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Partitions with difference conditions and Alder’s conjecture. (English) Zbl 1064.05021

Summary: In 1956, H. L. Alder [Research Problem 4, Bull. Am. Math. Soc. 62 (1956), p. 76] conjectured that the number of partitions of \(n\) into parts differing by at least \(d\) is greater than or equal to that of partitions of \(n\) into parts \(\pm 1\pmod {d+3}\) for \(d\geq 4\). In 1971, G. E. Andrews [Pac. J. Math. 36, 279–284 (1971; Zbl 0195.31201)] proved that the conjecture holds for \(d= 2^r-1\), \(r\geq 4\). We sketch a proof of the conjecture for all \(d\geq 32\).

MSC:

05A17 Combinatorial aspects of partitions of integers
11P81 Elementary theory of partitions

Citations:

Zbl 0195.31201
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References:

[1] BULL AM MATH SOC 62 pp 76– (1956) · doi:10.1090/S0002-9904-1956-09993-8
[2] PACIFIC J MATH 36 pp 279– (1971) · Zbl 0195.31201 · doi:10.2140/pjm.1971.36.279
[3] AM J MATH 91 pp 18– (1969) · Zbl 0186.30203 · doi:10.2307/2373264
[4] BULL AM MATH SOC 54 pp 712– (1948) · Zbl 0035.31201 · doi:10.1090/S0002-9904-1948-09062-0
[5] BULL AM MATH SOC 52 pp 538– (1946) · Zbl 0060.10101 · doi:10.1090/S0002-9904-1946-08605-X
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