A239223 -id:A239223 - cp's OEIS frontend

A238616 Number of partitions of n having standard deviation σ < 1.

1, 2, 3, 4, 6, 8, 10, 12, 15, 19, 23, 25, 33, 41, 44, 51, 58, 67, 78, 84, 99, 117, 124, 132, 155, 186, 202, 219, 244, 268, 290, 317, 344, 396, 427, 449, 501, 557, 597, 639, 714, 752, 824, 885, 948, 1031, 1084, 1185, 1308, 1390, 1452, 1589, 1692, 1788, 1919

Offset: 1

Clark Kimberling, Mar 01 2014

Here, "standard deviation" means "population standard deviation" (denoted by σ), not "sample standard deviation" (denoted by s); σ is the square root of variance, so that σ of a list t = (t(k)), such as the partitions of a positive integer, is given by the formula sqrt((Sum_{k=1..#t} (t(k) - mean(t))^2)/(#t)), where #t is the number of terms in t(k). (The distinction between σ and s is discussed in most probability and statistics textbooks. The command "StandardDeviation" in Mathematica gives s, not σ.)

			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) = 8.

Cf. A238617-A238620, A238655-A238662.

Column k=0 of A239223.

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..55);  # Alois P. Heinz, Mar 12 2014

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]; Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Nov 20 2015, after Alois P. Heinz *)
(* The interest of this 3rd program is just to show how Mathematica's StandardDeviation can be used, with a correction factor, to compute sigma, the population standard deviation. *)
sigma[t_] := If[Length[t] == 1, 0, StandardDeviation[t]*Sqrt[(Length[t]-1)/ Length[t]]];
a[n_] := Count[IntegerPartitions[n], p_ /; sigma[p] < 1];
Array[a, 30] (* Jean-François Alcover, May 28 2021 *)

a(n) = A000041(n) - A238620(n).

A239228 Number T(n,k) of partitions of n into distinct parts with standard deviation σ in the half-open interval [k,k+1); triangle T(n,k), n>=1, 0<=k<=max(0,floor(n/2)-1), read by rows.

Original entry on oeis.org

1, 1, 2, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 3, 1, 1, 3, 2, 2, 1, 1, 4, 3, 1, 1, 2, 4, 3, 2, 1, 2, 4, 5, 2, 1, 1, 2, 5, 6, 2, 2, 1, 1, 5, 8, 4, 2, 1, 1, 3, 5, 9, 5, 3, 1, 1, 1, 7, 9, 7, 4, 2, 1, 1, 2, 6, 12, 9, 4, 3, 1, 1, 2, 5, 15, 11, 6, 3, 2, 1, 1, 2, 6, 16

Offset: 1

Alois P. Heinz, Mar 12 2014

			Triangle T(n,k) begins:
  1;
  1;
  2;
  1, 1;
  2, 1;
  2, 1, 1;
  2, 2, 1;
  1, 3, 1, 1;
  3, 2, 2, 1;
  1, 4, 3, 1, 1;
  2, 4, 3, 2, 1;
  2, 4, 5, 2, 1, 1;

Alois P. Heinz, Rows n = 1..100, flattened
Wikipedia, Standard deviation

Column k=0 gives A239140.
Row sums give A000009.
Maximal index in row n is A140106(n).
Cf. A239223.

b:= proc(n, i, m, s, c) `if`(n>i*(i+1)/2, 0, `if`(n=0,
      x^floor(sqrt(s/c-(m/c)^2)), b(n, i-1, m, s, c)+
      `if`(i>n, 0, b(n-i, i-1, m+i, s+i^2, c+1))))
    end:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$3)):
seq(T(n), n=1..20);

b[n_, i_, m_, s_, c_] := If[n > i*(i + 1)/2, 0, If[n == 0, x^Floor[Sqrt[ s/c - (m/c)^2]], b[n, i - 1, m, s, c] + If[i > n, 0, b[n - i, i - 1, m + i, s + i^2, c + 1]]]];
T[n_] := Table[Coefficient[#, x, i], {i, 0, Exponent[#, x]}]&[b[n, n, 0, 0, 0]];
Table[T[n], {n, 1, 20}] // Flatten (* Jean-François Alcover, May 22 2018, translated from Maple *)

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A238616 Number of partitions of n having standard deviation σ < 1.

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