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

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

Author

Robert A. Russell, Oct 14 2018

Keywords

• nonn
• easy

Comments

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.

Examples

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

Links

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

Crossrefs

Col. 4 of A320525.

Cf. A000453 (oriented), A056328 (unoriented), A304974 (achiral).

Programs

Mathematica

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]

Formula

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).