Jun 09 2018

An equivalent color pattern is obtained when we permute the colors. Thus all permutations of ABCDE are equivalent, as are AABCDEF and BBCDEFA. A color pattern is achiral if it is equivalent to its reversal. Rotations of the colors of a cycle are equivalent, so for cycles AABCCDEF = BCCDEFAA = CCDEFAAB.

			For a(5) = 12, the achiral patterns for both rows and cycles are AAAAA, AABAA, ABABA, ABBBA, AABCC, ABACA, ABBBC, ABCAB, ABCBA, ABCBD, ABCDA, and ABCDE.

Index entries for linear recurrences with constant coefficients, signature (1,11,-11,-36,36,36,-36).

Sixth column of A305749.
Cf. A056273 (oriented), A056325 (unoriented), A320936 (chiral), for rows.
Cf. A056294 (oriented), A056356 (unoriented), A320746 (chiral), for cycles.

seq(coeff(series((1-10*x^2+x^3+29*x^4-6*x^5-25*x^6+8*x^7)/((1-x)*(1-2*x^2)*(1-3*x^2)*(1-6*x^2)),x,n+1), x, n), n = 0 .. 35); # Muniru A Asiru, Oct 30 2018

Table[If[EvenQ[n], StirlingS2[(n+12)/2, 6] - 19 StirlingS2[(n+10)/2, 6] + 140 StirlingS2[(n+8)/2, 6] - 501 StirlingS2[(n+6)/2, 6] + 887 StirlingS2[(n+4)/2, 6] - 692 StirlingS2[(n+2)/2, 6] + 160 StirlingS2[n/2, 6], StirlingS2[(n+11)/2, 6] - 18 StirlingS2[(n+9)/2, 6] + 124 StirlingS2[(n+7)/2, 6] - 404 StirlingS2[(n+5)/2, 6] + 613 StirlingS2[(n+3)/2, 6] - 340 StirlingS2[(n+1)/2, 6]], {n, 0, 40}]
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[Ach[n, j], {j, 0, k}], {n, 0, 40}]
CoefficientList[Series[(1-10x^2+x^3+29x^4-6x^5-25x^6+8x^7) / ((1-x)(1-2x^2)(1-3x^2)(1-6 x^2)), {x, 0, 40}], x]
LinearRecurrence[{1,11,-11,-36,36,36,-36},{1,1,2,3,7,12,31,58},40]
Join[{1}, Table[If[EvenQ[n], (36 + 45 2^(n/2) + 40 3^(n/2) + 19 6^(n/2)) / 180, (72 + 45 2^((n+1)/2) + 40 3^((n+1)/2) + 13 6^((n+1)/2)) / 360], {n,40}]]

a(n) = Sum_{j=0..6} Ach(n,j), where Ach(n,k) = [n>1] * (k*T(n-2,k) + T(n-2,k-1) + T(n-2,k-2)) + [0<=n<2 & n==k].
G.f.: (1-10x^2+x^3+29x^4-6x^5-25x^6+8x^7) / ((1-x)*(1-2x^2)*(1-3x^2)*(1-6x^2)).
a(2m) = S2(m+6,6) - 19*S2(m+5,6) + 140*S2(m+4,6) - 501*S2(m+3,6) + 887*S2(m+2,6) - 692*S2(m+1,6) + 160*S2(m,6);
a(2m-1) = S2(m+5,6) - 18*S2(m+4,6) + 124*S2(m+3,6) - 404*S2(m+2,6) + 613*S2(m+1,6) - 340*S2(m,6), where S2(n,k) is the Stirling subset number A008277.
For n>0, a(2m) = (36 + 45*2^m + 40*3^m + 19*6^m) / 180.
a(2m-1) = (72 + 45*2^m + 40*3^m + 13*6^m) / 360.
a(n) = 2*A056325(n) - A056273(n) = A056273(n) - 2*A320936(n) = A056325(n) - A320936(n).
a(n) = 2*A056356(n) - A056294(n) = A056294(n) - 2*A320746(n) = A056356(n) - A320936(n).
a(n) = A057427(n) + A052551(n-2) + A304973(n) + A304974(n) + A304975(n) + A304976(n).
a(n) = a(n-1) + 11*a(n-2) - 11*a(n-3) - 36*a(n-4) + 36*a(n-5) + 36*a(n-6) - 36*a(n-7). - Muniru A Asiru, Oct 30 2018

