A238661 - OEIS

%I #14 Mar 11 2014 13:40:19

%S 0,0,0,0,0,0,1,2,5,7,12,18,29,42,61,85,118,164,223,299,399,530,693,

%T 888,1157,1488,1901,2403,3044,3807,4783,5935,7368,9097,11197,13721,

%U 16806,20441,24868,30133,36494,43895,52880,63424,75900,90609,108088,128404

%N Number of partitions of n having standard deviation σ > 2.

%C Regarding "standard deviation" see Comments at A238616.

%F a(n) + A238659(n) = A000041(n).

%e There are 22 partitions of 8, whose standard deviations are given by these approximations:  0., 3., 2., 2.35702, 1., 1.69967, 1.73205, 0., 1.24722, 0.942809, 1.22474, 1.2, 0.471405, 1., 0.707107, 0.8, 0.745356, 0., 0.489898, 0.471405, 0.349927, 0, so that a(8) = 2.

%t z = 50; g[n_] := g[n] = IntegerPartitions[n]; c[t_] := c[t] = Length[t]; s[t_] := s[t] = Sqrt[Sum[(t[[k]] - Mean[t])^2, {k, 1, c[t]}]/c[t]];

%t Table[Count[g[n], p_ /; s[p] < 2], {n, z}]   (*A238658*)

%t Table[Count[g[n], p_ /; s[p] <= 2], {n, z}]  (*A238659*)

%t Table[Count[g[n], p_ /; s[p] == 2], {n, z}]  (*A238660*)

%t Table[Count[g[n], p_ /; s[p] > 2], {n, z}]   (*A238661*)

%t Table[Count[g[n], p_ /; s[p] >= 2], {n, z}]  (*A238662*)

%t t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]

%t ListPlot[Sort[t[30]]] (*plot of st deviations of partitions of 30*)

%Y Cf. A238616, A238658-A238660, A238662.

%K nonn,easy

%O 1,8

%A _Clark Kimberling_, Mar 03 2014