A238616 -id:A238616 - cp's OEIS frontend

A238655 Number of partitions of n having standard deviation σ > 3.

0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 5, 5, 10, 15, 23, 33, 49, 68, 99, 133, 179, 246, 334, 432, 583, 756, 974, 1261, 1618, 2076, 2657, 3336, 4228, 5270, 6592, 8190, 10182, 12567, 15533, 19008, 23307, 28410, 34622, 42041, 50959, 61487, 74259, 88734, 106666, 127587

Offset: 1

Clark Kimberling, Mar 03 2014

Regarding "standard deviation" see Comments at A238616.

			There are 30 partitions of 9, whose standard deviations are given by these approximations:  0., 3.5, 2.5, 2.82843, 1.5, 2.16025, 2.16506, 0.5, 1.63299, 1.41421, 1.63936, 1.6, 1.41421, 0.816497, 1.29904, 1.08972, 1.16619, 1.11803, 0., 0.829156, 0.979796, 0.433013, 0.748331, 0.763763, 0.699854, 0.4, 0.5, 0.451754, 0.330719, 0, so that a(9) = 1.

Cf. A238616, A238661, A238656, A238657.

z = 53; 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] > 3], {n, z}]   (*A238655*)
Table[Count[g[n], p_ /; s[p] > 4], {n, z}]   (*A238656*)
Table[Count[g[n], p_ /; s[p] > 5], {n, z}]   (*A238657*)
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*)

A239141 Number of strict partitions of n having standard deviation <= 1.

Original entry on oeis.org

1, 1, 2, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2, 3, 2, 2

Offset: 1

Clark Kimberling, Mar 11 2014

Regarding standard deviation, see Comments at A238616.

			The standard deviations of the strict partitions of 9 are 0.0, 3.5, 2.5, 1.5, 2.16025, 0.5, 1.63299, 0.816497, so that a(9) = 3.

Antti Karttunen, Table of n, a(n) for n = 1..10005
Index entries for linear recurrences with constant coefficients, signature (0,0,1).

Cf. A000009, A238616, A239140, A239142, A239143.

z = 30; g[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; 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}]   (* A239140 *)
Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (* A239141 *)
Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (* periodic 01 *)
Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (* A239142 *)
Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (* A239143 *)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]
ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)
(* Peter J. C. Moses, Mar 03 2014 *)
Join[{1, 1, 2},LinearRecurrence[{0, 0, 1},{2, 2, 3},83]] (* Ray Chandler, Aug 25 2015 *)

A239141(n) = (1+(n>3)+!(n%3)); \\ Antti Karttunen, May 24 2021

a(n) + A239142(n) = A000009(n) for n >= 1.

G.f.: -(x^5 + x^4 + x^3 + 2*x^2 + x + 1)*x / ((x-1)*(x^2 + x + 1)). - Alois P. Heinz, Mar 14 2014

A239142 Number of strict partitions of n having standard deviation sigma > 1.

Original entry on oeis.org

0, 0, 0, 0, 1, 1, 3, 4, 5, 8, 10, 12, 16, 20, 24, 30, 36, 43, 52, 62, 73, 87, 102, 119, 140, 163, 189, 220, 254, 293, 338, 388, 445, 510, 583, 665, 758, 862, 979, 1111, 1258, 1423, 1608, 1814, 2045, 2302, 2588, 2907, 3262, 3656, 4094, 4580, 5118, 5715, 6376

Offset: 1

Clark Kimberling, Mar 11 2014

Regarding standard deviation, see Comments at A238616.

			The standard deviations of the strict partitions of 9 are 0., 3.5, 2.5, 1.5, 2.16025, 0.5, 1.63299, 0.816497, so that a(9) = 5.

Cf. A239140, A239141, A239142, A000009, A238616.

z = 30; g[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; 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}]   (* A239140 *)
Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (* A239141 *)
Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (* periodic 01 *)
Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (* A239142 *)
Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (* A239143 *)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]
ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)
(* Peter J. C. Moses, Mar 03 2014 *)

a(n) + A239141(n) = A000009(n) for n >=1.

G.f.: Product_{m>=1} (1+x^m) -1 +(x^5+x^4+x^3+2*x^2+x+1)*x / ((x-1)*(x^2+x+1)). - Alois P. Heinz, Mar 14 2014

A239143 Number of strict partitions of n having standard deviation σ >= 1.

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 3, 5, 5, 9, 10, 13, 16, 21, 24, 31, 36, 44, 52, 63, 73, 88, 102, 120, 140, 164, 189, 221, 254, 294, 338, 389, 445, 511, 583, 666, 758, 863, 979, 1112, 1258, 1424, 1608, 1815, 2045, 2303, 2588, 2908, 3262, 3657, 4094, 4581, 5118, 5716, 6376

Offset: 1

Clark Kimberling, Mar 11 2014

Regarding standard deviation, see Comments at A238616.

			The standard deviations of the strict partitions of 9 are 0., 3.5, 2.5, 1.5, 2.16025, 0.5, 1.63299, 0.816497, so that a(9) = 5.

Cf. A239140, A239141, A239142, A000009, A238616.

z = 30; g[n_] := Select[IntegerPartitions[n], Max[Length /@ Split@#] == 1 &]; 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}]   (* A239140 *)
Table[Count[g[n], p_ /; s[p] <= 1], {n, z}]  (* A239141 *)
Table[Count[g[n], p_ /; s[p] == 1], {n, z}]  (* periodic 01 *)
Table[Count[g[n], p_ /; s[p] > 1], {n, z}]   (* A239142 *)
Table[Count[g[n], p_ /; s[p] >= 1], {n, z}]  (* A239143 *)
t[n_] := t[n] = N[Table[s[g[n][[k]]], {k, 1, PartitionsQ[n]}]]
ListPlot[Sort[t[30]]] (*plot of st.dev's of strict partitions of 30*)
(* Peter J. C. Moses, Mar 03 2014 *)

a(n) + A239140(n) = A000009(n) for n >=1.

