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)
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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)
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(3)
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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)
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(5)
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(6)
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Defining as the number of positive integers for which , Gordon and Ono (1997) proved that if , then
(7)
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for all , where .