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A238618 Number of partitions of n having standard deviation σ = 1.

%I A238618 #23 Nov 16 2015 14:58:21
%S A238618 0,0,0,1,0,1,0,2,0,1,0,5,0,1,0,4,0,4,0,7,0,1,0,16,0,1,0,10,0,11,0,10,
%T A238618 0,1,0,26,0,1,0,26,0,24,0,13,0,1,0,60,0,5,0,17,0,19,0,52,0,1,0,117,0,
%U A238618 1,0,36,0,46,0,23,0,29,0,160,0,1,0,30,0,61,0,140
%N A238618 Number of partitions of n having standard deviation σ = 1.
%C A238618 Regarding "standard deviation" see Comments at A238616.
%e A238618 There are 11 partitions of 6, whose standard deviations are given by these approximations:  0., 2., 1., 1.41421, 0., 0.816497, 0.866025, 0., 0.5, 0.4, 0, so that a(6) = 1.
%p A238618 b:= proc(n, i, m, s, c) `if`(n=0, `if`(s/c-(m/c)^2=1, 1, 0),
%p A238618       `if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,
%p A238618        m+i*j, s+i^2*j, c+j), j=0..n/i)))
%p A238618     end:
%p A238618 a:= n-> b(n$2, 0$3):
%p A238618 seq(a(n), n=1..50);  # _Alois P. Heinz_, Mar 11 2014
%t A238618 z = 55; g[n_] := g[n] = IntegerPartitions[n]; s[t_] := s[t] = Sqrt[Sum[(t[[k]] - Mean[t])^2, {k, 1, Length[t]}]/Length[t]]
%t A238618 Table[Count[g[n], p_ /; s[p] < 1], {n, z}]   (*A238616*)
%t A238618 Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (*A238617*)
%t A238618 Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (*A238618*)
%t A238618 Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (*A238619*)
%t A238618 Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (*A238620*)
%t A238618 t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
%t A238618 ListPlot[Sort[t[30]]] (*plot of st.dev's of partitions of 30*)
%t A238618 b[n_, i_, m_, s_, c_] := b[n, i, m, s, c] = If[n == 0, If[s/c - (m/c)^2 == 1, 1, 0], If[i == 1, b[0, 0, m + n, s + n, c + n], Sum[b[n - i*j, i - 1, m + i*j, s + i^2*j, c + j], {j, 0, n/i}]]]; a[n_] := b[n, n, 0, 0, 0]; Table[a[n], {n, 1, 50}] (* _Jean-François Alcover_, Nov 16 2015, after _Alois P. Heinz_ *)
%Y A238618 Cf. A238616.
%K A238618 nonn,easy
%O A238618 1,8
%A A238618 _Clark Kimberling_, Mar 01 2014
%E A238618 a(56)-a(80) from _Alois P. Heinz_, Mar 11 2014