# Partition of integer and its conjugate

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

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?

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

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

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