A320527 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 4, 28, 167, 824, 3840, 16920, 72655, 305140, 1265264, 5193188, 21173607, 85887984, 347150080, 1399355440, 5629755935, 22615859180, 90754215024, 363888497148, 1458169977847, 5840524999144, 23385639542720, 93613165023560, 374664497695215, 1499293455643620, 5999080285068784, 24002040333605908

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(5)=4, the chiral pairs are AABCD-ABCDD, ABACD-ABCDC, ABBCD-ABCCD and ABCAD-ABCDB.

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

Col. 4 of A320525.

Cf. A000453 (oriented), A056328 (unoriented), 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}]
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 = 4; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{8, -12, -44, 121, 12, -228, 144}, {0, 0, 0, 0, 4, 28, 167}, 30]

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].
G.f.: (x^4 / Product_{k=1..4} (1 - k*x) - x^4*(1 + x)^2*(1 - 2 x^2) / Product_{k=1..4} (1 - k*x^2)) / 2.
a(n) = (A000453(n) - A304974(n)) / 2 = A000453(n) - A056328(n) = A056328(n) - A304974(n).

A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132, 123050, 688850, 3752350, 20032446, 105372624, 548066568, 2826316248, 14478890712, 73794322750, 374602205590, 1895629599050, 9568906539786, 48208435317284, 242500368793628, 1218342441784468, 6115097961883092, 30669103347259650, 153720181809997530, 770100204404335350, 3856500105221902326

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(6)=6, the chiral pairs are AABCDE-ABCDEE, ABACDE-ABCDED, ABCADE-ABCDEC, ABCDAE-ABCDEB, ABBCDE-ABCDDE, and ABCBDE-ABCDCE.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (13,-48,-36,551,-683,-1542,3546,80,-4280,2400).

Col. 5 of A320525.

Cf. A000481 (oriented), A056329 (unoriented), A304975 (achiral).

I:=[0,0,0,0,0,6,64,508,3428,21132]; [n le 10 select I[n] else 13*Self(n-1)-48*Self(n-2)-36*Self(n-3)+551*Self(n-4)-683*Self(n-5) -1542*Self(n-6)+3546*Self(n-7)+80*Self(n-8)-4280*Self(n-9) +2400*Self(n-10): n in [1..30]]; // G. C. Greubel, Oct 20 2018

k=5; Table[(StirlingS2[n,k] - If[EvenQ[n], 3StirlingS2[n/2+2,5] - 11StirlingS2[n/2+1,5] + 6StirlingS2[n/2,5], StirlingS2[(n+5)/2,5] - 3StirlingS2[(n+3)/2,5]])/2, {n,30}]
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 = 5; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132}, 30]

m=30; v=concat([0,0,0,0,0,6,64,508,3428,21132], vector(m-10)); for(n=11, m, v[n] = 13*v[n-1]-48*v[n-2]-36*v[n-3]+551*v[n-4]-683*v[n-5] -1542*v[n-6] +3546*v[n-7] +80*v[n-8] -4280*v[n-9] +2400*v[n-10]); v \\ G. C. Greubel, Oct 20 2018

a(n) = (S2(n,k) - A(n,k))/2, where k=5 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^5 / Product_{k=1..5} (1 - k*x) - x^5 (1 + x) (1 - 3 x^2) (1 + 2 x - 2 x^2) / Product_{k=1..5} (1 - k*x^2)) / 2.
a(n) = (A000481(n) - A304975(n)) / 2 = A000481(n) - A056329(n) = A056329(n) - A304975(n).
a(n) = 13*a(n-1) - 48*a(n-2) - 36*a(n-3) + 551*a(n-4) - 683*a(n-5) - 1542*a(n-6) + 3546*a(n-7) + 80*a(n-8) - 4280*a(n-9) + 2400*a(n-10) for n>10. - Colin Barker, May 12 2020

