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A085436 Number of partitions of n without rotational symmetry (or 1-fold symmetry). 1
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 (list; graph; refs; listen; history; text; internal format)
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 A194048

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

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Last modified March 2 09:23 EST 2021. Contains 341746 sequences. (Running on oeis4.)