Every fifth one divides

MacMahon computed the partition counts to two hundred by hand, using the pentagonal recurrence — the only method available that does not require expanding the product itself — and gave the table to Ramanujan. What Ramanujan noticed in it is the most surprising thing on this ladder.

$$p(4) = 5, \quad p(9) = 30, \quad p(14) = 135, \quad p(19) = 490, \quad p(24) = 1575.$$

Every one of them divisible by five. And it continues: at every $n$ leaving remainder four on division by five, the count is divisible by five, without exception, forever.

There is no reason for it in anything this ladder has said. The counting has been by dot diagrams and by products, neither of which knows what five is; the growth is governed by a square root and a $π$; and now an arithmetic progression of the index controls an arithmetic property of the value.

Three of them #

Ramanujan found three, and published them in 1919:

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

The offsets — four, five, six — look arbitrary and are not: in each case the offset is chosen so that $24n + 1$ is divisible by the modulus, which is the shape of every statement in this area and is the first hint that the natural variable is $24n - 1$ rather than $n$.

Nothing similar is known for two or three. Whether $p(n)$ is even or odd about half the time each is conjectured, unproven, and has resisted everything; the small primes are the hard case here, which is the opposite of the usual situation.

What it takes to see it #

The pattern is invisible in the list of values and obvious in the list of remainders, which is the whole difficulty of finding it.

Here are the counts to nineteen, in two rows: $1, 1, 2, 3, 5, 7, 11, 15, 22, 30$ and then $42, 56, 77, 101, 135, 176, 231, 297, 385, 490$. Nothing announces itself. Here they are modulo five: $1, 1, 2, 3, 0, 2, 1, 0, 2, 0, 2, 1, 2, 1, 0, 1, 1, 2, 0, 0$. The zeroes fall at positions $4, 9, 14, 19$ — and also at $7$ and $18$, which are not part of any pattern and are exactly the sort of noise that makes reading a table hard.

So the discovery required computing a long table, reducing it modulo a prime nobody had a reason to choose, and picking a regular subsequence of zeroes out of an irregular set of them. Ramanujan did it for three primes at once. Nobody has since found a fourth congruence of the same small shape, because there is none: the progressions for larger primes exist, by Ono’s theorem, and their common differences run to thousands.

This is the standing hazard with tables, and this collection has met it from the other side: a pattern in the small cases that is real, against a pattern that is coincidence, and no way to tell from the table which is which. Ramanujan’s three were real. He also conjectured a fourth family, for powers, that turned out to need correcting — the general statement he wrote for arbitrary powers of $5$, $7$ and $11$ is false as written for $7$, and Watson found the right version.

Ramanujan’s own proof #

His argument is an identity, and it is worth writing down because it settles the divisibility outright rather than explaining it:

$$\sum_{n \ge 0} p(5n+4) \, q^n = 5 \prod_{k \ge 1} \frac{(1 - q^{5k})^5}{(1 - q^k)^6}.$$

The right-hand side has an explicit factor of five in front and a product with whole-number coefficients behind it, so every coefficient on the left is divisible by five. That is a complete proof, and it explains nothing at all: the identity is verified by manipulating series, and the five appears at the end of the manipulation rather than at the start.

There is a matching identity for seven, with a factor of seven and a similar product. For eleven there is one too, and it is considerably harder.

Why the offsets are what they are #

The remainders four, five and six look like three unrelated numbers and are one number in disguise.

Multiply each by $24$ and add one: $24 \cdot 4 + 1 = 97$, which leaves $2$ modulo five — that is not it. Try the other direction: the condition is that $24n \equiv 1$ modulo the prime. For five: $24 \equiv 4$, and $4n \equiv 1 \pmod 5$ gives $n \equiv 4$. For seven: $24 \equiv 3$, and $3n \equiv 1 \pmod 7$ gives $n \equiv 5$. For eleven: $24 \equiv 2$, and $2n \equiv 1 \pmod{11}$ gives $n \equiv 6$. Three offsets, one rule.

The $24$ is not a coincidence either, and it comes from the same place as everything else in the upper half of this ladder. The generating function for partitions becomes a modular object only after multiplying by $q^{1/24}$ — the Dedekind eta function is $q^{1/24} \prod (1 - q^k)$ — so the natural variable in every statement about it is $24n - 1$ rather than $n$. Ramanujan’s congruences, stated in that variable, all read the same way: $p(n)$ is divisible by $\ell$ whenever $24n - 1$ is divisible by $\ell$.

That reframing is what makes Ono’s general theorem statable at all, and it is a good example of a pattern this collection keeps meeting: a change of variable that turns three separate facts into one fact, and which is forced by the object rather than chosen for tidiness.

Dyson’s rank #

Freeman Dyson, aged twenty and still an undergraduate, wanted an explanation of the kind the identity does not give. If $p(5n+4)$ is divisible by five, perhaps the partitions themselves fall into five equal heaps, and perhaps there is a statistic that sorts them.

He proposed one and named it the rank:

$$\text{rank} = (\text{largest part}) - (\text{number of parts}).$$

The conjecture: the partitions of $5n + 4$, sorted by rank modulo five, fall into five classes of exactly equal size — which would make the divisibility a corollary of a partition into equal heaps rather than an accident of an identity.

It works for five and it works for seven. The figures build the classes by listing every partition of the number drawn, computing each rank, and counting — so the equality of the classes is measured rather than quoted, at every parameter any essay uses.

Atkin and Swinnerton-Dyer proved Dyson’s conjecture in 1954, ten years after he stated it, and the proof is analytic rather than combinatorial: no bijection between the classes is exhibited, and to this day the equality of the heaps is known without anybody being able to pair their members up.

