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i would like to understand basic definition of conjugate partition,this is what is said in my book

Let $υ = (u_1, u_2, . . . , u_n)$ be a sequence of integers such that $u_1 ≥ u_2 ≥ · · · ≥ u_n ≥ 0.$ The conjugate partition of $υ$ is $υ∗ = (u_1^*, u_2^*. . . , u_t^* )$, where $u_i^*$ is the number of js such that $u_j ≥ i$ . $t$ is sometimes taken to be $u_1$, but is sometimes greater (obtained by extending with $0$s).

Examples: $(4, 3, 2, 2, 1)^* = (5, 4, 2, 1).$

this is what was said in my book,so as i understand from this conjugate partition does not mean complement of set(partition),but set with missed some element,for example in given example we have missed ${3,2}$,so it means that we can create a lot of conjugate partition from given set right?so it depend on person's aim right?for exmaple we may get another partition by exclude $5$ instead of $3$ right?please help me to clarify everything

dato datuashvili
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1 Answers1

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****
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**
*

This is a way of representing the partition of a number $4+3+2+2+1 = 12$

If we flip this on its side, we get:

*****
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*

This now represents the partition $5 + 4 + 2 + 1 = 12$. Do you see how this works?

muzzlator
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  • so it means that result should be same?i mean it is something like transformation where for example length is the same?? – dato datuashvili Apr 14 '13 at 19:20
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    There is a unique conjugate partition for a given partition. It is not a property of a bunch of partitions but rather a specific partition obtained by doing the above process. Much like the conjugate of a complex number is a single complex number obtained by a certain process (negating the imaginary component) – muzzlator Apr 14 '13 at 19:22
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    nice then i understand it, iw as thinking that it would be $5+2+2+1+1+1=12$ another one – dato datuashvili Apr 14 '13 at 19:27