- FindStat

Identifier

Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001106: Posets ⟶ ℤ

Values

[1] => ([],1) => ([(0,1)],2) => ([(0,1)],2) => 1
[1,2] => ([(0,1)],2) => ([(0,2),(2,1)],3) => ([(0,2),(2,1)],3) => 1
[2,1] => ([],2) => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,2,3] => ([(0,2),(2,1)],3) => ([(0,3),(2,1),(3,2)],4) => ([(0,3),(2,1),(3,2)],4) => 1
[1,3,2] => ([(0,1),(0,2)],3) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => 2
[2,1,3] => ([(0,2),(1,2)],3) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => ([(0,2),(0,3),(2,4),(3,4),(4,1)],5) => 2
[2,3,1] => ([(1,2)],3) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[3,1,2] => ([(1,2)],3) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6) => 3
[1,2,3,4] => ([(0,3),(2,1),(3,2)],4) => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,4),(2,3),(3,1),(4,2)],5) => 1
[1,2,4,3] => ([(0,3),(3,1),(3,2)],4) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6) => 2
[1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6) => 2
[2,1,3,4] => ([(0,3),(1,3),(3,2)],4) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6) => 2
[2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7) => 4
[1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6) => 1
[1,2,4,3,5] => ([(0,3),(1,4),(2,4),(3,1),(3,2)],5) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => ([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7) => 2

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$F_{1} = q$

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

Description

The number of supergreedy linear extensions of a poset.
A linear extension of a poset P with elements

$\{x_1,\dots,x_n\}$ is supergreedy, if it can be obtained by the following algorithm:

Step 1. Choose a minimal element $x_1$ .
Step 2. Suppose $X=\{x_1,\dots,x_i\}$ have been chosen, let $M$ be the set of minimal elements of $P\setminus X$ . If there is an element of $M$ which covers an element $x_j$ in $X$ , then let $x_{i+1}$ be one of these such that $j$ is maximal; otherwise, choose $x_{i+1}$ to be any element of $M$ .

This statistic records the number of supergreedy linear extensions.

Map

order ideals

Description

The lattice of order ideals of a poset.
An order ideal

$\mathcal I$ in a poset

$P$ is a downward closed set, i.e.,

$a \in \mathcal I$ and

$b \leq a$ implies

$b \in \mathcal I$ . This map sends a poset to the lattice of all order ideals sorted by inclusion with meet being intersection and join being union.

Map

to poset

Description

Return the poset corresponding to the lattice.

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

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