A320526 a(n) is the number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 3 colors (subsets).

0, 0, 0, 2, 10, 40, 141, 464, 1480, 4600, 14145, 43052, 130480, 393820, 1186521, 3568784, 10725760, 32213200, 96714465, 290284052, 871142800, 2613981700, 7843080201, 23531425304, 70598731840, 211804847800, 635432109585, 1906330676252, 5719061512720, 17157321139180

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(4)=2, the two chiral pairs are AABC-ABCC and ABAC-ABCB.
		

Links

• Index entries for linear recurrences with constant coefficients, signature (6, -6, -24, 49, 6, -66, 36).

Crossrefs

Column 3 of A320525.

Cf. A000392 (oriented), A056327 (unoriented), A304973 (achiral).

Programs

Magma

I:=[0,0,0,2,10,40,141]; [n le 7 select I[n] else 6*Self(n-1) -6*Self(n-2) -24*Self(n-3) +49*Self(n-4) +6*Self(n-5) -66*Self(n-6) +36*Self(n-7): n in [1..40]]; // G. C. Greubel, Oct 16 2018

Mathematica

k=3; Table[(StirlingS2[n,k] - If[EvenQ[n], 2StirlingS2[n/2+1,3] - 2StirlingS2[n/2,3], StirlingS2[(n+3)/2,3] - StirlingS2[(n+1)/2,3]])/2, {n, 1, 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 = 3; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{6, -6, -24, 49, 6, -66, 36}, {0, 0, 0, 2, 10, 40,
  141}, 40]

PARI

m=40; v=concat([0,0,0,2,10,40,141], vector(m-7)); for(n=8, m, v[n] = 6*v[n-1] -6*v[n-2] -24*v[n-3] +49*v[n-4] +6*v[n-5] -66*v[n-6] +36*v[n-7] ); v \\ G. C. Greubel, Oct 16 2018

Formula

a(n) = (S2(n,k) - A(n,k))/2, where k=3 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^3 / Product_{k=1..3} (1 - k*x) - x^3*(1 + 2 x)/((1 - 2 x^2)*(1 - 3 x^2))) / 2.

a(n) = (A000392(n) - A304973(n)) / 2 = A000392(n) - A056327(n) = A056327(n) - A304973(n).