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

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

Author

Robert A. Russell, Oct 27 2018

Keywords

• nonn
• easy

Comments

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.

Examples

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

Links

• 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).

Crossrefs

Column 6 of A320751.

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

Programs

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,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]

PARI

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

Formula

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)