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Encyclopedia Britannica



EA(nar...), (p.s,.,.)

This article appears in Volume V06, Page 753 of the Encyclopedia Britannica.

Encyclopedia Britannica - Main :: DRO-ECG
EA(nar...), (p.s,.,.) . (pqr...),
where the numbers pi, qi, 1.1 . . . are fixed and assumed to be in descending
order
  of magnitude, the summation being for every
partition
  (pqr . . . ) of the number n, is defined to be the
distribution
 
function
  of the
objects
  defined by (pqr . . . ) into the parcels defined by ( M i n.. . ). It gives a
complete
  enumeration of n
objects
  of whatever
species
  into parcels of the given
species
 .
1. One-to-One
Distribution
 . Parcels m in number (i.e. m = n).
Let h, be the homogeneous product-sum of degree s of
the quantities a, ,B, y, . . . so that
(1 ax. 113x. 1 --x. ...)1=1 +hix -f-h2x2+1a3x3 +...
hi = Ea = (1)
h2 = Ea2+Ea'a = (2) +(12)
h3 =Ea3+Ea2f+Eafly = (3) +(21) +(13). Form the product hpinglhri .. .
Any
term
  in he, may be regarded as derived from pi objects distributed into pi similar parcels, one
object
  in each
parcel
 , since the
order
  of occurrence of the letters a, 0, y, . in any
term
  is immaterial. Moreover, every selection of pi letters from the letters in a5/39y' .. . will occur in some term of hy1, every further selection of q1 letters will occur in some term of h51, and so on.
Therefore in the product h51h51hrl ... the term al'i32y' . . ., and there-fore also the symmetric
function
  (pgr ... ), will occur as many times as it is possible to distribute objects defined by (pqr .) into parcels defined by (Awl ...) one
object
  in each
parcel
 . Hence


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