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Some general problems on the number of parts in partitions

100%

Richmond L.

Acta Arithmetica

1994

tom 66

nr 4

297-313

Growth of the product $∏^n_{j=1} (1-x^{a_j})$

51%

Bell J., Borwein P., Richmond L.

Acta Arithmetica

1998

tom 86

nr 2

155-170

We estimate the maximum of $∏^n_{j=1} |1 - x^{a_j}|$ on the unit circle where 1 ≤ a₁ ≤ a₂ ≤ ... is a sequence of integers. We show that when $a_j$ is $j^k$ or when $a_j$ is a quadratic in j that takes on positive integer values, the maximum grows as exp(cn), where c is a positive constant. This complements results of Sudler and Wright that show exponential growth when $a_j$ is j. In contrast we show, under fairly general conditions, that the maximum is less than $2^n/n^r$, where r is an arbitrary positive number. One consequence is that the number of partitions of m with an even number of parts chosen from $a₁,...,a_n$ is asymptotically equal to the number of such partitions with an odd number of parts when $a_i$ satisfies these general conditions.

Ograniczanie wyników

2 Acta Arithmetica

2 Richmond L.

1 Bell J.

1 Borwein P.

1 1998

1 1994

Wyniki wyszukiwania

Some general problems on the number of parts in partitions

Growth of the product $∏^n_{j=1} (1-x^{a_j})$