| 释义 |
Partition Function P gives the number of ways of writing the Integer  as a sum of Positive Integers without regard to order. For example, since 4 can be written
so  . satisfies
 | (2) |
 , 2, ..., are 1, 2, 3, 5, 7, 11, 15, 22, 30, 42, ...(Sloane's A000041). The following table gives the value of for selected small . | | | 50 | 204226 | | 100 | 190569292 | | 200 | 3972999029388 | | 300 | 9253082936723602 | | 400 | 6727090051741041926 | | 500 | 2300165032574323995027 | | 600 | 458004788008144308553622 | | 700 | 60378285202834474611028659 | | 800 | 5733052172321422504456911979 | | 900 | 415873681190459054784114365430 | | 1000 | 24061467864032622473692149727991 |
for which is A046063). Numbers whichcannot be written as a Product of are 13, 17, 19, 23, 26, 29, 31, 34, 37, 38, 39, ... (Sloane's A046064),corresponding to numbers of nonisomorphic Abelian Groups which are not possible for any group order.
When explicitly listing the partitions of a number , the simplest form is the so-called natural representationwhich simply gives the sequence of numbers in the representation (e.g., (2, 1, 1) for the number  ). The multiplicity representation instead gives the number of times each number occurs together with that number (e.g., (2, 1),(1, 2) for  ). The Ferrers Diagram is a pictorial representation of a partition.
Euler  invented a Generating Function which gives rise to a Power Series in ,
 | (3) |
 | (4) |
 is the Euler also showed that, for
where the exponents are generalized Pentagonal Numbers 0, 1, 2, 5, 7, 12, 15, 22, 26, 35, ...(Sloane's A001318) and the sign of the  th term (counting 0 as the 0th term) is  (with  theFloor Function), the partition numbers are given by the Generating Function
 | (7) |
 | (8) |
 and the sign of the th term is, as above.
In 1916-1917, Hardy and Ramanujan used the Circle Method and elliptic Modular Functions to obtain the approximate solution
 | (9) |
 | (10) |
is the Floor Function, and  runs through the Integers less than and RelativelyPrime to  (when  ,  ). The remainder after  terms is
 | (17) |
 and  are fixed constants.
With  as defined above, Ramanujan also showed that
 | (18) |
 | (19) |
 | (20) |
 | (21) |
Let  be the number of partitions of containing Odd numbers only and  be the number of partitions of without duplication, then  | |  | (22) |
as discovered by Euler (Honsberger 1985). The first few values of  are 1, 1, 1, 2, 2, 3, 4, 5, 6, 8,10, ... (Sloane's A000009).
Let  be the number of partitions of Even numbers only, and let  ( ) be the number of partitionsin which the parts are all Even (Odd) and all different. The first few values of are 1, 1, 0, 1, 1, 1, 1, 1,2, 2, 2, 2, 3, 3, 3, 4, ... (Sloane's A000700). Some additional Generating Functions are givenby Honsberger (1985, pp. 241-242) Some additional interesting theorems following from these (Honsberger 1985, pp. 64-68 and 143-146) are:- 1. The number of partitions of in which no Even part is repeated is the same as the number of partitionsof in which no part occurs more than three times and also the same as the number of partitions in which no part isdivisible by four.
- 2. The number of partitions of in which no part occurs more often than
 times is the same as the number ofpartitions in which no term is a multiple of  . - 3. The number of partitions of in which each part appears either 2, 3, or 5 times is the same as the number of partitions in which each part is Congruent mod 12 to either 2, 3, 6, 9, or 10.
- 4. The number of partitions of in which no part appears exactly once is the same as the number of partitionsof in which no part is Congruent to 1 or 5 mod 6.
- 5. The number of partitions in which the parts are all Even and different is equal to the absolute differenceof the number of partitions with Odd and Even parts.
 , also written  , is the number of ways of writing as a sum of terms, and can be computed from the Recurrence Relation
 | (29) |
The function can be given explicitly for the first few values of ,
where is the Floor Function and  is the Nint function (Honsberger 1985, pp. 40-45). See also Alcuin's Sequence, Elder's Theorem, Euler's Pentagonal Number Theorem, Ferrers Diagram,Partition Function Q, Pentagonal Number, r(n), Rogers-Ramanujan Identities,Stanley's Theorem
References
Abramowitz, M. and Stegun, C. A. (Eds.). ``Unrestricted Partitions.'' §24.2.1 in Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables, 9th printing. New York: Dover, p. 825, 1972.Adler, H. ``Partition Identities--From Euler to the Present.'' Amer. Math. Monthly 76, 733-746, 1969. Adler, H. ``The Use of Generating Functions to Discover and Prove Partition Identities.'' Two-Year College Math. J. 10, 318-329, 1979. Andrews, G. Encyclopedia of Mathematics and Its Applications, Vol. 2: The Theory of Partitions. Cambridge, England: Cambridge University Press, 1984. Berndt, B. C. Ramanujan's Notebooks, Part IV. New York: Springer-Verlag, 1994. Conway, J. H. and Guy, R. K. The Book of Numbers. New York: Springer-Verlag, pp. 94-96, 1996. Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., pp. 40-45 and 64-68, 1985. Honsberger, R. More Mathematical Morsels. Washington, DC: Math. Assoc. Amer., pp. 237-239, 1991. Jackson, D. and Goulden, I. Combinatorial Enumeration. New York: Academic Press, 1983. MacMahon, P. A. Combinatory Analysis. New York: Chelsea, 1960. Rademacher, H. ``On the Partition Function  .'' Proc. London Math. Soc. 43, 241-254, 1937. Ruskey, F. ``Information of Numerical Partitions.'' http://sue.csc.uvic.ca/~cos/inf/nump/NumPartition.html. Sloane, N. J. A. Sequences A000009/M0281, A000041/M0663, A000700/M0217, A001318/M1336, A046063, and A046064 in ``An On-Line Version of the Encyclopedia of Integer Sequences.''http://www.research.att.com/~njas/sequences/eisonline.html.
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