Generator

The partition 5 + 4 + 2 + 1 and its conjugate

A generator in the number library, called 58 times across 11 essays. Below: what it draws with nothing chosen and at each mode an essay asks for, what it checks while drawing, and everywhere it is used.

ferrers is one function. Everything below came out of it during this build, at parameters taken from the essays rather than invented for this page — so a figure here is the same figure a reader meets in an essay, and if the generator changes, this page changes with it.

With nothing chosen

The partition 5 + 4 + 2 + 1 and its conjugate. A row of dots for each part, and the same dots read down the columns instead.

Partitions of 8: odd parts against distinct parts

The Durfee square of 6 + 5 + 5 + 3 + 2 + 1

A partition in two directions is a pile of cubes

A second direction raises the exponent

Wright's formula, with ζ(3) where π was

What it checks while it draws

Collected by running the family and recording what it asserted, not written here. The count is how many separate times the claim was put to the test while these drawings were made.

Ramanujan: 5 divides p(4) ×342857
Gordon's identity, i = 1, at n = 0 ×93
first identity at 0 ×41
second identity at 0 ×41
as many partitions of 30 have largest part 1 as have 1 parts ×30
the coefficient at 0 is what the theorem says ×27
p(0) is 1 ×21
the product's coefficient at 0 is p(0) ×19
the two products agree at 0 ×19
p(4) is divisible by 5 ×11
log p(10) is below π√(2n/3), as it is for every n ×10
runs of length 1 are as common as a coin's ×10
the estimate at 10 is within a factor of two of the count ×10
modulo 5, class 0 has its fair share of zeros ×5
modulo 5, class 1 has excess zeros ×5
the ranks modulo 5 split the 30 partitions of 9 into 5 equal classes ×4
the number partitioned is a whole number between 50 and 5000 ×3
the partitions of 8 into odd parts and into distinct parts come out equal ×3
modulo 2 every residue is equally common ×2
modulo 5 zero is at least as common as the congruence plus an even spread makes it ×2
the number of coefficients shown is a whole number between 6 and 20 ×2
the number sorted by rank leaves 4 on division by 5 ×2
the number whose partitions are sorted by rank is a whole number between 4 and 14 ×2
the range is a whole number between 12 and 60 ×2
|τ(p)| ≤ 2p^(11/2) ×1
232,848 plane partitions fit in the 4 × 4 × 4 box ×1
a partition whose parts differ by at least 2 ×1
and at this size it is under 0.04 everywhere ×1
and changes the number of parts by exactly one ×1
and it is the largest one that fits ×1
and it meets the diagonal at ln 2 / c ×1
and the estimate is closer at the far end than at the near one ×1
and the partitions the move cannot touch are exactly what survives ×1
and the ratio climbs towards that ceiling rather than away ×1
at least one of the divisible values is in the table ×1
counting by Durfee square accounts for every partition exactly once ×1
Deligne's bound |τ(n)| ≤ d(n) n^(11/2) ×1
doing it twice gives back what it started as ×1
each doubling of n brings the ratio closer to ln 2 / c ×1
each sample is a partition of n ×1
every listed partition adds to the number it partitions ×1
half a million partition numbers are within half a percent of half even ×1
MacMahon's box formula matches the count by transfer ×1
MacMahon's product counts the plane partitions found one by one ×1
partitions of 10: as many with gaps of two as with parts ≡ ±1 (mod 5) ×1
Ramanujan's value at e^(−2π): √((5 + √5)/2) − (1 + √5)/2 ×1
removing the staircase leaves a partition into at most k parts ×1
the angles fit the Sato–Tate density ×1
the average distance to the curve falls at each size ×1
the columns never increase ×1
the congruence drawn is the one modulo 5 or modulo 7 ×1
the conjugate is a partition of the same number ×1
the continued fraction equals q^(1/5) times the ratio of the two products ×1
the cube's count grows like e^(c n²) with c = (9/2) log 3 − 6 log 2 ×1
the curve is its own reflection in the diagonal ×1
the depth of the continued fraction is a whole number between 20 and 200 ×1
the drawn plane partition has 29 cubes ×1
the even count far exceeds the square root ×1
the first values are 1, −24, 252, −1472, 4830, −6048, −16744, 84480 ×1
the forced class is all zeros ×1
the gap to the Gumbel law shrinks as n grows ×1
the guess first fails at 6, where it predicts 141 and there are 140 ×1
the largest number counted is a whole number between 20 and 120 ×1
the largest number in the table is a whole number between 12 and 40 ×1
the marked values are whole numbers inside the range drawn ×1
the move gives another partition of the same number into distinct parts ×1
the number being partitioned is a whole number between 3 and 12 ×1
the number is a whole number between 5 and 14 ×1
the number of samples is a whole number between 1 and 8 ×1
the number whose distinct partitions are paired off is a whole number between 3 and 14 ×1
the numbers outside the congruence class are not all divisible either ×1
the partition view is one the family draws ×1
the partitions listed are all of them ×1
the parts are a descending list of whole numbers ×1
the ratio climbs towards 1 from below ×1
the recurrence divides exactly ×1
the recursion divides exactly ×1
the restricted counts are not the unrestricted one ×1
the rows never increase ×1
the same count against parts ≡ ±1 mod 6 fails somewhere ×1
the series drawn is the partition product or Euler's identity ×1
the signed count of distinct-part partitions is the product's coefficient ×1
the square, the arm and the leg account for every dot ×1
the staircase 1 + 3 + 5 + … holds k² dots ×1
there is a Durfee square to draw ×1
three sizes ×1
turning the diagram over twice gives it back ×1
volume v and volume 64 − v are equally common ×1
τ(mn) = τ(m)τ(n) when m and n share no factor ×1
τ(n) ≡ n σ₃(n) mod 7 ×1
τ(n) ≡ n σ₉(n) mod 5 ×1
τ(n) ≡ σ₁₁(n) mod 691 ×1
τ(n) is not zero ×1
τ(p) mod 23 is 2, −1 or 0 as x³ − x − 1 has three, no or one root mod p ×1
τ(p^(k+1)) = τ(p)τ(p^k) − p¹¹τ(p^(k−1)) ×1

