- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00310: Permutations —toric promotion⟶ Permutations
St000367: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = 1$

$F_{2} = 2$

$F_{3} = 4 + q$

$F_{4} = 11 + 3\ q$

$F_{5} = 27 + 9\ q + 2\ q^{2} + 3\ q^{3} + q^{4}$

$F_{6} = 68 + 23\ q + 9\ q^{2} + 13\ q^{3} + 10\ q^{4} + 5\ q^{5} + q^{6} + 2\ q^{7} + q^{8}$

Description

The number of simsun double descents of a permutation.
The restriction of a permutation

$\pi$ to

$[k] = \{1,\ldots,k\}$ is given in one-line notation by the subword of

$\pi$ of letters in

$[k]$ .
A simsun double descent of a permutation

$\pi$ is a double descent of any restriction of

$\pi$ to

$[1,\ldots,k]$ for some

$k$ . (Note here that the same double descent can appear in multiple restrictions!)

Map

toric promotion

Description

Toric promotion of a permutation.
Let

$\sigma\in\mathfrak S_n$ be a permutation and let

$\tau_{i, j}(\sigma) = \begin{cases} \sigma & \text{if $|\sigma^{-1}(i) - \sigma^{-1}(j)| = 1$}\\ (i, j)\circ\sigma & \text{otherwise}. \end{cases}$
The toric promotion operator is the product

$\tau_{n,1}\tau_{n-1,n}\dots\tau_{1,2}$ .
This is the special case of toric promotion on graphs for the path graph. Its order is

$n-1$ .

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.).