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As seen in the following:

$$\large \lambda=(3,2)\vdash 5$$

I looked it up and in logic the symbol means that what is on the right is provable by what is on the left, but what does it mean in the mathematical context? Does it mean "corresponds to"?

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  • $\begingroup$ What is the context, where did you see this? $\endgroup$ – DMcMor May 14 '17 at 19:38
  • $\begingroup$ Representation theory, the $\lambda$ is a cycle shape corresponding to an element in $S_n$ consisting of a 3-cycle and a 2-cycle (for example $(132)(45)$). I think I know but I want to make sure, since this is all new to me. The symbol is used in this paper a lot acsu.buffalo.edu/~yinsu/… $\endgroup$ – Nażysław Zbyłutowicz May 14 '17 at 19:41
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    $\begingroup$ I admittedly know nothing about representation theory, but a quick search yielded this paper. See Definition 1. $\endgroup$ – DMcMor May 14 '17 at 19:50
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$\lambda\vdash n$ means that:

$$\lambda=(\lambda_1,\dots,\lambda_k)$$

and

$$\sum_\limits{i=1}^k \lambda_i = n$$

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I hope it can help you:

  • A partition of a positive integer n is a multiset of positive integers that sum to n. We denote the number of partitions of n by $p_{n}$

  • We use Greek letters to denote partitions often $\lambda$,$\mu$ and $\nu$

􏰁We’ll write:

$λ:n=n_{1}+n_{2}+···+n_{k}$ or $λ⊢n$.

The notation λ ⊢ n means that λ is a partition of n.


For example:

The partitions of 5 are:

$$5$$ $$4+1$$ $$3+2$$ $$3+1+1$$ $$2+2+1$$ $$2+1+1+1$$ $$1+1+1+1+1$$

$p_{5}=7$

  • $3+1+1$, $1+3+1$, and $1+1+3$ are all the same partition, so we will write the numbers in non-increasing order.

$λ:5=3+1+1$, or $λ=311$, or $λ=3^11^2$, or $311 ⊢5$.

$ (3,2) ⊢5 $ means that $λ:5=3+2$ is a partition of $5$

  • In some sources partitions are treated as the sequence of summands, rather than as an expression with plus signs. For example, the partition $2 + 2 + 1$ might instead be written as the tuple $(2, 2, 1)$ or in the even more compact form $(2^2, 1)$ where the superscript indicates the number of repetitions of a term.

Partition_(number_theory)

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