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EA(nar...), (p.s„,.,.)

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Originally appearing in Volume V06, Page 753 of the 1911 Encyclopedia Britannica.
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EA(nar...), (p.s„,.,.) . (pqr...), where the See also:numbers pi, qi, 1.1 . . . are fixed and assumed to be in descending See also:order of magnitude, the summation being for every See also:partition (pqr . . . ) of the number n, is defined to be the See also:distribution See also:function of the See also:objects defined by (pqr . . . ) into the parcels defined by ( M i n.. . ). It gives a See also:complete enumeration of n objects of whatever See also: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. 1—13x. 1 --x. ...)—1=1 +hix -f-h2x2+1a3x3 +... hi = Ea = (1) h2 = Ea2+Ea'a = (2) +(12) h3 =Ea3+Ea2f+Eafly = (3) +(21) +(13). See also:

Form the product hpinglhri .. . Any See also:term in he, may be regarded as derived from pi objects distributed into pi similar parcels, one See also:object in each See also: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 (See also:

Awl ...) one object in each parcel.

End of Article: EA(nar...), (p.s„,.,.)

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