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# Partition Function b_k

The number of partitions of in which no parts are multiples of is sometimes denoted (Gordon and Ono 1997). is also the number of partitions of into at most copies of each part.

There is a special case

 (1)

where is the partition function Q, and is the number of irreducible -modular representations of the symmetric group . The generating function for is given by

 (2) (3)

where is a q-Pochhammer symbol.

The following table gives the first few values of for small .

 OEIS 2 A000009 1, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, ... 3 A000726 1, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, ... 4 A001935 1, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, ... 5 A035959 1, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, ...

Gordon and Ono (1997) show that

 (4) (5) (6)

Defining as the number of positive integers for which , Gordon and Ono (1997) proved that if , then

 (7)

for all , where .

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## References

Andrews, G. E. The Theory of Partitions. Cambridge, England: Cambridge University Press, p. 109, 1998.Carlitz, L. "Generating Functions and Partition Problems." In Theory of Numbers (Ed. A. L. Whiteman). Providence, RI: Amer. Math. Soc., pp. 144-169, 1965.Cayley, A. "A Memoir on the Transformation of Elliptic Functions." Collected Mathematical Papers, Vol. 9. London: Cambridge University Press, p. 128, 1889-1897.Honsberger, R. Mathematical Gems III. Washington, DC: Math. Assoc. Amer., p. 241, 1985.Gordon, B. and Ono, K. "Divisibility of Certain Partition Functions By Powers of Primes." Ramanujan J. 1, 25-34, 1997.Sloane, N. J. A. Sequences A000009/M0281, A000726/M0316, A001935/M0566, and A035959 in "The On-Line Encyclopedia of Integer Sequences."

## Cite this as:

Weisstein, Eric W. "Partition Function b_k." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/PartitionFunctionb.html