On the positive moments of ranks of partitions

Abstract: We study combinatorial and asymptotic properties of the rank of strongly unimodal sequences. We find a generating function for the rank enumeration function and give a new combinatorial interpretation of the ospt-function introduced by Andrews, Chan, and Kim. We conjecture that the enumeration function for the number of unimodal sequences of a fixed size and varying rank is log-concave, and we prove an asymptotic result in support of this conjecture.

Finally, we determine the asymptotic behavior of the rank for strongly unimodal sequences, and we prove that its values when appropriately renormalized are normally distributed with mean 0 in the asymptotic limit. References [Enhancements On Off] What's this? Reprint of the original. MR [2] George E. AndrewsConcave and convex compositionsRamanujan J. MR [4] George E.

Andrews and Bruce C. MR [5] George E. Theory Ser. Ano. AndrewsFreeman J. Dysonand Dean HickersonPartitions and indefinite quadratic formsInvent. Andrews and F. ApostolModular functions and Dirichlet series in number theory2nd ed. MR [9] A. Atkin and P. Swinnerton-DyerSome properties of partitionsProc. London Math. AuluckOn some new types of partitions associated with generalized Ferrers graphsProc. Cambridge Philos. MR [11] Patrick BillingsleyProbability and measure3rd ed.Hao and H.

Yang Context-free grammars and stable multivariate polynomials over Stirling permutationsIn: V. Pillwein and C. Schneider eds. PDF 14 Citations. Yang A context-free grammar for the Ramanujan-Shor polynomialsAdv. PDF 2 Citations. Du and J. Zhao Finding modular functions for Ramanujan-type identitiesAnn.

The First Positive Rank and Crank Moments for Overpartitions

Jia and L. PDF 8 Citations. Discrete Math. PDF 1 Citation. The spt-function of AndrewsIn: A. Claesson, M. Dukes, S. Kitaev, D. Manlove and K. Meeks eds. Press, Cambridge, Ji and W.

Zang Nearly equal distributions of the rank and the crank of partitionsIn: G. Andrews and F. Garvan eds. Fu Context-free grammars for permutations and increasing treesAdv. PDF 25 Citations. Guo, P. Guo, H. Huang and T. Hou and D. Zeilberger Automated discovery and proof of congruence theorems for partial sums of combinatorial sequencesJ.

Difference Equ. Chen On permutations with bounded drop sizeEuropean J. Hou, L. Sun and L. Zhang Ramanujan-type congruences for overpartitions modulo 16Ramanujan J. PDF 9 Citations. Huang and L. Wang Average size of a self-conjugate st -core partitionProc.Comedian Joel McHale calls on everyone to help local restaurants and wear masks with whatever they want to say out loud publicly, says White House might not want more tests out there but hospitals are full of sick people, says his Skype backdrop is going for the look of an RA who had everything taken away in a dorm room.

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Let's remember Hortencia Laurens and others out there who are suffering. They deserve respect, remembrance and better than they are getting now from us. Click here to subscribe to The Hill's Coronavirus Report. Click here to subscribe to our Overnight Healthcare Newsletter to stay up-to-date on all things coronavirus.

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But a trustworthy partnership between private equity and the public must be established. Stephanie Murphy D-Fla. Have a question for our speakers?Inthe first author and Garvan defined the crank of a partition as either the largest part, if 1 does not occur as a part, or the difference between the number of parts larger than the number Methods similar to those used here may be used to establish all the theorems and conjectures of [10].

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He denoted by N r,m,n the number of partitions of n with rank congruent to r modulo m and remarked that several relations appeared to hold between the This series is the generating function for the number of partitions of n into distinct parts with even rank minus the number with odd rank. These conjectures were established As another application, we obtain the overpartition rank generating function.

For an overpartition, we can define a rank in the same way [7]. Let N m,n be the Documents: Advanced Search Include Citations.

Some guesses in the theory of partitions, Eureka 8 by F J Dyson. Add To MetaCart. Rank differences for overpartitions by Jeremy Lovejoy, Robert Osburn In this paper, we prove formulas for the generating functions for rank differences for overpartitions. These are in terms of modular functions and generalized Lambert series. This paper has a two-fold purpose. In this paper we obtain several new results by attaching weights to Rogers-Ramanujan partitions.

In all instances the weights are defined multiplicatively. Citation Context InAtkin and Garvan initiated the study of rank and crank moments for ordinary partitions. These moments satisfy a strict inequality. We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in We prove that a strict inequality also holds for the first rank and crank moments of overpartitions and consider a new combinatorial interpretation in this setting.

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Relations between the rank and the crank modulo 9 by Richard Lewis - J. London Math. Soc In [10] various conjectural identities between the ranks and the cranks of partitions modulo 8, 9 and 12 were proposed.

These conjectures have now all been proved [14, 15, 11], with three exceptions. Here, we prove these three exceptions. Methods similar to those used here may be used to establish a Abstract - Cited by 3 0 self - Add to MetaCart In [10] various conjectural identities between the ranks and the cranks of partitions modulo 8, 9 and 12 were proposed.

What is the probability that the smallest part of a random integer partition of N is odd? What is the expected value of the smallest part of a random integer partition of N?

