- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00239: Permutations —Corteel⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001881: Lattices ⟶ ℤ

Values

[1,0] => [1] => [1] => ([(0,1)],2) => 1
[1,0,1,0] => [1,2] => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,1,0,0] => [2,1] => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4) => 2
[1,0,1,0,1,0] => [1,2,3] => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,0,1,1,0,0] => [1,3,2] => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,1,0,0,1,0] => [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,1,0,1,0,0] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7) => 1
[1,1,1,0,0,0] => [3,2,1] => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6) => 1
[1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9) => 1
[1,0,1,1,0,0,1,0] => [1,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9) => 1
[1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9) => 1
[1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[1,1,0,0,1,0,1,0] => [2,1,3,4] => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9) => 1
[1,1,0,0,1,1,0,0] => [2,1,4,3] => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8) => 1
[1,1,0,1,0,0,1,0] => [2,3,1,4] => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9) => 1
[1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9) => 1
[1,1,0,1,1,0,0,0] => [2,4,3,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[1,1,1,0,0,0,1,0] => [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8) => 1
[1,1,1,0,1,0,0,0] => [3,4,2,1] => [4,3,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9) => 1
[1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,4,1,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8) => 1
[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => [5,2,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 1
[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => [4,2,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,7),(5,6),(7,6)],8) => 1
[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7) => 1
[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => [1,5,3,6,2,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 1
[1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [1,4,6,2,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 1
[1,1,0,1,0,1,1,1,0,0,0,0] => [2,3,6,5,4,1] => [5,2,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 1
[1,1,0,1,1,0,1,0,0,1,0,0] => [2,4,5,3,6,1] => [6,2,4,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 1
[1,1,0,1,1,1,0,0,0,0,1,0] => [2,5,4,3,1,6] => [4,2,5,1,3,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 1
[1,1,0,1,1,1,0,0,0,1,0,0] => [2,5,4,3,6,1] => [4,2,6,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1
[1,1,0,1,1,1,0,0,1,0,0,0] => [2,5,4,6,3,1] => [5,2,6,4,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1
[1,1,0,1,1,1,0,1,0,0,0,0] => [2,5,6,4,3,1] => [5,2,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1
[1,1,0,1,1,1,1,0,0,0,0,0] => [2,6,5,4,3,1] => [4,2,5,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 1
[1,1,1,0,0,1,1,0,1,0,0,0] => [3,2,5,6,4,1] => [2,6,3,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1
[1,1,1,0,0,1,1,1,0,0,0,0] => [3,2,6,5,4,1] => [2,5,3,6,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1
[1,1,1,0,1,0,0,1,1,0,0,0] => [3,4,2,6,5,1] => [5,3,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1
[1,1,1,1,0,0,0,1,0,0,1,0] => [4,3,2,5,1,6] => [3,5,1,4,2,6] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(8,7)],9) => 1
[1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [3,6,1,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 1
[1,1,1,1,0,0,0,1,1,0,0,0] => [4,3,2,6,5,1] => [3,5,1,4,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1
[1,1,1,1,0,0,1,0,0,1,0,0] => [4,3,5,2,6,1] => [4,6,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8) => 1
[1,1,1,1,0,1,0,0,0,1,0,0] => [4,5,3,2,6,1] => [4,3,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 1
[1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [3,4,6,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9) => 1
[1,1,0,1,1,0,1,0,0,1,1,0,0,0] => [2,4,5,3,7,6,1] => [6,2,4,1,5,7,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,0,1,1,1,0,0,0,1,1,0,0,0] => [2,5,4,3,7,6,1] => [4,2,6,1,5,7,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,0,1,1,1,0,0,1,0,0,1,0,0] => [2,5,4,6,3,7,1] => [5,2,7,4,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,0,1,1,1,0,0,1,1,0,0,0,0] => [2,5,4,7,6,3,1] => [5,2,7,4,6,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,0,1,1,1,0,1,0,0,0,1,0,0] => [2,5,6,4,3,7,1] => [5,2,4,7,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,0,1,1,1,0,1,0,0,1,0,0,0] => [2,5,6,4,7,3,1] => [6,2,4,7,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,0,1,1,1,1,0,0,0,0,1,0,0] => [2,6,5,4,3,7,1] => [4,2,5,7,1,6,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,0,0,1,1,0,1,0,0,1,0,0] => [3,2,5,6,4,7,1] => [2,7,3,5,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,0,0,1,1,1,0,0,0,1,0,0] => [3,2,6,5,4,7,1] => [2,5,3,7,1,6,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,0,0,1,1,1,0,0,1,0,0,0] => [3,2,6,5,7,4,1] => [2,6,3,7,5,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,0,0,1,1,1,0,1,0,0,0,0] => [3,2,6,7,5,4,1] => [2,6,3,5,7,1,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,0,1,0,0,1,1,1,0,0,0,0] => [3,4,2,7,6,5,1] => [6,3,1,4,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,1,0,0,0,1,1,0,1,0,0,0] => [4,3,2,6,7,5,1] => [3,7,1,4,6,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,1,0,0,0,1,1,1,0,0,0,0] => [4,3,2,7,6,5,1] => [3,6,1,4,7,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,1,0,0,1,0,0,1,1,0,0,0] => [4,3,5,2,7,6,1] => [4,6,3,1,5,7,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,1,0,0,1,1,0,0,0,1,0,0] => [4,3,6,5,2,7,1] => [4,7,3,5,1,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,1,0,1,0,0,1,0,0,1,0,0] => [4,5,3,6,2,7,1] => [5,3,7,4,1,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1
[1,1,1,1,1,0,0,0,1,0,0,1,0,0] => [5,4,3,6,2,7,1] => [3,5,7,4,1,6,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9) => 1

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$F_{1} = q$

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

$F_{3} = 5\ q$

Description

The number of factors of a lattice as a Cartesian product of lattices.
Since the cardinality of a lattice is the product of the cardinalities of its factors, this statistic is one whenever the cardinality of the lattice is prime.

Map

lattice of intervals

Description

The lattice of intervals of a permutation.
An interval of a permutation

$\pi$ is a possibly empty interval of values that appear in consecutive positions of

$\pi$ . The lattice of intervals of

$\pi$ has as elements the intervals of

$\pi$ , ordered by set inclusion.

Map

to 312-avoiding permutation

Description

Sends a Dyck path to the 312-avoiding permutation according to Bandlow-Killpatrick.
This map is defined in [1] and sends the area (St000012The area of a Dyck path.) to the inversion number (St000018The number of inversions of a permutation.).

Map

Corteel

Description

Corteel's map interchanging the number of crossings and the number of nestings of a permutation.
This involution creates a labelled bicoloured Motzkin path, using the Foata-Zeilberger map. In the corresponding bump diagram, each label records the number of arcs nesting the given arc. Then each label is replaced by its complement, and the inverse of the Foata-Zeilberger map is applied.