For the partition $(6,4,4,2)$ of integer $16$, if we draw its Young diagram with four rows of boxes, one below the other, of size $6$, $4$, $4$, and $2$, then flipping the resulting Young diagram along diagram gives another Young diagram, with row sizes $(4,4,3,3,1,1)$, and this partition of $16$ is called the conjugate partition of $(6,4,4,2)$.

enter image description here

Question. Is it possible to define or compute the conjugate of a partition, without looking it in terms of Young diagram?

In other words, to find, or define conjugate of a partition, is it necessary to represent it by Young diagram?

  • $\begingroup$ Did you mean (6, 4, 4, 2) instead of (6, 6, 2, 2)? $\endgroup$ Apr 13, 2021 at 15:43
  • $\begingroup$ Thanks Steven for pointing the mistake. $\endgroup$ Apr 13, 2021 at 17:47

1 Answer 1


Yes. Count the number of nonzero entries and subtract $1$ from each. Repeat until everything is $0$.

For your example:

  1. $(6,4,4,2)$ has $\color{red}{4}$ nonzero entries.
  2. $(5,3,3,1)$ has $\color{red}{4}$ nonzero entries.
  3. $(4,2,2,0)$ has $\color{red}{3}$ nonzero entries.
  4. $(3,1,1,0)$ has $\color{red}{3}$ nonzero entries.
  5. $(2,0,0,0)$ has $\color{red}{1}$ nonzero entry.
  6. $(1,0,0,0)$ has $\color{red}{1}$ nonzero entry.
  7. $(0,0,0,0)$ is all $0$.

If you instead start with $(4,4,3,3,1,1)$, you will get $(6,4,4,2)$.


Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.