cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-4 of 4 results.

A238662 Number of partitions of n having population standard deviation >= 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 1, 3, 5, 9, 12, 20, 29, 43, 62, 88, 118, 169, 223, 306, 403, 532, 693, 907, 1160, 1490, 1910, 2423, 3044, 3845, 4783, 5957, 7401, 9104, 11209, 13805, 16806, 20449, 24920, 30223, 36494, 44022, 52880, 63511, 76003, 90631, 108088, 128708
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2014

Keywords

Comments

Regarding "population standard deviation" see Comments at A238616.

Examples

			There are 22 partitions of 8, whose population 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) = 3.

Crossrefs

Cf. A238616, A238658, A238660, A238661.

Programs

  • Maple
    b:= proc(n, i, m, s, c) `if`(n=0, `if`(s/c-(m/c)^2>=4, 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 = 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]];
Table[Count[g[n], p_ /; s[p] < 2], {n, z}]   (*A238658*)
Table[Count[g[n], p_ /; s[p] <= 2], {n, z}]  (*A238659*)
Table[Count[g[n], p_ /; s[p] == 2], {n, z}]  (*A238660*)
Table[Count[g[n], p_ /; s[p] > 2], {n, z}]   (*A238661*)
Table[Count[g[n], p_ /; s[p] >= 2], {n, z}]  (*A238662*)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
ListPlot[Sort[t[30]]] (* plot of st deviations 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 >= 4, 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, May 27 2021, after Alois P. Heinz *)

Formula

a(n) + A238658(n) = A000041(n).

A238658 Number of partitions of n having population standard deviation < 2.

Original entry on oeis.org

1, 2, 3, 5, 7, 10, 14, 19, 25, 33, 44, 57, 72, 92, 114, 143, 179, 216, 267, 321, 389, 470, 562, 668, 798, 946, 1100, 1295, 1521, 1759, 2059, 2392, 2742, 3206, 3674, 4172, 4831, 5566, 6265, 7115, 8089, 9152, 10381, 11664, 13131, 14927, 16666, 18565, 20977
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2014

Keywords

Examples

			There are 22 partitions of 8, whose population 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) = 19.

Crossrefs

Cf. A000041, A238616, A238659, A238660, A238661, A238662.

Programs

  • Mathematica
    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]];
Table[Count[g[n], p_ /; s[p] < 2], {n, z}]   (* A238658 *)
Table[Count[g[n], p_ /; s[p] <= 2], {n, z}]  (* A238659 *)
Table[Count[g[n], p_ /; s[p] == 2], {n, z}]  (* A238660 *)
Table[Count[g[n], p_ /; s[p] > 2], {n, z}]   (* A238661 *)
Table[Count[g[n], p_ /; s[p] >= 2], {n, z}]  (* A238662 *)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
ListPlot[Sort[t[30]]] (* plot of st deviations 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 >= 4, 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_] := PartitionsP[n] - b[n, n, 0, 0, 0];
Array[a, 50] (* Jean-François Alcover, May 27 2021, after Alois P. Heinz *)

Formula

a(n) + A238662(n) = A000041(n).

A238660 Number of partitions of n having population standard deviation = 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 1, 0, 1, 0, 2, 0, 2, 0, 1, 1, 3, 0, 5, 0, 7, 4, 2, 0, 19, 3, 2, 9, 20, 0, 38, 0, 22, 33, 7, 12, 84, 0, 8, 52, 90, 0, 127, 0, 87, 103, 22, 0, 304, 9, 74, 131, 153, 0, 214, 139, 390, 192, 59, 0, 1219, 0, 73, 460, 372, 383, 908, 0, 501, 439, 832, 0
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2014

Keywords

Comments

Regarding "standard deviation" see Comments at A238616.

Examples

			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) = 1.

Crossrefs

Cf. A238616, A238658-A238660, A238662.

Programs

  • Maple
    b:= proc(n, i, m, s, c) `if`(n=0, `if`(s/c-(m/c)^2=4, 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 = 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]];
Table[Count[g[n], p_ /; s[p] < 2], {n, z}]   (*A238658*)
Table[Count[g[n], p_ /; s[p] <= 2], {n, z}]  (*A238659*)
Table[Count[g[n], p_ /; s[p] == 2], {n, z}]  (*A238660*)
Table[Count[g[n], p_ /; s[p] > 2], {n, z}]   (*A238661*)
Table[Count[g[n], p_ /; s[p] >= 2], {n, z}]  (*A238662*)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
ListPlot[Sort[t[30]]] (*plot of st deviations 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 == 4, 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 *)

Extensions

a(51)-a(71) from Alois P. Heinz, Mar 11 2014

A238661 Number of partitions of n having standard deviation σ > 2.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 1, 2, 5, 7, 12, 18, 29, 42, 61, 85, 118, 164, 223, 299, 399, 530, 693, 888, 1157, 1488, 1901, 2403, 3044, 3807, 4783, 5935, 7368, 9097, 11197, 13721, 16806, 20441, 24868, 30133, 36494, 43895, 52880, 63424, 75900, 90609, 108088, 128404
Offset: 1

Views

Author

Clark Kimberling, Mar 03 2014

Keywords

Comments

Regarding "standard deviation" see Comments at A238616.

Examples

			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.

Crossrefs

Cf. A238616, A238658-A238660, A238662.

Programs

  • Mathematica
    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]];
Table[Count[g[n], p_ /; s[p] < 2], {n, z}]   (*A238658*)
Table[Count[g[n], p_ /; s[p] <= 2], {n, z}]  (*A238659*)
Table[Count[g[n], p_ /; s[p] == 2], {n, z}]  (*A238660*)
Table[Count[g[n], p_ /; s[p] > 2], {n, z}]   (*A238661*)
Table[Count[g[n], p_ /; s[p] >= 2], {n, z}]  (*A238662*)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsP[n]}]]
ListPlot[Sort[t[30]]] (*plot of st deviations of partitions of 30*)

Formula

a(n) + A238659(n) = A000041(n).
Showing 1-4 of 4 results.

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