login
This site is supported by donations to The OEIS Foundation.

 

Logo


Hints
(Greetings from The On-Line Encyclopedia of Integer Sequences!)
A238616 Number of partitions of n having standard deviation σ < 1. 18
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 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,2

COMMENTS

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[(t(k) - mean(t))^2: k = 1..#t)/(#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.)

LINKS

Table of n, a(n) for n=1..55.

FORMULA

a(n) + A238620(n) = A000041(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) = 8.

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..55);  # Alois P. Heinz, Mar 12 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]; Table[a[n], {n, 1, 55}] (* Jean-François Alcover, Nov 20 2015, after Alois P. Heinz *)

CROSSREFS

Cf. A238617-A238620, A238655-A238662.

Column k=0 of A239223.

Sequence in context: A001972 A005705 A139542 * A093717 A279029 A002093

Adjacent sequences:  A238613 A238614 A238615 * A238617 A238618 A238619

KEYWORD

nonn,easy

AUTHOR

Clark Kimberling, Mar 01 2014

STATUS

approved

Lookup | Welcome | Wiki | Register | Music | Plot 2 | Demos | Index | Browse | More | WebCam
Contribute new seq. or comment | Format | Style Sheet | Transforms | Superseeker | Recent | More pages
The OEIS Community | Maintained by The OEIS Foundation Inc.

License Agreements, Terms of Use, Privacy Policy .

Last modified June 28 03:21 EDT 2017. Contains 288813 sequences.