- FindStat

Identifier

Mp00080: Set partitions —to permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St000327: Posets ⟶ ℤ

Values

{{1,2}} => [2,1] => [2,1] => ([],2) => 0
{{1},{2}} => [1,2] => [1,2] => ([(0,1)],2) => 1
{{1,2,3}} => [2,3,1] => [3,2,1] => ([],3) => 0
{{1,2},{3}} => [2,1,3] => [2,1,3] => ([(0,2),(1,2)],3) => 2
{{1,3},{2}} => [3,2,1] => [3,2,1] => ([],3) => 0
{{1},{2,3}} => [1,3,2] => [1,3,2] => ([(0,1),(0,2)],3) => 2
{{1},{2},{3}} => [1,2,3] => [1,2,3] => ([(0,2),(2,1)],3) => 2
{{1,2,3,4}} => [2,3,4,1] => [4,2,3,1] => ([(2,3)],4) => 1
{{1,2,3},{4}} => [2,3,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 3
{{1,2,4},{3}} => [2,4,3,1] => [4,3,2,1] => ([],4) => 0
{{1,2},{3,4}} => [2,1,4,3] => [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => 4
{{1,2},{3},{4}} => [2,1,3,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => 3
{{1,3,4},{2}} => [3,2,4,1] => [4,2,3,1] => ([(2,3)],4) => 1
{{1,3},{2,4}} => [3,4,1,2] => [4,3,2,1] => ([],4) => 0
{{1,3},{2},{4}} => [3,2,1,4] => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4) => 3
{{1,4},{2,3}} => [4,3,2,1] => [4,3,2,1] => ([],4) => 0
{{1},{2,3,4}} => [1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 3
{{1},{2,3},{4}} => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => 4
{{1,4},{2},{3}} => [4,2,3,1] => [4,3,2,1] => ([],4) => 0
{{1},{2,4},{3}} => [1,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4) => 3
{{1},{2},{3,4}} => [1,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => 3
{{1},{2},{3},{4}} => [1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => 3
{{1,2,3,4,5}} => [2,3,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 2
{{1,2,3,4},{5}} => [2,3,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,2,3,5},{4}} => [2,3,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5) => 2
{{1,2,3},{4,5}} => [2,3,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 6
{{1,2,3},{4},{5}} => [2,3,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 4
{{1,2,4,5},{3}} => [2,4,3,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 2
{{1,2,4},{3,5}} => [2,4,5,1,3] => [4,5,3,1,2] => ([(1,4),(2,3)],5) => 2
{{1,2,4},{3},{5}} => [2,4,3,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
{{1,2,5},{3,4}} => [2,5,4,3,1] => [5,4,3,2,1] => ([],5) => 0
{{1,2},{3,4,5}} => [2,1,4,5,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 6
{{1,2},{3,4},{5}} => [2,1,4,3,5] => [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5) => 6
{{1,2,5},{3},{4}} => [2,5,3,4,1] => [5,4,3,2,1] => ([],5) => 0
{{1,2},{3,5},{4}} => [2,1,5,4,3] => [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5) => 6
{{1,2},{3},{4,5}} => [2,1,3,5,4] => [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5) => 4
{{1,2},{3},{4},{5}} => [2,1,3,4,5] => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5) => 4
{{1,3,4,5},{2}} => [3,2,4,5,1] => [5,2,3,4,1] => ([(2,3),(3,4)],5) => 2
{{1,3,4},{2,5}} => [3,5,4,1,2] => [5,4,3,2,1] => ([],5) => 0
{{1,3,4},{2},{5}} => [3,2,4,1,5] => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5) => 4
{{1,3,5},{2,4}} => [3,4,5,2,1] => [5,4,3,2,1] => ([],5) => 0
{{1,3},{2,4,5}} => [3,4,1,5,2] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 2
{{1,3},{2,4},{5}} => [3,4,1,2,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
{{1,3,5},{2},{4}} => [3,2,5,4,1] => [5,2,4,3,1] => ([(2,3),(2,4)],5) => 2
{{1,3},{2,5},{4}} => [3,5,1,4,2] => [5,4,3,2,1] => ([],5) => 0
{{1,3},{2},{4,5}} => [3,2,1,5,4] => [3,2,1,5,4] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 6
{{1,3},{2},{4},{5}} => [3,2,1,4,5] => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5) => 4
{{1,4,5},{2,3}} => [4,3,2,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 2
{{1,4},{2,3,5}} => [4,3,5,1,2] => [5,4,3,2,1] => ([],5) => 0
{{1,4},{2,3},{5}} => [4,3,2,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
{{1,5},{2,3,4}} => [5,3,4,2,1] => [5,4,3,2,1] => ([],5) => 0
{{1},{2,3,4,5}} => [1,3,4,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 4
{{1},{2,3,4},{5}} => [1,3,4,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 6
{{1,5},{2,3},{4}} => [5,3,2,4,1] => [5,4,3,2,1] => ([],5) => 0
{{1},{2,3,5},{4}} => [1,3,5,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 4
{{1},{2,3},{4,5}} => [1,3,2,5,4] => [1,3,2,5,4] => ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5) => 6
{{1},{2,3},{4},{5}} => [1,3,2,4,5] => [1,3,2,4,5] => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 5
{{1,4,5},{2},{3}} => [4,2,3,5,1] => [5,3,2,4,1] => ([(2,4),(3,4)],5) => 2
{{1,4},{2,5},{3}} => [4,5,3,1,2] => [5,4,3,2,1] => ([],5) => 0
{{1,4},{2},{3,5}} => [4,2,5,1,3] => [5,4,3,2,1] => ([],5) => 0
{{1,4},{2},{3},{5}} => [4,2,3,1,5] => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5) => 4
{{1,5},{2,4},{3}} => [5,4,3,2,1] => [5,4,3,2,1] => ([],5) => 0
{{1},{2,4,5},{3}} => [1,4,3,5,2] => [1,5,3,4,2] => ([(0,2),(0,3),(0,4),(4,1)],5) => 4
{{1},{2,4},{3,5}} => [1,4,5,2,3] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 4
{{1},{2,4},{3},{5}} => [1,4,3,2,5] => [1,4,3,2,5] => ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5) => 6
{{1,5},{2},{3,4}} => [5,2,4,3,1] => [5,4,3,2,1] => ([],5) => 0
{{1},{2,5},{3,4}} => [1,5,4,3,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 4
{{1},{2},{3,4,5}} => [1,2,4,5,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 4
{{1},{2},{3,4},{5}} => [1,2,4,3,5] => [1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 5
{{1,5},{2},{3},{4}} => [5,2,3,4,1] => [5,4,3,2,1] => ([],5) => 0
{{1},{2,5},{3},{4}} => [1,5,3,4,2] => [1,5,4,3,2] => ([(0,1),(0,2),(0,3),(0,4)],5) => 4
{{1},{2},{3,5},{4}} => [1,2,5,4,3] => [1,2,5,4,3] => ([(0,4),(4,1),(4,2),(4,3)],5) => 4
{{1},{2},{3},{4,5}} => [1,2,3,5,4] => [1,2,3,5,4] => ([(0,3),(3,4),(4,1),(4,2)],5) => 4
{{1},{2},{3},{4},{5}} => [1,2,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => 4

