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

0,9

Links

Table of n, a(n) for n=0..39.

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).

Example

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

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}]]

Prog

(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

Crossrefs

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

Sequence in context: A007230 A290234 A327734 * A059053 A032025 A088662

Adjacent sequences: A308703 A308704 A308705 * A308707 A308708 A308709

Keyword

nonn

Author

Robert A. Russell, Jun 18 2019

Status

approved