A320934 Number of chiral pairs of color patterns (set partitions) for a row of length n using 4 or fewer colors (subsets).

0, 0, 1, 4, 20, 80, 336, 1344, 5440, 21760, 87296, 349184, 1397760, 5591040, 22368256, 89473024, 357908480, 1431633920, 5726601216, 22906404864, 91625881600, 366503526400, 1466015154176, 5864060616704, 23456246661120, 93824986644480, 375299963355136, 1501199853420544, 6004799480791040

Offset

1,4

Comments

Two color patterns are equivalent if the colors are permuted.

A chiral row is not equivalent to its reverse.

There are nonrecursive formulas, generating functions, and computer programs for A124303 and A305750, which can be used in conjunction with the first formula.

Links

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

Index entries for linear recurrences with constant coefficients, signature (4,4,-16).

Formula

a(n) = (A124303(n) - A305750(n))/2.

a(n) = A124303(n) - A056323(n).

a(n) = A056323(n) - A305750(n).

a(n) = A122746(n-2) + A320526(n) + A320527(n).

a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=4 is the maximum number of colors, S2 is the Stirling subset number A008277, and Ach(n,k) = [n>=0 & n<2 & n==k] + [n>1]*(k*Ach(n-2,k) + Ach(n-2,k-1) + Ach(n-2,k-2)).

a(2*m) = (16^m - 4*4^m)/48.

a(2*m-1) = (16^m - 4*4^m)/192.

a(n) = (4^n - 4^floor(n/2+1))/48.

G.f.: x^2/((-1 + 4*x)*(-1 + 4*x^2)). - Stefano Spezia, Oct 29 2018

a(n) = 2^n*(2^n - (-1)^n - 3)/48. - Bruno Berselli, Oct 31 2018

Example

For a(4)=4, the chiral pairs are AAAB-ABBB, AABA-ABAA, AABC-ABCC, and ABAC-ABCB.

Mathematica

Table[(4^n - 4^Floor[n/2+1])/48, {n, 40}] (* or *)
LinearRecurrence[{4, 4, -16}, {0, 0, 1}, 40] (* or *)
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[Sum[StirlingS2[n, j]-Ach[n, j], {j, k}]/2, {n, 40}]
CoefficientList[Series[x^2/((-1 + 4 x) (-1 + 4 x^2)), {x, 0, 50}], x] (* Stefano Spezia, Oct 29 2018 *)

Crossrefs

Column 4 of A320751.

Cf. A124303 (oriented), A056323 (unoriented), A305750 (achiral).

Sequence in context: A255050 A371408 A292540 * A344063 A055296 A140532

Adjacent sequences: A320931 A320932 A320933 * A320935 A320936 A320937

Keyword

nonn,easy

Author

Robert A. Russell, Oct 27 2018

Status

approved