A320529 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 6 colors (subsets).

0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638, 210332227, 1367416232, 8752773288, 55343303064, 346540112781, 2153037307846, 13292835205606, 81652655795106, 499484899831775, 3045117929546220, 18513208314957356, 112297592929814292, 679900657841661529, 4110073054119135194, 24814158520762637754

Offset

1,7

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.

Links

Table of n, a(n) for n=1..29.

Index entries for linear recurrences with constant coefficients, signature (21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600).

Formula

a(n) = (S2(n,k) - A(n,k))/2, where k=6 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^6 / (Product_{k=1..6} (1 - k*x)) - x^6 *(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3) / Product_{k=1..6} (1 - k*x^2)) / 2.

a(n) = (A000770(n) - A304976(n)) / 2 = A000770(n) - A056330(n) = A056330(n) - A304976(n).

Example

For a(7)=9, the chiral pairs are AABCDEF-ABCDEFF, ABACDEF-ABCDEFE, ABCADEF-ABCDEFD, ABCDAEF-ABCDEFC, ABCDEAF-ABCDEFB, ABBCDEF-ABCDEEF, ABCBDEF-ABCDEDF, ABCDBEF-ABCDECF, and ABCCDEF-ABCDDEF.

Mathematica

k=6; Table[(StirlingS2[n, k] - If[EvenQ[n], StirlingS2[n/2+3, 6] - 3StirlingS2[n/2+2, 6] - 8StirlingS2[n/2+1, 6] + 16StirlingS2[n/2, 6], 3StirlingS2[(n+5)/2, 6] - 17StirlingS2[(n+3)/2, 6] + 20StirlingS2[(n+1)/2, 6]])/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 = 6; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{21, -159, 399, 1085, -8085, 9555, 34125, -98644, 5544, 253764, -248724, -136800, 317520, -129600}, {0, 0, 0, 0, 0, 0, 9, 124, 1300, 11316, 89513, 660978, 4658738, 31711638}, 30]

Prog

(PARI) x='x+O('x^30); concat(vector(6), Vec((x^6/prod(k=1, 6, 1-k*x) - x^6* (1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/prod(k=1, 6, (1-k*x^2)))/2)) \\ G. C. Greubel, Oct 19 2018
(Magma) m:=30; R<x>:=PowerSeriesRing(Integers(), m); [0, 0, 0, 0, 0, 0] cat Coefficients(R!((x^6/(&*[1-k*x: k in [1..6]]) - x^6*(1+x)*(1-4*x^2)*(1+2*x-x^2-4*x^3)/(&*[1-k*x^2: k in [1..6]]) )/2)); // G. C. Greubel, Oct 19 2018

Crossrefs

Column 6 of A320525.

Cf. A000770 (oriented), A056330 (unoriented), A304976 (achiral).

Sequence in context: A211101 A269627 A209504 * A280896 A364940 A138438

Adjacent sequences: A320526 A320527 A320528 * A320530 A320531 A320532

Keyword

nonn,easy

Author

Robert A. Russell, Oct 14 2018

Status

approved