A085436 Number of partitions of n without rotational symmetry (or 1-fold symmetry).

1, 0, 1, 1, 5, 2, 13, 8, 21, 17, 54, 31, 99, 70, 139, 131, 295, 207, 488, 387, 698, 657, 1253, 995, 1923, 1707, 2785, 2670, 4563, 3900, 6840, 6287, 9606, 9445, 14746, 13517, 21635, 20614, 30000, 29903, 44581, 42067, 63259

Offset: 1

Author

Wouter Meeussen, Aug 14 2003

Keywords

• nonn

Comments

A partition of n is considered to have d-fold symmetry if it consists of runs of (one or more) equal integers that add up to d or a multiple of n/d.

Set partitions with rotational d-fold symmetry (A084423) have block lengths that are d-fold symmetrical partitions of n, (d|n), as defined above.

Examples

			a(6)=2 since the 11 partitions of 6 consist of 4 having 6-fold symmetry: {6},{3,3},{2,2,2},{1,1,1,1,1,1}; 1 with 3-fold: {3,1,1,1}; 4 with 2-fold: {4,2},{4,1,1},{2,2,1,1},{2,1,1,1,1}; and only 2 with 1-fold symmetry (= no rotational symmetry): {5,1} and {3,2,1}.
		

Crossrefs

Cf. A084423.

Programs

Mathematica

Needs["DiscreteMath`Combinatorica`"]; f := Function[{n, d}, Cases[ Partitions[n], q_List /; (Union[ Mod[ (First[ # ] Length[ # ] &) /@ Split[q], d]] == {0})]]; fixp[j_] := Table[d = Part[ Divisors[n], k]; Length@f[n, d], {n, j}, {k, DivisorSigma[0, n]}]; Do[ Print[ Last[ Table[ Fold[ Plus, 0, MoebiusMu[ n/ Divisors[n]] Reverse[ fixp[i][[i]] ]], {n, i}]]], {i, 1, 43}]

Extensions

Edited and extended by Robert G. Wilson v, Aug 15 2003