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A320527
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Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 4 colors (subsets).
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5
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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
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OFFSET
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1,5
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COMMENTS
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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.
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LINKS
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FORMULA
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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.
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EXAMPLE
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For a(5)=4, the chiral pairs are AABCD-ABCDD, ABACD-ABCDC, ABBCD-ABCCD and ABCAD-ABCDB.
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MATHEMATICA
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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]
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CROSSREFS
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KEYWORD
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nonn,easy
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AUTHOR
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STATUS
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approved
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