A320528 Number of chiral pairs of color patterns (set partitions) in a row of length n using exactly 5 colors (subsets).

0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132, 123050, 688850, 3752350, 20032446, 105372624, 548066568, 2826316248, 14478890712, 73794322750, 374602205590, 1895629599050, 9568906539786, 48208435317284, 242500368793628, 1218342441784468, 6115097961883092, 30669103347259650, 153720181809997530, 770100204404335350, 3856500105221902326

Offset

1,6

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

Colin Barker, Table of n, a(n) for n = 1..1000

Index entries for linear recurrences with constant coefficients, signature (13,-48,-36,551,-683,-1542,3546,80,-4280,2400).

Formula

a(n) = (S2(n,k) - A(n,k))/2, where k=5 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^5 / Product_{k=1..5} (1 - k*x) - x^5 (1 + x) (1 - 3 x^2) (1 + 2 x - 2 x^2) / Product_{k=1..5} (1 - k*x^2)) / 2.

a(n) = (A000481(n) - A304975(n)) / 2 = A000481(n) - A056329(n) = A056329(n) - A304975(n).

a(n) = 13*a(n-1) - 48*a(n-2) - 36*a(n-3) + 551*a(n-4) - 683*a(n-5) - 1542*a(n-6) + 3546*a(n-7) + 80*a(n-8) - 4280*a(n-9) + 2400*a(n-10) for n>10. - Colin Barker, May 12 2020

Example

For a(6)=6, the chiral pairs are AABCDE-ABCDEE, ABACDE-ABCDED, ABCADE-ABCDEC, ABCDAE-ABCDEB, ABBCDE-ABCDDE, and ABCBDE-ABCDCE.

Mathematica

k=5; Table[(StirlingS2[n, k] - If[EvenQ[n], 3StirlingS2[n/2+2, 5] - 11StirlingS2[n/2+1, 5] + 6StirlingS2[n/2, 5], StirlingS2[(n+5)/2, 5] - 3StirlingS2[(n+3)/2, 5]])/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 = 5; Table[(StirlingS2[n, k] - Ach[n, k])/2, {n, 1, 30}]
LinearRecurrence[{13, -48, -36, 551, -683, -1542, 3546, 80, -4280, 2400}, {0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132}, 30]

Prog

(PARI) m=30; v=concat([0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132], vector(m-10)); for(n=11, m, v[n] = 13*v[n-1]-48*v[n-2]-36*v[n-3]+551*v[n-4]-683*v[n-5] -1542*v[n-6] +3546*v[n-7] +80*v[n-8] -4280*v[n-9] +2400*v[n-10]); v \\ G. C. Greubel, Oct 20 2018
(Magma) I:=[0, 0, 0, 0, 0, 6, 64, 508, 3428, 21132]; [n le 10 select I[n] else 13*Self(n-1)-48*Self(n-2)-36*Self(n-3)+551*Self(n-4)-683*Self(n-5) -1542*Self(n-6)+3546*Self(n-7)+80*Self(n-8)-4280*Self(n-9) +2400*Self(n-10): n in [1..30]]; // G. C. Greubel, Oct 20 2018

Crossrefs

Col. 5 of A320525.

Cf. A000481 (oriented), A056329 (unoriented), A304975 (achiral).

Sequence in context: A222596 A067447 A083225 * A237357 A230282 A186668

Adjacent sequences: A320525 A320526 A320527 * A320529 A320530 A320531

Keyword

nonn,easy

Author

Robert A. Russell, Oct 14 2018

Status

approved