G.f.: Product_{m>=1} (1+x^m) -1 +(x^6+x^5+x^4+2*x^3+3*x^2+2*x+1)*x / ((x-1)*(x+1)*(x^2+x+1)). - Alois P. Heinz, Mar 14 2014

A238656 Number of partitions of n having standard deviation σ > 4.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 4, 5, 9, 14, 19, 28, 41, 57, 80, 109, 149, 199, 265, 351, 457, 599, 780, 1011, 1299, 1664, 2121, 2682, 3377, 4252, 5345, 6660, 8279, 10277, 12733, 15596, 19245, 23556, 28761, 35066, 42723, 51615, 62657, 75494, 90978

Offset: 1

Clark Kimberling, Mar 03 2014

Regarding "standard deviation" see Comments at A238616.

			There are 30 partitions of 9, whose standard deviations are given by these approximations:  0., 3.5, 2.5, 2.82843, 1.5, 2.16025, 2.16506, 0.5, 1.63299, 1.41421, 1.63936, 1.6, 1.41421, 0.816497, 1.29904, 1.08972, 1.16619, 1.11803, 0., 0.829156, 0.979796, 0.433013, 0.748331, 0.763763, 0.699854, 0.4, 0.5, 0.451754, 0.330719, 0, so that a(9) = 0.

Cf. A238616, A238661, A238655, A238657.

z = 53; 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] > 3], {n, z}]   (*A238655*)
Table[Count[g[n], p_ /; s[p] > 4], {n, z}]   (*A238656*)
Table[Count[g[n], p_ /; s[p] > 5], {n, z}]   (*A238657*)
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*)

A238657 Number of partitions of n having standard deviation σ > 5.

Original entry on oeis.org

0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 0, 1, 2, 3, 5, 9, 11, 16, 25, 34, 45, 64, 87, 121, 160, 212, 279, 369, 481, 614, 797, 1027, 1308, 1670, 2102, 2661, 3345, 4189, 5224, 6494, 8069, 9982, 12281, 15093, 18508, 22731, 27564, 33639, 40757, 49496, 59838, 72228

Offset: 1

Clark Kimberling, Mar 03 2014

Regarding "standard deviation" see Comments at A238616.

			There are 30 partitions of 9, whose standard deviations are given by these approximations:  0., 3.5, 2.5, 2.82843, 1.5, 2.16025, 2.16506, 0.5, 1.63299, 1.41421, 1.63936, 1.6, 1.41421, 0.816497, 1.29904, 1.08972, 1.16619, 1.11803, 0., 0.829156, 0.979796, 0.433013, 0.748331, 0.763763, 0.699854, 0.4, 0.5, 0.451754, 0.330719, 0, so that a(9) = 0.

Cf. A238616, A238661, A238655, A238657.

z = 53; 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] > 3], {n, z}]   (*A238655*)
Table[Count[g[n], p_ /; s[p] > 4], {n, z}]   (*A238656*)
Table[Count[g[n], p_ /; s[p] > 5], {n, z}]   (*A238657*)
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*)

A239223 Number T(n,k) of partitions of n 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, 2, 3, 4, 1, 6, 1, 8, 2, 1, 10, 4, 1, 12, 7, 2, 1, 15, 10, 4, 1, 19, 14, 6, 2, 1, 23, 21, 7, 4, 1, 25, 32, 14, 3, 2, 1, 33, 39, 19, 6, 3, 1, 41, 51, 27, 10, 3, 2, 1, 44, 70, 39, 13, 7, 2, 1, 51, 92, 52, 21, 9, 3, 2, 1, 58, 121, 69, 30, 10, 6, 2, 1, 67, 149

Offset: 1

Alois P. Heinz, Mar 12 2014

			Triangle T(n,k) begins:
   1;
   2;
   3;
   4,  1;
   6,  1;
   8,  2,  1;
  10,  4,  1;
  12,  7,  2, 1;
  15, 10,  4, 1;
  19, 14,  6, 2, 1;
  23, 21,  7, 4, 1;
  25, 32, 14, 3, 2, 1;

Alois P. Heinz, Rows n = 1..65, flattened
Wikipedia, Standard deviation

Column k=0 gives A238616.
Row sums give A000041.
Maximal index in row n is A140106(n).
Cf. A239228.

b:= proc(n, i, m, s, c) `if`(n=0, x^floor(sqrt(s/c-(m/c)^2)),
      `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:
T:= n->(p->seq(coeff(p, x, i), i=0..degree(p)))(b(n$2, 0$3)):
seq(T(n), n=1..18);

b[n_, i_, m_, s_, c_] := b[n, i, m, s, c] = If[n==0, x^Floor[Sqrt[s/c - (m/c)^2]], 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}]]]; T[n_] := Function[p, Table[Coefficient[p, x, i], {i, 0, Exponent[p, x]}]][b[n, n, 0, 0, 0]]; Table[T[n], {n, 1, 18}] // Flatten (* Jean-François Alcover, Nov 17 2015, translated from Maple *)

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A238655 Number of partitions of n having standard deviation σ > 3.

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A239141 Number of strict partitions of n having standard deviation <= 1.

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A239142 Number of strict partitions of n having standard deviation sigma > 1.

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A239143 Number of strict partitions of n having standard deviation σ >= 1.

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A238656 Number of partitions of n having standard deviation σ > 4.

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A238657 Number of partitions of n having standard deviation σ > 5.

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A239223 Number T(n,k) of partitions of n 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.

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