A320644 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 4 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 2, 17, 84, 388, 1586, 6405, 24927, 96404, 368641, 1407515, 5357974, 20403120, 77699323, 296229485, 1130614092, 4321324766, 16539645539, 63397442097, 243352167691, 935420468092, 3600493932070, 13876442107403, 53546144395718, 206864753332164, 800067806813323, 3097590602034137, 12004772596768984, 46568647645538594, 180809553280920680

Offset: 1

Robert A. Russell, Oct 18 2018

Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056297 and A304974, which can be used in conjunction with the first formula.

			For a(6)=2, the chiral pairs are AABACD-AABCAD and AABCBD-AABCDC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 4 of A320647.

Cf. A056297 (oriented), A056359 (unoriented), A304974 (achiral).

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 *)
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]]
k=4; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]

a(n) = (A056297(n) - A304974(n)) / 2 = A056297(n) - A056359(n) = A056359(n) - A304974(n).

a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=4 is number of colors or sets, 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)), and 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)).

A320936 Number of chiral pairs of color patterns (set partitions) for a row of length n using 6 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859, 1699012, 9808848, 57335124, 338073107, 2004955824, 11936998016, 71253827696, 426061036747, 2550545918300, 15280090686256, 91588065861292, 549159350303235, 3293482358956552, 19755007003402944

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056273 and A305752, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Colin Barker, Table of n, a(n) for n = 1..1000
Index entries for linear recurrences with constant coefficients, signature (16,-84,84,685,-2140,180,7200,-8244, -4176,11664,-5184).

Column 6 of A320751.

Cf. A056273 (oriented), A056325 (unoriented), A305752 (achiral).

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,k}]/2,{n,40}]
LinearRecurrence[{16, -84, 84, 685, -2140, 180, 7200, -8244, -4176, 11664, -5184}, {0, 0, 1, 4, 20, 86, 409, 1976, 10168, 54208, 299859}, 40]

concat([0,0], Vec(x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)) + O(x^40))) \\ Colin Barker, Nov 22 2018

a(n) = (A056273(n) - A305752(n))/2.
a(n) = A056273(n) - A056325(n).
a(n) = A056325(n) - A305752(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n) + A320529(n).
a(n) = Sum_{j=1..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)).
From Colin Barker, Nov 22 2018: (Start)
G.f.: x^3*(1 - 12*x + 40*x^2 + 18*x^3 - 308*x^4 + 376*x^5 + 364*x^6 - 882*x^7 + 378*x^8) / ((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 4*x)*(1 - 6*x)*(1 - 2*x^2)*(1 - 3*x^2)*(1 - 6*x^2)).
a(n) = 16*a(n-1) - 84*a(n-2) + 84*a(n-3) + 685*a(n-4) - 2140*a(n-5) + 180*a(n-6) + 7200*a(n-7) - 8244*a(n-8) - 4176*a(n-9) + 11664*a(n-10) - 5184*a(n-11) for n>11.
(End)

A320645 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 5 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 4, 51, 339, 2010, 10900, 56700, 286888, 1426542, 7014746, 34229050, 166197824, 804243285, 3883608940, 18729354638, 90266471623, 434946282498, 2096010533584, 10104160993993, 48733654211358, 235195966291020, 1135892493220025, 5490005931157446, 26555178320890184, 128550000630553133, 622790399873432344, 3019641804537586657

Offset: 1

Robert A. Russell, Oct 18 2018

Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056298 and A304975, which can be used in conjunction with the first formula.

