Affiliation: Pennsylvania State University
Email: yee@math.psu.edu
Title Of Talk: Partitions with difference conditions and Alder's conjecture
URL: http://www.math.psu.edu/yee/research.html
Abstract: The well-known Rogers-Ramanujan identities may be stated partition-theoretically as follows. If $c=1$ or $2$, then the number of partitions of $n$ into parts $\equiv \pm c \pmod{5}$ equals the number of partitions of $n$ into parts $\ge c$ with minimal difference 2 between parts. In 1926, Schur proved that the number of partitions of $n$ with minimal difference $2$ between parts and no consecutive multiples of $3$ equals the number of partitions into parts $\equiv \pm 1 \pmod{6}$, and Bressoud proved the Schur theorem by establishing a bijection in 1981. In 1956, Alder conjectured that the number of partitions into parts differing by at least $d$ is greater than or equal to the number of partitions of $n$ into parts $\equiv \pm 1 \pmod{d+3}$. Euler's identity, the first Rogers-Ramanujan identity, and the Schur's theorem show that the conjecture is true for $d=1,2,3$, respectively. In 1971, Andrews proved the conjecture holds for $d=2^r-1, r\ge 4$. In this talk, we will discuss the conjecture for $d\ge 32$.
Last update made Tue Nov 16 19:46:23 EST 2004.
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