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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).
5
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
1,5
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.
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).
EXAMPLE
For a(5)=4, the chiral pairs are AABCD-ABCDD, ABACD-ABCDC, ABBCD-ABCCD and ABCAD-ABCDB.
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]
CROSSREFS
Col. 4 of A320525.
Cf. A000453 (oriented), A056328 (unoriented), A304974 (achiral).
Sequence in context: A272936 A053524 A270943 * A272281 A182190 A317232
KEYWORD
nonn,easy
AUTHOR
Robert A. Russell, Oct 14 2018
STATUS
approved