Dyson’s Crank Distribution Conjecture

نویسنده

  • DANIEL PARRY
چکیده

Bringmann and Dousse recently established a conjecture of Dyson dealing with the limiting asymptotics of the Andrews-Garvan crank statistic for integer partitions. A direct “sieving” technique is used to establish this conjecture and establish the range of validity. Unlike the approach of Bringmann and Dousse, the technique readily yields the analogous result for Dyson’s partition rank and all of Garvan’s k-rank statistics.

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

منابع مشابه

On Dyson’s Crank Conjecture and the Uniform Asymptotic Behavior of Certain Inverse Theta Functions

In this paper we prove a longstanding conjecture by Freeman Dyson concerning the limiting shape of the crank generating function. We fit this function in a more general family of inverse theta functions which play a key role in physics.

متن کامل

Some Observations on Dyson's New Symmetries of Partitions

We utilize Dyson’s concept of the adjoint of a partition to derive an infinite family of new polynomial analogues of Euler’s Pentagonal Number Theorem. We streamline Dyson’s bijection relating partitions with crank ≤ k and those with k in the Rank-Set of partitions. Also, we extend Dyson’s adjoint of a partition to MacMahon’s “modular” partitions with modulus 2. This way we find a new combinato...

متن کامل

On the Distribution of the spt-Crank

Andrews, Garvan and Liang introduced the spt-crank for vector partitions. We conjecture that for any n the sequence {NS(m,n)}m is unimodal, where NS(m,n) is the number of S-partitions of size n with crank m weight by the spt-crank. We relate this conjecture to a distributional result concerning the usual rank and crank of unrestricted partitions. This leads to a heuristic that suggests the conj...

متن کامل

Congruences for Andrews’ Smallest Parts Partition Function and New Congruences for Dyson’s Rank

Let spt(n) denote the total number of appearances of smallest parts in the partitions of n. Recently, Andrews showed how spt(n) is related to the second rank moment, and proved some surprising Ramanujan-type congruences mod 5, 7 and 13. We prove a generalization of these congruences using known relations between rank and crank moments. We obtain an explicit Ramanujan-type congruence for spt(n) ...

متن کامل

Nearly Equal Distributions of the Rank and the Crank of Partitions

Let N(≤ m,n) denote the number of partitions of n with rank not greater than m, and let M(≤ m,n) denote the number of partitions of n with crank not greater than m. Bringmann and Mahlburg observed that N(≤ m,n) ≤ M(≤ m,n) ≤ N(≤ m + 1, n) for m < 0 and 1 ≤ n ≤ 100. They also pointed out that these inequalities can be restated as the existence of a re-ordering τn on the set of partitions of n suc...

متن کامل

ذخیره در منابع من


  با ذخیره ی این منبع در منابع من، دسترسی به آن را برای استفاده های بعدی آسان تر کنید

برای دانلود متن کامل این مقاله و بیش از 32 میلیون مقاله دیگر ابتدا ثبت نام کنید

ثبت نام

اگر عضو سایت هستید لطفا وارد حساب کاربری خود شوید

عنوان ژورنال:

دوره   شماره 

صفحات  -

تاریخ انتشار 2014