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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 $(6,4,4,2)$.

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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?

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

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