# Rank of a partition

The **rank of a partition** is a partition statistic, introduced by [Freeman Dyson](https://www.edgepedia.org/freeman-dyson) in 1944, defined as the largest part of the partition minus its number of parts. It was the first statistic shown to give a combinatorial explanation of two of [Ramanujan's congruences](https://www.edgepedia.org/ramanujans-congruences) for the partition function, and its generating function became one of the central examples in the theory of mock modular forms.<sup>[1](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)</sup>

| Key fact | Detail |
|---|---|
| Definition | For a partition \(\lambda\) with largest part \(\lambda_1\) and \(\ell(\lambda)\) parts, the rank is \(\lambda_1 - \ell(\lambda)\)<sup>[1](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)</sup> |
| Introduced by | Freeman Dyson, "Some Guesses in the Theory of Partitions," *Eureka* (Cambridge) 8 (1944), pp. 10–15<sup>[1](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)</sup> |
| Possible values | For partitions of \(n\), the rank takes the values \(n-1, n-3, n-4, \ldots, 0, -1, -2, \ldots, 4-n, 3-n, 1-n\) and no others<sup>[1](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)</sup> |
| Dyson's conjecture | \(N(0,5,5n+4)=N(1,5,5n+4)=\cdots=N(4,5,5n+4)=p(5n+4)/5\), and similarly \(N(0,7,7n+5)=\cdots=N(6,7,7n+5)=p(7n+5)/7\)<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S0747717124000506)</sup> |
| Proof of the conjecture | Atkin and Swinnerton-Dyer, 1954, who established generating functions for every rank difference for \(\ell=5,7\)<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S0747717124000506)</sup> |
| Modulus 11 | The rank does not equally dissect the Ramanujan congruence modulo 11; Dyson conjectured a different statistic, the crank, given by Garvan in 1988 and defined by Andrews and Garvan<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC1266116/)</sup> |
| Generating function | \(R(w;q)=1+\sum_{n\geq 1}\sum_{m\in\mathbb{Z}} N(m,n)w^m q^n = 1+\sum_{n\geq 1}\frac{q^{n^2}}{\prod_{j=1}^{n}(1-(w+w^{-1})q^j+q^{2j})}\)<sup>[4](https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p08-p.pdf)</sup> |

## Definition and basic properties

A partition of a positive integer \(n\) is a weakly decreasing sequence of positive integers \(\pi_1 \geq \pi_2 \geq \cdots \geq \pi_r\) whose sum is \(n\); the \(\pi_i\) are called the parts, and \(p(n)\) denotes the number of partitions of \(n\), with \(p(0)=1\).<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S0747717124000506)</sup> Dyson defined the rank of a partition to be its largest part minus its number of parts.<sup>[1](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)</sup> For example, the partition \(5+2+2+1\) of 10 has largest part 5 and four parts, so its rank is \(5-4=1\).<sup>[5](https://malors.com/wp-content/uploads/2024/08/august-12-math-camp-2019.pdf)</sup>

The rank is bounded in a narrow range. Dyson noted that the ranks of partitions of \(n\) take the values \(n-1, n-3, n-4, \ldots, 0, -1, -2, \ldots, 4-n, 3-n, 1-n\), and no others.<sup>[1](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)</sup> The statistic is also symmetric: conjugating a partition, which reflects its Ferrers diagram across the diagonal, exchanges the largest part with the number of parts and so negates the rank. Dyson used this duality to show \(N(m,n)=N(-m,n)\), where \(N(m,n)\) denotes the number of partitions of \(n\) with rank \(m\).<sup>[1](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)</sup>

A separate convention in combinatorics calls the side length of the Durfee square, the largest square that fits inside the partition's diagram, a "rank" as well, so the intended statistic must be named explicitly.

## Dyson's conjectures and the Ramanujan congruences

Ramanujan discovered in 1919 three congruences for the partition function:

\[ p(5n+4) \equiv 0 \pmod{5}, \qquad p(7n+5) \equiv 0 \pmod{7}, \qquad p(11n+6) \equiv 0 \pmod{11}. \]

Dyson wanted a direct combinatorial proof, one that would demonstrate "by cross-examination of the partitions themselves" the existence of exclusive, exhaustive, and equally numerous subclasses, rather than a proof through generating functions.<sup>[1](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)</sup> Writing \(N(t,m,n)\) for the number of partitions of \(n\) whose rank is congruent to \(t\) modulo \(m\), he conjectured that

\[ N(0,5,5n+4)=N(1,5,5n+4)=\cdots=N(4,5,5n+4)=\frac{p(5n+4)}{5} \]

and

\[ N(0,7,7n+5)=N(1,7,7n+5)=\cdots=N(6,7,7n+5)=\frac{p(7n+5)}{7}. \]

