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1. Definition & Example

There are two standard ways to denote $\pi \in \mathfrak{S}_n$.

$$\pi(1)=5,\pi(2)=4,\pi(3)=2,\pi(4)=3,\pi(5)=1.$$

The six permutations of size 3

 [1,2,3] 

 [1,3,2] 

 [2,1,3] 

 [2,3,1] 

 [3,1,2] 

 [3,2,1] 

2. FindStat representation and coverage

3. Additional information

3.1. The Rothe diagram of a permutation

3.2. The Lehmer code for a permutation

3.3. Special classes of permutations

3.3.1. Pattern-avoiding permutations

$$ \#\big\{ \sigma \in \mathfrak{S}_n : \sigma \text{ avoids } \tau \big\} = \frac{1}{n+1}\binom{2n}{n}, $$

where we set $t_0=0$ and $t_k=n+1$.

3.4. Classes of permutations in Schubert calculus

The following classes of permutations play important roles in the theory of Schubert polynomials, see e.g. [Man01, Section 2.2]. A permutation $\sigma \in \mathfrak{S}_n$ is called

3.5. Properties

3.6. Remarks

4. References

5. Sage examples


CategoryCombinatorialCollection