A308706 Number of chiral pairs of set partitions of a primitive cycle of n elements having exactly two different elements.

0, 0, 0, 0, 0, 0, 0, 1, 2, 7, 12, 31, 58, 126, 233, 484, 904, 1800, 3395, 6643, 12612, 24457, 46655, 90157, 172750, 333498, 641214, 1238664, 2388618, 4620006, 8931536, 17302033, 33521792, 65042495, 126257160, 245361171, 477087772, 928510506, 1808145395, 3523813566

Offset: 0

Author

Robert A. Russell, Jun 18 2019

Keywords

• nonn

Examples

			For a(7)=1, the chiral pair is 0001011-0001101.  For a(8)=2, the chiral pairs are 00001011-00001101 and 00010011-00011001.
		

Crossrefs

Cf. A000048 (oriented), A000046 (unoriented), A179781 (achiral), A059053 (not primitive).

Programs

Mathematica

Join[{0}, Table[(DivisorSum[NestWhile[#/2 &, n, EvenQ], MoebiusMu[#] 2^(n/#) &]/(2 n) - DivisorSum[n, MoebiusMu[n/#] 2^Floor[#/2] &])/2, {n, 1, 40}]]

PARI

a(n) = if (n, (sumdiv(n, d, if (d%2, moebius(d)*2^(n/d)))/(2*n) - sumdiv(n, d, moebius(n/d)*2^(d\2)))/2, 0); \\ Michel Marcus, Jun 27 2019; corrected Jun 12 2022

Formula

a(n) = ((1/(2n)) * Sum_{odd d|n} mu(d)*2^(n/d) - Sum_{d|n} mu(n/d)*2^floor(d/2)) / 2, where mu is the Möbius function at A008683.

a(n) = A000048(n) - A000046(n) = (A000048(n) - A179781(n))/2 = A000046(n) - A179781(n).

A059053(n) = Sum_{d|n} a(d).