Sorting the partitions of \(5n+4\) by rank modulo 5 therefore splits them into five equal classes, which implies the congruence modulo 5, and the same holds for modulus 7.<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S0747717124000506)</sup>

**Proof in 1954.** Atkin and Swinnerton-Dyer confirmed both conjectures in their paper *Some Properties of Partitions*, published in the *Proceedings of the London Mathematical Society* in 1954.<sup>[6](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-4.1.84)</sup> In fact, they established the generating functions of every rank difference \(N(s,\ell,\ell n+d)-N(t,\ell,\ell n+d)\) for \(\ell=5,7\) and \(0 \leq s,t,d \leq \ell-1\).<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S0747717124000506)</sup>

## Failure modulo 11 and the crank

The rank does not equally dissect the Ramanujan congruence modulo 11; even the smallest examples show this.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC1266116/)</sup> Dyson observed the failure and conjectured the existence of another partition statistic, which he named the "crank", that would combinatorially explain the congruence modulo 11.<sup>[2](https://www.sciencedirect.com/science/article/abs/pii/S0747717124000506)</sup> He remarked that the crank was unique among arithmetical functions in having been named before it was discovered.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC1266116/)</sup>

Forty years later, in 1988, F. G. Garvan gave a crank in terms of restricted triples of partitions that interprets the congruence modulo 11 as well as giving new interpretations of the congruences modulo 5 and 7, though not in terms of ordinary partitions alone as Dyson had asked.<sup>[7](https://www.ams.org/tran/1988-305-01/S0002-9947-1988-0920146-8/S0002-9947-1988-0920146-8.pdf)</sup> Andrews and Garvan then defined the crank and proved that it simultaneously explains all three Ramanujan congruences modulo 5, 7, and 11; they also showed that the crank dissects the congruences modulo 5 and 7 in a different way than the rank does.<sup>[3](https://pmc.ncbi.nlm.nih.gov/articles/PMC1266116/)</sup>

## Generating function and mock modularity

If \(N(m,n)\) denotes the number of partitions of \(n\) with rank \(m\), the two-variable rank generating function is

\[ R(w;q) := 1 + \sum_{n=1}^{\infty} \sum_{m=-\infty}^{\infty} N(m,n) w^m q^n = 1 + \sum_{n=1}^{\infty} \frac{q^{n^2}}{\prod_{j=1}^{n}\left(1-(w+w^{-1})q^j+q^{2j}\right)}. \]

Setting \(w=1\) recovers the partition generating function; setting \(w=-1\) gives \(\mathcal{R}_1(-1;q)\), which is Ramanujan's mock theta function \(f(q)\), the first example of a mock theta function, mentioned in Ramanujan's last letter to Hardy.<sup>[4](https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p08-p.pdf)</sup><sup> • </sup><sup>[8](https://arxiv.org/abs/2101.01112)</sup>

The rank generating function does not have strict modular properties as a function of two variables, but after a minor normalization it is a mock Jacobi form, a fact established through work of Zwegers, Bringmann and Ono, and Zagier.<sup>[9](https://par.nsf.gov/servlets/purl/10526883)</sup> In a 2010 paper in the *Annals of Mathematics*, Kathrin Bringmann—at the time a researcher on mock modular forms—and [Emory University](https://www.edgepedia.org/emory-university)'s Ken Ono established that, for any root of unity ζ ≠ 1, the function R(ζ;q) is, up to inessential terms, the holomorphic part of a weak Maass form of weight 1/2 for some subgroup of SL₂(ℤ), and from this result they derived the modularity of the generating function for N(r,t;n), which counts the partitions of n with rank congruent to r modulo t.<sup>[4](https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p08-p.pdf)</sup> This realized Dyson's 1987 speculation that the mock theta functions should admit a group-theoretical structure involving theta functions.<sup>[4](https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p08-p.pdf)</sup>

