A320937 Number of chiral pairs of color patterns (set partitions) for a row of length n.

0, 0, 1, 4, 20, 86, 409, 1988, 10404, 57488, 338180, 2103378, 13814202, 95423766, 691415451, 5239857008, 41431883216, 341036489096, 2916365967707, 25862060748614, 237434856965694, 2253357681164288, 22076002386446896, 222979432604192844, 2319295160051570620

Offset

1,4

Comments

Two color patterns are equivalent if the colors are permuted.

A chiral row is not equivalent to its reverse.

Links

Andrew Howroyd, Table of n, a(n) for n = 1..200

Formula

a(n) = (A000110(n) + A080107(n)) / 2 = A000110(n) - A103293(n+1) = A103293(n+1) - A080107(n).

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

Example

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

Mathematica

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 *)
Table[Sum[StirlingS2[n, j]-Ach[n, j], {j, n}]/2, {n, 40}]

Prog

(PARI) \\ Ach is A304972 as square matrix.
Ach(n)={my(M=matrix(n, n, i, k, i>=k)); for(i=3, n, for(k=2, n, M[i, k]=k*M[i-2, k] + M[i-2, k-1] + if(k>2, M[i-2, k-2]))); M}
seq(n)={my(A=Ach(n)); vector(n, n, sum(k=1, n, stirling(n, k, 2) - A[n, k])/2)} \\ Andrew Howroyd, Sep 18 2019

Crossrefs

Row sums of triangle A320525.

Limit as k increases of column k of array A320751.

Cf. A000110 (oriented), A103293 (unoriented), A080107 (achiral).

Sequence in context: A343361 A320935 A320936 * A196953 A093357 A027156

Adjacent sequences: A320934 A320935 A320936 * A320938 A320939 A320940

Keyword

nonn,easy

Author

Robert A. Russell, Oct 27 2018

Status

approved