Where the rank fails #

For eleven the rank does not work. Sorting the partitions of $11n + 6$ by rank modulo eleven gives eleven classes of unequal sizes, and the congruence is true anyway.

Dyson said so in the paper that proposed the rank, which is the part of the story worth admiring. He had a statistic that explained two of the three congruences, he checked the third, it failed, and rather than quietly restricting his conjecture he predicted that some other statistic existed which would work for all three — and named it in advance:

“I hold in fact that there exists an arithmetical coefficient similar to, but more recondite than, the rank of a partition; I shall call this hypothetical coefficient the ‘crank’ of the partition.”

The crank was found forty-four years later, by Andrews and Garvan in 1988. Its definition is genuinely more recondite: it is the largest part when the partition has no ones, and otherwise the number of parts larger than the number of ones, minus the number of ones. Nobody would propose that from first principles; it was reverse-engineered from what the generating function needed.

And it works for all three. Sorted by crank modulo five, seven or eleven, the partitions of $5n+4$, $7n+5$ and $11n+6$ fall into equal classes in every case.

What the rank is, on a diagram #

The rank has a picture, and it is the first rung’s picture with two of its measurements subtracted.

A Ferrers diagram has a width — the largest part — and a height — the number of parts. The rank is width minus height. So a wide flat partition has a large positive rank, a tall thin one has a large negative rank, and conjugation, which exchanges width and height, negates the rank exactly.

That last fact is a real check on the statistic and a real constraint on what it can do. Since conjugation is an involution on the partitions of $n$ that negates ranks, the multiset of ranks is symmetric about zero — so the rank classes modulo an odd number $m$ satisfy the symmetry that class $k$ and class $-k$ have equal size, whatever $n$ is. That symmetry is free, and it already gives half of what Dyson needed: at $n = 4$ it forces classes $1$ and $4$ to match and classes $2$ and $3$ to match, leaving only class $0$ unconstrained.

What it does not give is the equality of all five, and that is the content of the conjecture. The symmetry is a consequence of a bijection anybody can perform; the equality is a theorem proved by generating functions with no bijection attached. The gap between those two is exactly the gap between what a diagram argument can reach and what it cannot, which is the boundary this ladder has been walking along since its second rung.

Splitting into equal classes, as a method #

The shape of Dyson’s idea is one this collection meets in several places, and it is worth naming: to show a count is divisible by $m$, exhibit an action or a statistic that splits the objects into $m$ classes of equal size.

Necklaces that prove a theorem is the cleanest instance — Fermat’s little theorem falls out because rotating a necklace of prime length either fixes it or produces $p$ distinct arrangements, so the non-constant ones come in groups of $p$. Counting colourings up to symmetry is the same instinct made systematic.

What is unusual about the rank is that the classes are equal and no map between them is known. In the necklace argument the group action supplies the correspondence explicitly; here the classes are proved equal by comparing generating functions, and a combinatorial bijection between rank classes would be a real advance. So the method’s usual advantage — that it explains rather than verifies — is only half delivered.

What came after #

The three congruences turned out to be the beginning of something much larger.

Watson and Atkin extended them to powers: $p(25n + 24)$ is divisible by $25$, $p(49n + 47)$ by $49$, and there is a general statement for arbitrary powers of five, seven and eleven. Those are genuinely stronger statements rather than repetitions — divisibility by twenty-five does not follow from divisibility by five, in the way that a number’s divisors do not follow from its smallest one.

Then, in 2000, Ono proved that congruences of this type exist for every prime modulus at least five — for each such prime there are arithmetic progressions on which $p(n)$ is always divisible by it — and Ahlgren and Ono extended that to every modulus coprime to six. So the phenomenon Ramanujan spotted at three small primes is universal, and the reason it is invisible is that the progressions for larger primes have enormous common differences. The one for thirteen has a common difference in the thousands; nothing in a table of two hundred values could have shown it.

The proofs run through modular forms, which is also where the asymptotic formula came from. That is the honest summary of this whole ladder’s upper half: the elementary machinery — diagrams, products, involutions — carries it as far as the pentagonal theorem, and everything past that is a consequence of the generating function being a modular object.

What the pictures cannot show #

The figures check divisibility on the values they draw and equality of classes at the numbers they sort. Both are finite verifications of infinite statements, and the drawing says which numbers it checked.

The classes are the interesting limitation. At $n = 9$ there are thirty partitions and five classes of six, and the figure counts them; at $n = 99$ there are $169{,}229{,}875$ partitions and no enumeration is possible. So the pictures can show the phenomenon exactly where a reader could have checked it themselves, and the general statement rests on the analytic proof.

Nothing here draws the crank. Its definition has a case split in it — one rule when the partition has no ones and another when it does — and a figure that sorted partitions by it would be a table of numbers with no visible structure, which is a fair description of what the crank is. That it works is a theorem; that it looks like nothing anybody would invent is the reason it took forty-four years.

What this anchor has come to #

Five rungs, and the partitions have been five things: dots to be rearranged, coefficients in a product, terms that cancel, a quantity with a size, and a set with an arithmetic of its own.

The thread is that each rung’s method fails at the next question and is replaced. Diagrams prove identities and cannot count; the product counts and cannot say how large; the analysis says how large and cannot see divisibility; the rank sees divisibility and fails at eleven. That is not a defect in the subject — it is what a subject looks like from inside, and the honest way to draw a ladder is with each rung standing on a method the one below could not supply.

What is left named and unwritten: the crank’s own generating function, which is where the modern subject starts; the Rogers–Ramanujan identities, which this ladder mentioned and did not develop; and the modular forms under all of it, which are a field rather than a rung.