Ferrers diagrams, conjugates, and Young tableaux
A partition of a positive integer n is a way of writing n as a sum of positive integers, where order does not matter. For example, 4 can be partitioned as 4, 3+1, 2+2, 2+1+1, and 1+1+1+1 -- giving p(4) = 5. The partition function p(n) grows rapidly and has deep connections to number theory, algebra, and even physics.
In this lesson, you will visualize partitions as Ferrers diagrams, explore the conjugation operation, and fill Young tableaux by hand.
A Ferrers diagram represents a partition as rows of dots or boxes, with row lengths in decreasing order. The partition (4, 2, 1) of 7 has a row of 4 dots, then 2, then 1. This visual representation makes many partition identities transparent. Browse all partitions of n and see the partition count p(n) grow.
A Ferrers diagram represents an integer partition as rows of boxes in descending order. Click any partition to enlarge it. The partition function p(5) counts all ways to write 5 as a sum of positive integers.
Key insight: The partition function p(n) has no simple closed form, but Ramanujan discovered remarkable congruences: p(5k+4) ≡ 0 (mod 5), p(7k+5) ≡ 0 (mod 7), and p(11k+6) ≡ 0 (mod 11).
The conjugate of a partition is obtained by transposing its Ferrers diagram -- reflecting it along the main diagonal. Rows become columns and vice versa. A partition that equals its own conjugate is called self-conjugate. The Durfee square is the largest square that fits in the top-left corner of the diagram.
The conjugate of a partition is obtained by reflecting its Ferrers diagram across the diagonal. The Durfee square (dashed yellow) is the largest k x k square fitting in the top-left corner. (6) → (1,1,1,1,1,1)
Key insight: Self-conjugate partitions of n are in bijection with partitions of n into distinct odd parts. The Durfee square size is a key invariant used in partition identities and q-series.
A standard Young tableau fills a Young diagram with the numbers 1 through n so that entries increase along each row (left to right) and down each column. The number of standard tableaux of a given shape is given by the elegant hook length formula: fλ = n! / Π h(u), where h(u) is the hook length at cell u.
Click cells to fill them with 1, 2, 3, ... Numbers must increase left-to-right and top-to-bottom. The hook length formula: f = 5! / (product of hook lengths) = 5.
Key insight: The hook length at a cell counts the cells directly to its right and below it, plus the cell itself. The hook length formula, discovered by Frame, Robinson, and Thrall in 1954, remains one of the most beautiful results in algebraic combinatorics.