- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
St000213: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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$F_{1} = q$

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

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

$F_{4} = q + 6\ q^{2} + 6\ q^{3} + q^{4}$

$F_{5} = q + 10\ q^{2} + 21\ q^{3} + 9\ q^{4} + q^{5}$

$F_{6} = q + 16\ q^{2} + 53\ q^{3} + 49\ q^{4} + 12\ q^{5} + q^{6}$

Description

The number of weak exceedances (also weak excedences) of a permutation.
This is defined as

$\operatorname{wex}(\sigma)=\#\{i:\sigma(i) \geq i\}.$
The number of weak exceedances is given by the number of exceedances (see St000155The number of exceedances (also excedences) of a permutation.) plus the number of fixed points (see St000022The number of fixed points of a permutation.) of

$\sigma$ .

Map

inverse first fundamental transformation

Description

Let

$\sigma = (i_{11}\cdots i_{1k_1})\cdots(i_{\ell 1}\cdots i_{\ell k_\ell})$ be a permutation given by cycle notation such that every cycle starts with its maximal entry, and all cycles are ordered increasingly by these maximal entries.
Maps

$\sigma$ to the permutation

$[i_{11},\ldots,i_{1k_1},\ldots,i_{\ell 1},\ldots,i_{\ell k_\ell}]$ in one-line notation.
In other words, this map sends the maximal entries of the cycles to the left-to-right maxima, and the sequences between two left-to-right maxima are given by the cycles.

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.).