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Concept

Partition

A way of writing a whole number as a sum of whole numbers with the order disregarded. The counts have no simple closed form and are handled instead through a product whose factors decide how many copies of each part are used.

Named by 12 essays across 3 fields — each of them below, with the objects they name alongside it.

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.

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 · Partitions A product of 3 polynomials, and what its coefficients count. The coefficients of a product of small polynomials, with the combinations of choices that reach one marked total written out beneath it.

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.

discrete · Generating functions The partition product's coefficients to q¹². A row of series coefficients computed by expanding a product, beside the same numbers obtained another way.

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 · Partitions The product of (1 − qᵏ), and what survives at 12. The coefficients of the pentagonal product drawn as signed bars, with the partitions into distinct parts that Franklin's move leaves unpaired.

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 · Partitions p(n) to 60, against the Hardy–Ramanujan estimate. The number of partitions of each number up to sixty on a logarithmic scale, with the asymptotic estimate drawn over it and the ratio of the two tabulated.

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 · Partitions Every fifth partition count divides, and the rank that says why. A row of partition counts with the ones in a congruence class marked, and a histogram of partitions sorted by rank.

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 · Partitions The cube cut into 6 symmetric chains. The subsets of a set of 4 partitioned into 6 chains by the bracket rule, each chain running from size k to size 4 − k and passing once through the middle layer.

The cube cut into chains

Write a subset as a string of brackets, match them the ordinary way, and the unmatched ones say which chain it is on. Six chains cover all sixteen subsets of a four-element set, and the bound and the example arrive together.

discrete · Posets Permutations of n things, by how many pairs they put in the wrong order. A table whose row n and column k hold the number of permutations of size n whose inversions is k, with each row's total beside it — the plain count the one-variable series gives.

The coefficient that is a polynomial

Add a second variable to track a statistic and each coefficient stops being a number. Set the new variable to one and the old count comes back untouched; leave it in and the mean of the statistic is a derivative rather than an average.

discrete · Generating functions Random partitions of 1,000, scaled, against their limit shape. The outlines of 4 uniformly random partitions of 1000, scaled by the square root of 1000, lying close to the curve e^(−cx) + e^(−cy) = 1 with c = π/√6.

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 · Partitions Two ways of counting that agree at every number. For n up to 40, the counts of partitions with gaps of at least two against partitions into parts congruent to 1 or 4 mod 5, on a logarithmic scale, equal at every n, with the second identity's counts beside them.

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.

number · Partitions Which pairs of the square's symmetries commute: 40 of 64. A 8-by-8 grid over the elements of the 8 symmetries of a 4-gon, rows and columns grouped into 5 conjugacy classes, with a filled square wherever the two elements commute: 40 filled squares, and each row's count of filled squares written at its end.

Five-eighths of the pairs, and no more

Pick two symmetries of a square at random and do them in both orders: forty times in sixty-four the result is the same. No group that fails to commute does better. The reason is a count of pairs that turns into a count of conjugacy classes, and a two-line argument about the centre that caps the answer at five-eighths — reached by the square and the quaternions, approached from above by nothing, and approached from below by groups that commute a little more than half the time.

algebra · Symmetry groups A partition in two directions is a pile of cubes. Plane partition 5 4 3 1 / 4 3 2 / 3 2 1 / 1 of 29, drawn as stacked cubes; PL(29) = 3759612.

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.

number · Partitions

Named alongside it

The objects these essays reach for when they reach for this one.

Generating functionBijectionCounting two waysAsymptoticsBinomial coefficientFerrers diagramRecursionAlgebraic identityCatalan numbersConjugate partitionDurfee squareFormal power series

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