## Congruences and moments for the rank

The rank supports its own arithmetic. Bringmann and Ono proved that Dyson's rank function has infinitely many Ramanujan-type congruences, and Bringmann used the circle method in 2009 to prove an asymptotic formula for the Fourier coefficients of the rank generating functions.<sup>[10](https://arxiv.org/abs/0710.5793)</sup><sup> • </sup><sup>[11](https://arxiv.org/abs/2406.06294)</sup> The \(k\)-th rank moment is \(N_k(n) := \sum_m m^k N(m,n)\), and Andrews showed how his smallest parts function \(\mathrm{spt}(n)\), which counts the total number of appearances of smallest parts in the partitions of \(n\), is related to the second rank moment; he proved Ramanujan-type congruences for \(\mathrm{spt}(n)\) modulo 5, 7, and 13, later generalized using relations between rank and crank moments.<sup>[12](https://qseries.org/fgarvan/talks/talk-spt.pdf)</sup><sup> • </sup><sup>[10](https://arxiv.org/abs/0710.5793)</sup>

The rank has also been generalized. [A. O. L. Atkin](https://www.edgepedia.org/a-o-l-atkin) defined the \(k\)-th rank of a partition as the \(k\)-th largest part minus the number of parts greater than or equal to \(k\).<sup>[13](https://uwaterloo.ca/combinatorics-and-optimization/events/tutte-colloquium-bruce-richmond-0)</sup> The higher Dyson systems \(\mathcal{R}_m(z;q)\) for \(m \geq 2\) are a multivariable refinement of the rank combining \(m\) graded rank contributions, with \(m=1\) recovering Dyson's rank generating function; unlike the classical case, these higher systems are not expected to fit the mock-modular framework.<sup>[14](https://arxiv.org/abs/2607.13159)</sup>

## References

1. [F. J. Dyson (1944). Some Guesses in the Theory of Partitions. *Eureka* (Cambridge) 8, 10–15.](https://qseries.org/fgarvan/references/dyson/dyson-1944.pdf)
2. [The deviation on cranks of partitions, ScienceDirect](https://www.sciencedirect.com/science/article/abs/pii/S0747717124000506)
3. [K. Mahlburg. Partition congruences and the Andrews-Garvan-Dyson crank, PNAS (PMC full text)](https://pmc.ncbi.nlm.nih.gov/articles/PMC1266116/)
4. [K. Bringmann and K. Ono (2010). Dyson's ranks and Maass forms. *Annals of Mathematics* 171(1).](https://annals.math.princeton.edu/wp-content/uploads/annals-v171-n1-p08-p.pdf)
5. [University of Toronto math camp notes on partitions and the rank](https://malors.com/wp-content/uploads/2024/08/august-12-math-camp-2019.pdf)
6. [A. O. L. Atkin and P. Swinnerton-Dyer (1954). Some Properties of Partitions. *Proc. London Math. Soc.* s3-4: 84–106.](https://londmathsoc.onlinelibrary.wiley.com/doi/abs/10.1112/plms/s3-4.1.84)
7. [F. G. Garvan (1988). New combinatorial interpretations of Ramanujan's partition congruences mod 5, 7 and 11. *Trans. Amer. Math. Soc.* 305(1).](https://www.ams.org/tran/1988-305-01/S0002-9947-1988-0920146-8/S0002-9947-1988-0920146-8.pdf)
8. [Congruences modulo powers of 5 for the rank parity function, arXiv:2101.01112](https://arxiv.org/abs/2101.01112)
9. [Equidistribution and partition polynomials, NSF PAR](https://par.nsf.gov/servlets/purl/10526883)
10. [Congruences for Andrews' Smallest Parts Partition Function and New Congruences for Dyson's Rank, arXiv:0710.5793](https://arxiv.org/abs/0710.5793)
11. [Exact formulae for ranks of partitions, alphaXiv](https://arxiv.org/abs/2406.06294)
12. [F. G. Garvan. The Smallest Parts Partition Function (talk slides)](https://qseries.org/fgarvan/talks/talk-spt.pdf)
13. [Tutte Colloquium, Bruce Richmond, University of Waterloo](https://uwaterloo.ca/combinatorics-and-optimization/events/tutte-colloquium-bruce-richmond-0)
14. [C. Alfes, K. Ono, A. Swaminathan. Beyond Mock Modularity: Elliptic Corrections for Higher Dyson Ranks, arXiv:2607.13159](https://arxiv.org/abs/2607.13159)

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*Topic: Encyclopedia › Physical world and mathematics › Mathematics and statistics › Numbers and algebra › Arithmetic and number systems › Integer sequences and partitions › Partitions › Partition congruences*

*Initially written Sep 17, 2026 · Reviewed: Oct 15, 2026 · Edited: Oct 15, 2026 · Last review: Oct 15, 2026*

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