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

1,5

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.

Links

Table of n, a(n) for n=1..43.

Example

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}.

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}]

Crossrefs

Cf. A084423.

Sequence in context: A013946 A261327 A330613 * A277710 A286148 A369369

Adjacent sequences: A085433 A085434 A085435 * A085437 A085438 A085439

Keyword

nonn

Author

Wouter Meeussen, Aug 14 2003

Extensions

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

Status

approved