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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 $±1\phantom{\rule{4.44443pt}{0ex}}\left(mod\phantom{\rule{0.277778em}{0ex}}d+3\right)$ for $d\ge 4$. In 1971, G. E. Andrews [Pac. J. Math. 36, 279–284 (1971; Zbl 0195.31201)] proved that the conjecture holds for $d={2}^{r}-1$, $r\ge 4$. We sketch a proof of the conjecture for all $d\ge 32$.

MSC:
 05A17 Partitions of integers (combinatorics) 11P81 Elementary theory of partitions