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Partition Function b_k


The number of partitions of n in which no parts are multiples of k is sometimes denoted b_k(n) (Gordon and Ono 1997). b_k(n) is also the number of partitions of n into at most k-1 copies of each part.

There is a special case

 b_2(n)=Q(n),
(1)

where Q(n) is the partition function Q, and b_p(n) is the number of irreducible p-modular representations of the symmetric group S_n. The generating function for b_k(n) is given by

sum_(n=0)^(infty)b_k(n)x^n=product_(n=1)^(infty)(1-x^(kn))/(1-x^n)
(2)
=((x^k)_infty)/((x)_infty),
(3)

where (q)_k is a q-Pochhammer symbol.

The following table gives the first few values of b_k(n) for small k.

kOEISb_k(n)
2A0000091, 1, 2, 2, 3, 4, 5, 6, 8, 10, 12, 15, 18, 22, ...
3A0007261, 2, 2, 4, 5, 7, 9, 13, 16, 22, 27, 36, 44, 57, ...
4A0019351, 2, 3, 4, 6, 9, 12, 16, 22, 29, 38, 50, 64, 82, ...
5A0359591, 2, 3, 5, 6, 10, 13, 19, 25, 34, 44, 60, 76, 100, ...

Gordon and Ono (1997) show that

b_5(5n+4)=0 (mod 5)
(4)
b_7(7n+5)=0 (mod 7)
(5)
b_(11)(11n+6)=0 (mod 11).
(6)

Defining S_k(N;M) as the number of positive integers n<=N for which b_k(n)=0 (mod M), Gordon and Ono (1997) proved that if p_i^(a_i)>=sqrt(k), then

 lim_(N->infty)(S_k(N;p_i^j))/N=1
(7)

for all j, where k=p_1^(a_1)p_2^(a_2)...p_m^(a_m).


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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

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