It is straightforward to see that the answers to these questions are both 1, to leading order. This paper shows that the prec Abstract - Cited by 2 1 self - Add to MetaCart What is the probability that the smallest part of a random integer partition of N is odd? This paper shows that the precise asymptotic expansion of each answer is dictated by special values of an arithmetic L-function. Alternatively, the asymptotics are dictated by the asymptotic expansions of quantum modular forms. A quantum modular form is a function on the rational numbers which has pseudo-modular properties and nice asymptotic expansions near each root of unity.It is a measure of rank correlation : the similarity of the orderings of the data when ranked by each of the quantities.

It is named after Maurice Kendallwho developed it in[1] though Gustav Fechner had proposed a similar measure in the context of time series in Intuitively, the Kendall correlation between two variables will be high when observations have a similar or identical for a correlation of 1 rank i.

The Kendall rank coefficient is often used as a test statistic in a statistical hypothesis test to establish whether two variables may be regarded as statistically dependent.

This test is non-parametricas it does not rely on any assumptions on the distributions of X or Y or the distribution of XY. The precise distribution cannot be characterized in terms of common distributions, but may be calculated exactly for small samples; for larger samples, it is common to use an approximation to the normal distributionwith mean zero and variance.

The Tau-a statistic tests the strength of association of the cross tabulations. Both variables have to be ordinal. Tau-a will not make any adjustment for ties.

It is defined as:. The Tau-b statistic, unlike Tau-a, makes adjustments for ties. A value of zero indicates the absence of association. Be aware that some statistical packages, e. SPSS, use alternative formulas for computational efficiency, with double the 'usual' number of concordant and discordant pairs.

on the positive moments of ranks of partitions

Tau-c also called Stuart-Kendall Tau-c [7] is more suitable than Tau-b for the analysis of data based on non-square i.

For instance, one variable might be scored on a 5-point scale very good, good, average, bad, very badwhereas the other might be based on a finer point scale. The Kendall Tau-c coefficient is defined as: [8].

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For a 2-tailed test, multiply that number by two to obtain the p -value. If the p -value is below a given significance level, one rejects the null hypothesis at that significance level that the quantities are statistically independent.

This is sometimes referred to as the Mann-Kendall test. The number of Bubble Sort swaps is equal to:. From Wikipedia, the free encyclopedia.

For the astronomical radio source, see Taurus A. It is not to be confused with Tau distribution. Mathematics portal. Journal of the American Statistical Association. Analysis of Ordinal Categorical Data Second ed.In mathematicsparticularly in the fields of number theory and combinatoricsthe rank of a partition of a positive integer is a certain integer associated with the partition.

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In fact at least two different definitions of rank appear in the literature. The first definition, with which most of this article is concerned, is that the rank of a partition is the number obtained by subtracting the number of parts in the partition from the largest part in the partition. The concept was introduced by Freeman Dyson in a paper published in the journal Eureka.

A different concept, sharing the same name, is used in combinatorics, where the rank is taken to be the size of the Durfee square of the partition. The ranks of the partitions of n take the following values and no others: [1]. The following notations are used to specify how many partitions have a given rank. Let nq be a positive integers and m be any integer.

Srinivasa Ramanujan in a paper published in proved the following congruences involving the partition function p n : [2]. In commenting on this result, Dyson noted that ". We require a proof which will not appeal to generating functions. Using this new idea, he made the following conjectures:. These conjectures were proved by Atkin and Swinnerton-Dyer in From Wikipedia, the free encyclopedia.

Main article: Durfee square. Dyson Eureka Cambridge. Proceedings of the Cambridge Philosophical Society. XIX : — Atkin; H. Swinnerton-Dyer Proceedings of the London Mathematical Society. Hardy and E. Wright An introduction to the theory of numbers.

London: Oxford University Press.

on the positive moments of ranks of partitions

International Journal of Number Theory. Retrieved 24 November Proceedings of the American Mathematical Society. Cambridge University Press.Abstract: Andrews, Chan, and Kim recently introduced a modified definition of crank and rank moments for integer partitions that allows the study of both even and odd moments.

In this paper, we prove the asymptotic behavior of these moments in all cases. Our main result states that the two families of moment functions are asymptotically equal, but the crank moments are also asymptotically larger than the rank moments. Andrews, Chan, and Kim also gave a combinatorial description for the differences of the first crank and rank moments that they named the ospt-function.

Our main results therefore also give the asymptotic behavior of the ospt-function and its analogs for higher momentsand we further determine the behavior of the ospt-function modulo by relating its parity to Andrews' spt-function.

on the positive moments of ranks of partitions

References [Enhancements On Off] What's this? Lars V. AhlforsComplex analysis3rd ed. An introduction to the theory of analytic functions of one complex variable; International Series in Pure and Applied Mathematics. MR 2. George E.

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Reine Angew. MR 5. Theory Ser.

Rank of a partition

Ano. Andrews and F. Andrews, F. Garvan, and J. Liang, Self-conjugate vector partitions and the parity of the spt functionpreprint.

Tom M. ApostolModular functions and Dirichlet series in number theory2nd ed. MR 9. Orlando, FL: Academic Press, Atkin and F. GarvanRelations between the ranks and cranks of partitionsRamanujan J.