A238620 Number of partitions of n having population standard deviation >= 1.

0, 0, 0, 1, 1, 3, 5, 10, 15, 23, 33, 52, 68, 94, 132, 180, 239, 318, 412, 543, 693, 885, 1131, 1443, 1803, 2250, 2808, 3499, 4321, 5336, 6552, 8032, 9799, 11914, 14456, 17528, 21136, 25458, 30588, 36699, 43869, 52422, 62437, 74290, 88186, 104527, 123670

Offset

1,6

Comments

Regarding "standard deviation" see Comments at A238616.

Links

Table of n, a(n) for n=1..47.

Formula

a(n) = A000041(n) - A238616(n).

Example

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) = 3.

Maple

b:= proc(n, i, m, s, c) `if`(n=0, `if`(s/c-(m/c)^2>=1, 1, 0),
      `if`(i=1, b(0$2, m+n, s+n, c+n), add(b(n-i*j, i-1,
       m+i*j, s+i^2*j, c+j), j=0..n/i)))
    end:
a:= n-> b(n$2, 0$3):
seq(a(n), n=1..50);  # Alois P. Heinz, Mar 11 2014

Mathematica

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]]
Table[Count[g[n], p_ /; s[p] < 1], {n, z}]   (*A238616*)
Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (*A238617*)
Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (*A238618*)
Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (*A238619*)
Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (*A238620*)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
ListPlot[Sort[t[30]]] (*plot of st.dev's of partitions of 30*)
(* Second program: *)
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];
Array[a, 50] (* Jean-François Alcover, Jun 03 2021, after Alois P. Heinz *)

Crossrefs

Cf. A238616.

Sequence in context: A190557 A023602 A024452 * A074968 A308826 A090491

Adjacent sequences: A238617 A238618 A238619 * A238621 A238622 A238623

Keyword

nonn,easy

Author

Clark Kimberling, Mar 01 2014

Status

approved