- FindStat

Identifier

Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St000001: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,1,1,1,0,0,0,0] => [4,3,2,1] => 16

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => 16

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

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

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

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

[1,1,0,0,1,1,1,0,0,0] => [2,1,5,4,3] => 8

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

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

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

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

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

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

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

[1,1,0,1,1,0,1,0,0,0] => [2,4,5,3,1] => 9

[1,1,0,1,1,1,0,0,0,0] => [2,5,4,3,1] => 35

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

[1,1,1,0,0,0,1,1,0,0] => [3,2,1,5,4] => 8

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

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

[1,1,1,0,0,1,1,0,0,0] => [3,2,5,4,1] => 19

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

[1,1,1,0,1,0,0,1,0,0] => [3,4,2,5,1] => 9

[1,1,1,0,1,0,1,0,0,0] => [3,4,5,2,1] => 14

[1,1,1,0,1,1,0,0,0,0] => [3,5,4,2,1] => 70

[1,1,1,1,0,0,0,0,1,0] => [4,3,2,1,5] => 16

[1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => 35

[1,1,1,1,0,0,1,0,0,0] => [4,3,5,2,1] => 70

[1,1,1,1,0,1,0,0,0,0] => [4,5,3,2,1] => 168

[1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => 768

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

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

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

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

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

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

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

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

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

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

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

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

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

[1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => 16

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

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

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

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

[1,0,1,1,0,0,1,1,1,0,0,0] => [1,3,2,6,5,4] => 8

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

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

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

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

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

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

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

[1,0,1,1,0,1,1,0,1,0,0,0] => [1,3,5,6,4,2] => 9

[1,0,1,1,0,1,1,1,0,0,0,0] => [1,3,6,5,4,2] => 35

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

[1,0,1,1,1,0,0,0,1,1,0,0] => [1,4,3,2,6,5] => 8

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

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

[1,0,1,1,1,0,0,1,1,0,0,0] => [1,4,3,6,5,2] => 19

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

[1,0,1,1,1,0,1,0,0,1,0,0] => [1,4,5,3,6,2] => 9

[1,0,1,1,1,0,1,0,1,0,0,0] => [1,4,5,6,3,2] => 14

[1,0,1,1,1,0,1,1,0,0,0,0] => [1,4,6,5,3,2] => 70

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$F_{1} = q$

$F_{2} = 2\ q$

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

$F_{4} = 7\ q + 3\ q^{2} + 2\ q^{3} + q^{5} + q^{16}$

$F_{5} = 11\ q + 6\ q^{2} + 6\ q^{3} + 3\ q^{4} + 2\ q^{5} + 2\ q^{8} + 2\ q^{9} + q^{14} + 2\ q^{16} + q^{19} + 2\ q^{35} + 2\ q^{70} + q^{168} + q^{768}$

$F_{6} = 16\ q + 10\ q^{2} + 12\ q^{3} + 8\ q^{4} + 7\ q^{5} + 2\ q^{6} + 6\ q^{8} + 4\ q^{9} + 5\ q^{14} + 4\ q^{15} + 3\ q^{16} + 2\ q^{19} + 2\ q^{20} + 2\ q^{28} + 3\ q^{29} + 2\ q^{30} + 4\ q^{35} + q^{42} + 3\ q^{64} + 4\ q^{70} + q^{80} + 2\ q^{92} + 2\ q^{112} + 4\ q^{162} + 2\ q^{168} + 3\ q^{288} + 2\ q^{450} + 2\ q^{471} + 2\ q^{768} + 2\ q^{990} + q^{2112} + q^{2145} + 2\ q^{2310} + 2\ q^{5775} + 2\ q^{15015} + q^{48048} + q^{292864}$

Description

The number of reduced words for a permutation.
This is the number of ways to write a permutation as a minimal length product of simple transpositions. E.g., there are two reduced words for the permutation

$[3,2,1]$ , which are

$(1,2)(2,3)(1,2) = (2,3)(1,2)(2,3)$ .

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.).