			For a(7)=4, the chiral pairs are AABACDE-AABCDAE, AABCBDE-AABCDED, AABCDBE-AABCDEC, and ABACBDE-ABACDBE.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 5 of A320647.

Cf. A056298 (oriented), A056360 (unoriented), A304975 (achiral).

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 *)
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]]
k=5; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]

a(n) = (A056298(n) - A304975(n)) / 2 = A056298(n) - A056360(n) = A056360(n) - A304975(n).

a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=5 is number of colors or sets, 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)), and 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)).

A320743 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 4, 13, 46, 144, 420, 1221, 3474, 9856, 27794, 78632, 222156, 629760, 1787440, 5087797, 14509580, 41479867, 118811286, 341009901, 980488510, 2824029648, 8146494860, 23534997912, 68084154502, 197211336576, 571915188840, 1660405181149, 4825559508106, 14038010213051, 40875403561680, 119122661856133, 347441159864556, 1014152747485696

Offset: 1

Robert A. Russell, Oct 21 2018

Two color patterns are equivalent if the colors are permuted.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A002076 and A182522, which can be used in conjunction with the first formula.

			For a(6)=4, the chiral pairs are AAABBC-AAABCC, AABABC-AABCAC, AABACB-AABCAB, and AABACC-AABBAC.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 3 of A320742.

Cf. A002076 (oriented), A056353 (unoriented), A182522 (achiral).

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]]
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=3; Table[Sum[(DivisorSum[n,EulerPhi[#] Adnk[#,n/#,j]&]/n - Ach[n,j])/2, {j, k}], {n,40}]

a(n) = (A002076(n) - A182522(n)) / 2 = A002076(n) - A056353(n) = A056353(n) - A182522(n).
a(n) = Sum_{j=1..k} -Ach(n,j)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,j), where k=3 is the maximum number of colors, 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)), and 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)).
a(n) = A059053(n) + A320643(n).

A320935 Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).

Original entry on oeis.org

0, 0, 1, 4, 20, 86, 400, 1852, 8868, 42892, 210346, 1038034, 5150110, 25623486, 127740880, 637539592, 3184224728, 15910524632, 79520923966, 397508610454, 1987255480650, 9935410066186, 49674450471460, 248364429410332, 1241798688445588, 6208922948527572, 31044403310614786

Offset: 1

Robert A. Russell, Oct 27 2018

Two color patterns are equivalent if the colors are permuted.
A chiral row is not equivalent to its reverse.
There are nonrecursive formulas, generating functions, and computer programs for A056272 and A305751, which can be used in conjunction with the first formula.

			For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Index entries for linear recurrences with constant coefficients, signature (11,-34,-16,247,-317,-200,610,-300).

Column 5 of A320751.

Cf. A056272 (oriented), A056324 (unoriented), A305751 (achiral).

LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {0, 0, 1, 4, 20, 86, 400, 1852}, 40] (* or *)
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=5; Table[Sum[StirlingS2[n,j]-Ach[n,j],{j,k}]/2,{n,40}]

a(n) = (A056272(n) - A305751(n))/2.
a(n) = A056272(n) - A056324(n).
a(n) = A056324(n) - A305751(n).
a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n).
a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=5 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)).
G.f.: x^3*(1 - 7*x + 10*x^2 + 18*x^3 - 49*x^4 + 25*x^5)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 5*x)*(1 - 5*x^2)*(1 - 2*x^2)). - Bruno Berselli, Oct 31 2018

A320529 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 6 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638, 210332227, 1367416232, 8752773288, 55343303064, 346540112781, 2153037307846, 13292835205606, 81652655795106, 499484899831775, 3045117929546220, 18513208314957356, 112297592929814292, 679900657841661529, 4110073054119135194, 24814158520762637754

Offset: 1

Robert A. Russell, Oct 14 2018

Two color patterns are equivalent if we permute the colors. Chiral color patterns must not be equivalent if we reverse the order of the pattern.

