A239228 -id:A239228 - cp's OEIS frontend

A239140 Number of strict partitions of n having standard deviation σ < 1.

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

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) = 3.

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

Cf. A000009, A083039, A238616, A239141, A239142, A239143.

Column k=0 of A239228.

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[{-1, 0, 1, 1},{1, 2, 2, 2},83]] (* Ray Chandler, Aug 25 2015 *)

A083039(n) = (1+!(n%2)+!(n%3));
A239140(n) = if(n<=3,1+(3==n),A083039(n-3)); \\ Antti Karttunen, May 24 2021

a(n + 3) = A083039(n) for n >= 1 (periodic with period 6); a(n) + A239143(n) = A000009(n) for n >=1.

G.f.: -(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

A-number in the first formula corrected by Antti Karttunen, May 24 2021

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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A239140 Number of strict partitions of n having standard deviation σ < 1.

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