- FindStat

Identifier

Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
St001874: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 1 + q$

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

$F_{4} = 6 + 14\ q + 4\ q^{2}$

$F_{5} = 24 + 60\ q + 36\ q^{2}$

$F_{6} = 120 + 312\ q + 252\ q^{2} + 36\ q^{3}$

Description

Lusztig's a-function for the symmetric group.
Let

$x$ be a permutation corresponding to the pair of tableaux

$(P(x),Q(x))$
by the Robinson-Schensted correspondence and

$\operatorname{shape}(Q(x)')=( \lambda_1,...,\lambda_k)$
where

$Q(x)'$ is the transposed tableau.
Then

$a(x)=\sum\limits_{i=1}^{k}{\binom{\lambda_i}{2}}$ .
See exercise 10 on page 198 in the book by Björner and Brenti "Combinatorics of Coxeter Groups" for equivalent characterisations and properties.

Map

to 321-avoiding permutation

Description

Sends a Dyck path to a 321-avoiding permutation.
This bijection defined in [3, pp. 60] and in [2, Section 3.1].
It is shown in [1] that it sends the number of centered tunnels to the number of fixed points, the number of right tunnels to the number of exceedences, and the semilength plus the height of the middle point to 2 times the length of the longest increasing subsequence.

Map

left-to-right-maxima to Dyck path

Description

The left-to-right maxima of a permutation as a Dyck path.
Let

$(c_1, \dots, c_k)$ be the rise composition Mp00102rise composition of the path. Then the corresponding left-to-right maxima are

$c_1, c_1+c_2, \dots, c_1+\dots+c_k$ .
Restricted to 321-avoiding permutations, this is the inverse of Mp00119to 321-avoiding permutation (Krattenthaler), restricted to 312-avoiding permutations, this is the inverse of Mp00031to 312-avoiding permutation.