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%I #3 Mar 30 2012 18:37:43
%S 1,0,1,1,5,2,13,8,21,17,54,31,99,70,139,131,295,207,488,387,698,657,
%T 1253,995,1923,1707,2785,2670,4563,3900,6840,6287,9606,9445,14746,
%U 13517,21635,20614,30000,29903,44581,42067,63259
%N Number of partitions of n without rotational symmetry (or 1-fold symmetry).
%C 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.
%C Set partitions with rotational d-fold symmetry (A084423) have block lengths that are d-fold symmetrical partitions of n, (d|n), as defined above.
%e 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}.
%t 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}]
%Y Cf. A084423.
%K nonn
%O 1,5
%A _Wouter Meeussen_, Aug 14 2003
%E Edited and extended by _Robert G. Wilson v_, Aug 15 2003
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