A320935 Number of chiral pairs of color patterns (set partitions) for a row of length n using 5 or fewer colors (subsets).

0, 0, 1, 4, 20, 86, 400, 1852, 8868, 42892, 210346, 1038034, 5150110, 25623486, 127740880, 637539592, 3184224728, 15910524632, 79520923966, 397508610454, 1987255480650, 9935410066186, 49674450471460, 248364429410332, 1241798688445588, 6208922948527572, 31044403310614786

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 A056272 and A305751, which can be used in conjunction with the first formula.

Links

Table of n, a(n) for n=1..27.

Index entries for linear recurrences with constant coefficients, signature (11,-34,-16,247,-317,-200,610,-300).

Formula

a(n) = (A056272(n) - A305751(n))/2.

a(n) = A056272(n) - A056324(n).

a(n) = A056324(n) - A305751(n).

a(n) = A122746(n-2) + A320526(n) + A320527(n) + A320528(n).

a(n) = Sum_{j=1..k} (S2(n,j) - Ach(n,j)) / 2, where k=5 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)).

G.f.: x^3*(1 - 7*x + 10*x^2 + 18*x^3 - 49*x^4 + 25*x^5)/((1 - x)*(1 - 2*x)*(1 - 3*x)*(1 - 5*x)*(1 - 5*x^2)*(1 - 2*x^2)). - Bruno Berselli, Oct 31 2018

Example

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

Mathematica

LinearRecurrence[{11, -34, -16, 247, -317, -200, 610, -300}, {0, 0, 1, 4, 20, 86, 400, 1852}, 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=5; Table[Sum[StirlingS2[n, j]-Ach[n, j], {j, k}]/2, {n, 40}]

Crossrefs

Column 5 of A320751.

Cf. A056272 (oriented), A056324 (unoriented), A305751 (achiral).

Sequence in context: A262768 A250003 A343361 * A320936 A320937 A196953

Adjacent sequences: A320932 A320933 A320934 * A320936 A320937 A320938

Keyword

nonn,easy

Author

Robert A. Russell, Oct 27 2018

Status

approved