Dyson's crank of a partition

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Dyson's Crank of a Partition

holds. He was thus led to conjecture the existence of some other partition statistic (which he called the crank); this unknown statistic should provide a combinatorial interpretation of ^-p(lln + 6) in the same way that (1.1) and (1.2) treat the primes 5 and 7. In [4, 5], one of us was able to find a crank relative to vector partitions as follows: For a partition 7r, let #(7r) be the number of ...

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What is the crank of a partition?

The crank of an integer partition is an integer statistic which yields a combinatorial explanation of Ramanujan’s three famous congruences for the partition function. The existence of such a statistic was conjectured by Freeman Dyson in 1944 and realized 44 years later by George Andrews and Frank Gravan. Since then, groundbreaking work by Ken Ono, Karl Mahlburg, and others tells us that not onl...

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Partition congruences and the Andrews-Garvan-Dyson crank.

In 1944, Freeman Dyson conjectured the existence of a "crank" function for partitions that would provide a combinatorial proof of Ramanujan's congruence modulo 11. Forty years later, Andrews and Garvan successfully found such a function and proved the celebrated result that the crank simultaneously "explains" the three Ramanujan congruences modulo 5, 7, and 11. This note announces the proof of ...

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A Crank Analog on a Certain Kind of Partition Function Arising from the Cubic Continued Fraction

In a series of papers, H.-C. Chan has studied congruence properties of a certain kind of partition function that arises from Ramanujan’s cubic continued fraction. This partition function a(n), is defined by ∑∞ n=0 a(n)q n = 1 (q;q)∞(q2;q2)∞ . In particular, he proved that a(3n + 2) ≡ 0 (mod 3). As Chan mentioned in his paper, it is natural to ask if there exists an analog of the rank or the cra...

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Influence of crank length and crank-axle height on standing arm-crank (grinding) power.

OBJECTIVE To determine the optimal crank length and crank-axle height for maximum power production during standing arm cranking ("grinding"). METHODS Nine elite professional America's Cup grinders (age = 36 +/- 2 yr, body mass = 104 +/- 1 kg, body fat = 13% +/- 2%) performed eight maximal 6-s sprints on an adjustable standing arm-crank ergometer fitted with an SRM power crank. The protocol in...

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ژورنال

عنوان ژورنال: Bulletin of the American Mathematical Society

سال: 1988

ISSN: 0273-0979

DOI: 10.1090/s0273-0979-1988-15637-6