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

Author

Robert A. Russell, Oct 27 2018

Keywords

• nonn
• easy

Comments

Two color patterns are equivalent if the colors are permuted.

A chiral row is not equivalent to its reverse.

Examples

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

Links

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

Crossrefs

Row sums of triangle A320525.

Limit as k increases of column k of array A320751.

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

Programs

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}]

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

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)).