- FindStat

Identifier

Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000242: Permutations ⟶ ℤ

Values

[1,0] => [1,0] => [1] => 0

[1,0,1,0] => [1,1,0,0] => [2,1] => 0

[1,1,0,0] => [1,0,1,0] => [1,2] => 0

[1,0,1,0,1,0] => [1,1,1,0,0,0] => [3,2,1] => 1

[1,0,1,1,0,0] => [1,1,0,0,1,0] => [2,1,3] => 1

[1,1,0,0,1,0] => [1,0,1,1,0,0] => [1,3,2] => 1

[1,1,0,1,0,0] => [1,1,0,1,0,0] => [2,3,1] => 0

[1,1,1,0,0,0] => [1,0,1,0,1,0] => [1,2,3] => 0

[1,0,1,0,1,0,1,0] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => 2

[1,0,1,0,1,1,0,0] => [1,1,1,0,0,0,1,0] => [3,2,1,4] => 2

[1,0,1,1,0,0,1,0] => [1,1,0,0,1,1,0,0] => [2,1,4,3] => 2

[1,0,1,1,0,1,0,0] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => 1

[1,0,1,1,1,0,0,0] => [1,1,0,0,1,0,1,0] => [2,1,3,4] => 1

[1,1,0,0,1,0,1,0] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => 2

[1,1,0,0,1,1,0,0] => [1,0,1,1,0,0,1,0] => [1,3,2,4] => 1

[1,1,0,1,0,0,1,0] => [1,1,0,1,1,0,0,0] => [2,4,3,1] => 1

[1,1,0,1,0,1,0,0] => [1,1,1,0,1,0,0,0] => [3,4,2,1] => 3

[1,1,0,1,1,0,0,0] => [1,1,0,1,0,0,1,0] => [2,3,1,4] => 1

[1,1,1,0,0,0,1,0] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => 1

[1,1,1,0,0,1,0,0] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => 1

[1,1,1,0,1,0,0,0] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => 0

[1,1,1,1,0,0,0,0] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => 0

[1,0,1,0,1,0,1,0,1,0] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 3

[1,0,1,0,1,0,1,1,0,0] => [1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 3

[1,0,1,0,1,1,0,0,1,0] => [1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 3

[1,0,1,0,1,1,0,1,0,0] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 2

[1,0,1,0,1,1,1,0,0,0] => [1,1,1,0,0,0,1,0,1,0] => [3,2,1,4,5] => 2

[1,0,1,1,0,0,1,0,1,0] => [1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 3

[1,0,1,1,0,0,1,1,0,0] => [1,1,0,0,1,1,0,0,1,0] => [2,1,4,3,5] => 2

[1,0,1,1,0,1,0,0,1,0] => [1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 2

[1,0,1,1,0,1,0,1,0,0] => [1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 3

[1,0,1,1,0,1,1,0,0,0] => [1,1,1,0,0,1,0,0,1,0] => [3,2,4,1,5] => 2

[1,0,1,1,1,0,0,0,1,0] => [1,1,0,0,1,0,1,1,0,0] => [2,1,3,5,4] => 2

[1,0,1,1,1,0,0,1,0,0] => [1,1,0,0,1,1,0,1,0,0] => [2,1,4,5,3] => 2

[1,0,1,1,1,0,1,0,0,0] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => 1

[1,0,1,1,1,1,0,0,0,0] => [1,1,0,0,1,0,1,0,1,0] => [2,1,3,4,5] => 1

[1,1,0,0,1,0,1,0,1,0] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 3

[1,1,0,0,1,0,1,1,0,0] => [1,0,1,1,1,0,0,0,1,0] => [1,4,3,2,5] => 2

[1,1,0,0,1,1,0,0,1,0] => [1,0,1,1,0,0,1,1,0,0] => [1,3,2,5,4] => 2

[1,1,0,0,1,1,0,1,0,0] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => 2

[1,1,0,0,1,1,1,0,0,0] => [1,0,1,1,0,0,1,0,1,0] => [1,3,2,4,5] => 1

[1,1,0,1,0,0,1,0,1,0] => [1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 2

[1,1,0,1,0,0,1,1,0,0] => [1,1,0,1,1,0,0,0,1,0] => [2,4,3,1,5] => 2

[1,1,0,1,0,1,0,0,1,0] => [1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 3

[1,1,0,1,0,1,0,1,0,0] => [1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 3

[1,1,0,1,0,1,1,0,0,0] => [1,1,1,0,1,0,0,0,1,0] => [3,4,2,1,5] => 4

[1,1,0,1,1,0,0,0,1,0] => [1,1,0,1,0,0,1,1,0,0] => [2,3,1,5,4] => 2

[1,1,0,1,1,0,0,1,0,0] => [1,1,0,1,1,0,0,1,0,0] => [2,4,3,5,1] => 1

[1,1,0,1,1,0,1,0,0,0] => [1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 3

[1,1,0,1,1,1,0,0,0,0] => [1,1,0,1,0,0,1,0,1,0] => [2,3,1,4,5] => 1

[1,1,1,0,0,0,1,0,1,0] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => 2

[1,1,1,0,0,0,1,1,0,0] => [1,0,1,0,1,1,0,0,1,0] => [1,2,4,3,5] => 1

[1,1,1,0,0,1,0,0,1,0] => [1,0,1,1,0,1,1,0,0,0] => [1,3,5,4,2] => 2

[1,1,1,0,0,1,0,1,0,0] => [1,0,1,1,1,0,1,0,0,0] => [1,4,5,3,2] => 4

[1,1,1,0,0,1,1,0,0,0] => [1,0,1,1,0,1,0,0,1,0] => [1,3,4,2,5] => 1

[1,1,1,0,1,0,0,0,1,0] => [1,1,0,1,0,1,1,0,0,0] => [2,3,5,4,1] => 1

[1,1,1,0,1,0,0,1,0,0] => [1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 3

[1,1,1,0,1,0,1,0,0,0] => [1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 4

[1,1,1,0,1,1,0,0,0,0] => [1,1,0,1,0,1,0,0,1,0] => [2,3,4,1,5] => 1

[1,1,1,1,0,0,0,0,1,0] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => 1

[1,1,1,1,0,0,0,1,0,0] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => 1

[1,1,1,1,0,0,1,0,0,0] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => 1

[1,1,1,1,0,1,0,0,0,0] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => 0

[1,1,1,1,1,0,0,0,0,0] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

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

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

$F_{5} = 2 + 12\ q + 15\ q^{2} + 10\ q^{3} + 3\ q^{4}$

$F_{6} = 2 + 18\ q + 37\ q^{2} + 36\ q^{3} + 31\ q^{4} + 8\ q^{5}$

Description

The number of indices that are not cyclical small weak excedances.
A cyclical small weak excedance is an index

$i$ such that

$\pi_i \in \{ i,i+1 \}$ considered cyclically.

Map

Lalanne-Kreweras involution

Description

The Lalanne-Kreweras involution on Dyck paths.
Label the upsteps from left to right and record the labels on the first up step of each double rise. Do the same for the downsteps. Then form the Dyck path whose ascent lengths and descent lengths are the consecutives differences of the labels.

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.).