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$F_{2} = 1 + q$

$F_{3} = 2 + 3\ q^{2}$

$F_{4} = 4 + 2\ q + 7\ q^{3} + 2\ q^{4}$

$F_{5} = 13 + 9\ q^{2} + 20\ q^{4} + 2\ q^{5} + 8\ q^{6}$

Description

The number of cover relations in a poset.
Equivalently, this is also the number of edges in the Hasse diagram [1].

Map

Demazure product with inverse

Description

This map sends a permutation

$\pi$ to

$\pi^{-1} \star \pi$ where

$\star$ denotes the Demazure product on permutations.
This map is a surjection onto the set of involutions, i.e., the set of permutations

$\pi$ for which

$\pi = \pi^{-1}$ .

Map

to permutation

Description

Sends the set partition to the permutation obtained by considering the blocks as increasing cycles.

Map

permutation poset

Description

Sends a permutation to its permutation poset.
For a permutation

$\pi$ of length

$n$ , this poset has vertices

$\{ (i,\pi(i))\ :\ 1 \leq i \leq n \}$
and the cover relation is given by

$(w, x) \leq (y, z)$ if

$w \leq y$ and

$x \leq z$ .
For example, the permutation

$[3,1,5,4,2]$ is mapped to the poset with cover relations

$\{ (2, 1) \prec (5, 2),\ (2, 1) \prec (4, 4),\ (2, 1) \prec (3, 5),\ (1, 3) \prec (4, 4),\ (1, 3) \prec (3, 5) \}.$