Where it is called

Every figure on this list is drawn by the same rule, so a change to the rule changes all of them at once. That is why the list is published.

Number

A diagram turned on its side

Write a partition as rows of dots, then read the columns instead. Every theorem in this essay is that one move, and the move proves things that no formula suggests.

Number

A partition piled into a corner

A partition is a row of numbers that never increases. Let it never increase down the columns too, and it becomes a pile of cubes pushed into the corner of a room. MacMahon counted the piles with one change to Euler's product — each part size k in k colours — and in a box with a product over its cells. The second direction moves the growth from e^(√n) to e^(n^(2/3)), puts ζ(3) where π was, and makes a count that grows with a box's area. One direction further, his guess fails at six.

Discrete

A polynomial that counts

Hang a counting sequence on the powers of a variable and the two ways of combining choices — this and that, this or that — become multiplication and addition, so a recursion turns into an equation and the equation can be solved.

Number

Every fifth one divides

p(4) is 5, p(9) is 30, p(14) is 135, and every partition count at a number leaving four on division by five is divisible by five. Ramanujan read it off a table; the explanation is a way of splitting those partitions into five equal heaps.

Number

Every partition, hidden in a product

Multiply out one factor for each part size and the coefficient of q to the n is the number of partitions of n. Nothing is being approximated: the product is a bookkeeping device that does the counting by multiplying.

Number

Is the partition count even half the time?

The number of partitions of n is even for 50.0% of the n up to half a million, its runs of one parity are as long as a coin's, and nothing proves that the share is a half — the best theorems only show there are at least about √n of each. Modulo 5 and 7 the zeros carry Ramanujan's congruences and something more: an excess that follows whether 1 − 24n is a square.

Number

The product raised to the twenty-fourth power

Euler's product (1 − q)(1 − q²)(1 − q³)… has the partition numbers for its reciprocal. Raise it to the twenty-fourth power and multiply by q, and the coefficients that come out — 1, −24, 252, −1472, 4830, −6048 — multiply like the primes they are indexed by, stay inside a bound Ramanujan guessed and Deligne proved fifty-eight years later, agree with a divisor sum modulo 691, and know, modulo 23, how a cubic equation factors.

Number

The shape a random partition takes

There are about twenty-four thousand billion billion billion ways to write 1,000 as a sum of whole numbers. Pick one at random, draw its Ferrers diagram, shrink it by the square root of a thousand, and it is almost exactly the curve e^(−cx) + e^(−cy) = 1 with c = π/√6. So is the next one, and the next. A random partition of a large number has a shape, and the shape is known exactly.

Number

The size of a number with no formula

There is no closed expression for the number of partitions of n. There is an expression for how large it is — with a square root in the exponent and a π in front — and it is accurate enough that rounding a few terms of its refinement gives the exact count.

Number

The terms that cancel almost everything

Multiply out the product of 1 − q, 1 − q², 1 − q³ and so on, and nearly every coefficient is zero. What survives is a single plus or minus one at 1, 2, 5, 7, 12, 15 — and the reason is a way of pairing partitions off so that each pair cancels.

Number

Two counts that agree for no visible reason

Write 10 as a sum of whole numbers that differ from each other by at least two, and there are six ways. Write 10 as a sum of numbers that each leave 1 or 4 on division by 5, and there are six ways. The same happens for 20 (thirty-one each), for 40 (three hundred and seventy-four each), for every number anyone has checked and every number there is. The two lists look nothing alike, and no one has found a simple way to turn one into the other.

The whole library · What the figures prove