			For a(7)=9, the chiral pairs are AABCDEF-ABCDEFF, ABACDEF-ABCDEFE, ABCADEF-ABCDEFD, ABCDAEF-ABCDEFC, ABCDEAF-ABCDEFB, ABBCDEF-ABCDEEF, ABCBDEF-ABCDEDF, ABCDBEF-ABCDECF, and ABCCDEF-ABCDDEF.

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

Column 6 of A320525.

Cf. A000770 (oriented), A056330 (unoriented), A304976 (achiral).

m:=30; R:=PowerSeriesRing(Integers(), m); [0,0,0,0,0,0] cat Coefficients(R!((x^6/(&*[1-k*x: k in [1..6]]) - x^6*(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/(&*[1-k*x^2: k in [1..6]]) )/2)); // G. C. Greubel, Oct 19 2018

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}]
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[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638}, 30]

x='x+O('x^30); concat(vector(6), Vec((x^6/prod(k=1,6, 1-k*x) - x^6* (1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/prod(k=1,6,(1-k*x^2)))/2)) \\ G. C. Greubel, Oct 19 2018

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)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.
a(n) = (A000770(n) - A304976(n)) / 2 = A000770(n) - A056330(n) = A056330(n) - A304976(n).

A320646 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 6 colors (subsets).

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 9, 125, 1054, 7928, 54383, 356594, 2259504, 14008733, 85422360, 514773336, 3074341497, 18238301412, 107649939612, 632987843336, 3711471738408, 21716706883190, 126879832615600, 740528154956264, 4319137675225128, 25181504728152534, 146788320134425736, 855660631677225738, 4988501691655508510, 29089896998939710698

Offset: 1

Robert A. Russell, Oct 19 2018

Two color patterns are the same if the colors are permuted. A chiral cycle is different from its reverse.
Adnk[d,n,k] in Mathematica program is coefficient of x^k in A(d,n)(x) in Gilbert and Riordan reference.
There are nonrecursive formulas, generating functions, and computer programs for A056299 and A304976, which can be used in conjunction with the first formula.

			For a(8)=9, the chiral pairs are AABACDEF-AABCDEAF, AABCADEF-AABCDAEF, AABCBDEF-AABCDEFE, AABCDBEF-AABCDEFD, AABCDEBF-AABCDEFC, AABCDCEF-AABCDEDF, ABACDEBF-ABACDEBF, ABCADBEF-ABCADECF, and ABCDAEBF-ABCADBEF.

Andrew Howroyd, Table of n, a(n) for n = 1..200
E. N. Gilbert and J. Riordan, Symmetry types of periodic sequences, Illinois J. Math., 5 (1961), 657-665.

Column 6 of A320647.

Cf. A056299 (oriented), A056361 (unoriented), A304976 (achiral).

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 *)
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]]
k=6; Table[DivisorSum[n,EulerPhi[#]Adnk[#,n/#,k]&]/(2n) - Ach[n,k]/2,{n,40}]

a(n) = (A056299(n) - A304976(n)) / 2 = A056299(n) - A056361(n) = A056361(n) - A304976(n).

a(n) = -Ach(n,k)/2 + (1/2n)*Sum_{d|n} phi(d)*A(d,n/d,k), where k=5 is number of colors or sets, 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)), and 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)).

cp's OEIS Frontend

A305752 Number of achiral color patterns (set partitions) in a row or cycle of length n with 6 or fewer colors (subsets).

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A320527 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets).

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A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

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A320644 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 4 colors (subsets).

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A320936 Number of chiral pairs of color patterns (set partitions) for a row of length n using 6 or fewer colors (subsets).

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A320645 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using exactly 5 colors (subsets).

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A320743 Number of chiral pairs of color patterns (set partitions) in a cycle of length n using 3 or fewer colors (subsets).

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