- FindStat

Identifier

Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00088: Permutations —Kreweras complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001879: Posets ⟶ ℤ

Values

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

search for individual values

searching the database for the individual values of this statistic

Description

The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.

Map

upper permutation

Description

The upper bound in the Grassmann interval corresponding to the decorated permutation.
Let

$I$ be the anti-exceedance set of a decorated permutation

$w$ . Let

$v$ be the

$k$ -Grassmannian permutation determined by

$v[k] = w^{-1}(I)$ and let

$u$ be the permutation satisfying

$u = wv$ . Then

$[u, v]$ is the Grassmann interval corresponding to

$w$ .
This map returns

$v$ .

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

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

Map

Kreweras complement

Description

Sends the permutation

$\pi \in \mathfrak{S}_n$ to the permutation

$\pi^{-1}c$ where

$c = (1,\ldots,n)$ is the long cycle.