- FindStat

Identifier

Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001685: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[4,4,2] => [2,2,2,2,1,1] => [1,0,1,1,0,1,1,1,1,0,0,0,0,0] => [1,3,7,6,5,4,2] => 2

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

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

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

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

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

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

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

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

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

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

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

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

[6,2,1,1,1] => [3,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,4,5,7,6,3,2] => 3

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

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

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

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

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

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

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

[4,4,1,1,1] => [2,2,2,2,2,1] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => 2

>>> Load all 209 entries. <<<

[4,3,2,1,1] => [3,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,1,0,0,0,0] => [1,3,6,7,5,4,2] => 2

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

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

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

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

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

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

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

[6,4,2] => [3,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,1,0,0,0,0] => [1,3,5,7,6,4,2] => 3

[6,3,2,1] => [3,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,1,0,0,0,0] => [1,4,3,7,6,5,2] => 3

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

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

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

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

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

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

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

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

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

[4,4,3,1] => [3,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,6,7,5,4,3] => 2

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

[4,2,1,1,1,1,1,1] => [4,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,1,0,0,0] => [1,3,6,5,7,4,2] => 2

[3,2,1,1,1,1,1,1,1] => [4,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,4,5,6,7,3,2] => 1

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

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

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

[8,2,1,1,1] => [4,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,1,0,0,0] => [1,4,5,3,7,6,2] => 4

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

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

[6,4,1,1,1] => [3,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,5,7,6,4,3] => 3

[6,3,2,1,1] => [3,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,1,0,0,0,0] => [1,3,4,7,6,5,2] => 3

[6,2,1,1,1,1,1] => [4,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,1,0,0,0] => [1,4,3,6,7,5,2] => 3

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

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

[5,4,2,1,1] => [5,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,1,0,0] => [1,3,6,5,4,7,2] => 2

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

[5,3,2,1,1,1] => [5,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,4,5,6,3,7,2] => 1

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

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

[4,4,1,1,1,1,1] => [4,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,1,0,0,0] => [1,2,6,5,7,4,3] => 2

[4,3,3,2,1] => [6,2,2,1,1,1] => [1,0,1,1,1,0,1,1,0,0,0,0,1,0] => [1,4,6,5,3,2,7] => 2

[4,3,2,1,1,1,1] => [4,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,1,0,0,0] => [1,3,5,6,7,4,2] => 2

[8,4,2] => [4,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,1,0,0,0] => [1,3,5,4,7,6,2] => 4

[8,3,2,1] => [4,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,1,0,0,0] => [1,4,3,5,7,6,2] => 4

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

[6,6,2] => [3,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0] => [1,3,2,7,6,5,4] => 3

[6,5,2,1] => [5,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,1,0,0] => [1,4,3,6,5,7,2] => 3

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

[6,4,3,1] => [3,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,1,0,0,0,0] => [1,2,4,7,6,5,3] => 3

[6,2,1,1,1,1,1,1] => [4,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,1,0,0,0] => [1,3,4,6,7,5,2] => 3

[5,4,4,1] => [5,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => 2

[5,4,3,2] => [5,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,1,0,0] => [1,3,5,6,4,7,2] => 2

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

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

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

[5,2,1,1,1,1,1,1,1] => [5,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,4,5,3,6,7,2] => 1

[4,3,3,2,1,1] => [6,2,2,2,1,1] => [1,0,1,1,0,1,1,1,0,0,0,0,1,0] => [1,3,6,5,4,2,7] => 2

[4,3,1,1,1,1,1,1,1] => [4,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,5,6,7,4,3] => 2

[3,3,3,2,1,1,1] => [6,3,2,1,1,1] => [1,0,1,1,1,0,1,0,1,0,0,0,1,0] => [1,4,5,6,3,2,7] => 1

[10,2,1,1,1] => [5,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,1,0,0] => [1,4,5,3,2,7,6] => 5

[8,4,1,1,1] => [4,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,1,0,0,0] => [1,2,5,4,7,6,3] => 4

[8,3,2,1,1] => [4,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,1,0,0,0] => [1,3,4,5,7,6,2] => 4

[8,2,1,1,1,1,1] => [4,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0] => [1,4,3,2,7,6,5] => 4

[6,6,1,1,1] => [3,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => 3

[6,5,2,1,1] => [5,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,1,0,0] => [1,3,4,6,5,7,2] => 3

[6,4,1,1,1,1,1] => [4,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,1,0,0,0] => [1,2,4,6,7,5,3] => 3

[6,3,3,2,1] => [6,3,3,1,1,1] => [1,0,1,1,1,0,0,1,1,0,0,0,1,0] => [1,4,3,6,5,2,7] => 3

[6,3,2,1,1,1,1] => [4,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,1,0,0,0] => [1,3,2,6,7,5,4] => 3

[5,4,3,1,1,1] => [5,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,5,6,4,7,3] => 2

[5,4,2,1,1,1,1] => [5,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,1,0,0] => [1,3,5,4,6,7,2] => 2

[5,3,2,1,1,1,1,1] => [5,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,2] => 1

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

[4,4,3,3,1] => [6,2,2,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0] => [1,2,6,5,4,3,7] => 2

[4,3,3,3,2] => [6,3,2,2,1,1] => [1,0,1,1,0,1,1,0,1,0,0,0,1,0] => [1,3,5,6,4,2,7] => 2

[3,3,2,1,1,1,1,1,1,1] => [6,4,2,1,1,1] => [1,0,1,1,1,0,1,0,0,1,0,0,1,0] => [1,4,5,3,6,2,7] => 1

[10,4,2] => [5,5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,1,0,0] => [1,3,5,4,2,7,6] => 5

[10,3,2,1] => [5,5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,1,0,0] => [1,4,3,5,2,7,6] => 5

[8,6,2] => [4,4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,1,0,0,0] => [1,3,2,5,7,6,4] => 4

[8,5,2,1] => [5,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,1,0,0] => [1,4,3,2,6,7,5] => 4

[8,4,3,1] => [4,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,1,0,0,0] => [1,2,4,5,7,6,3] => 4

[8,2,1,1,1,1,1,1] => [4,4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,1,0,0,0] => [1,3,4,2,7,6,5] => 4

[6,5,4,1] => [5,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,1,0,0] => [1,2,4,6,5,7,3] => 3

[6,5,3,2] => [5,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,1,0,0] => [1,3,2,6,5,7,4] => 3

[6,3,3,2,1,1] => [6,3,3,2,1,1] => [1,0,1,1,0,1,0,1,1,0,0,0,1,0] => [1,3,4,6,5,2,7] => 3

[6,3,1,1,1,1,1,1,1] => [4,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,6,7,5,4] => 3

[5,4,1,1,1,1,1,1,1] => [5,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => 2

[5,3,3,2,1,1,1] => [6,5,2,1,1,1] => [1,0,1,1,1,0,1,0,0,0,1,0,1,0] => [1,4,5,3,2,6,7] => 1

[5,3,2,1,1,1,1,1,1] => [5,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,7,2] => 1

[4,3,3,3,1,1,1] => [6,3,2,2,2,1] => [1,0,1,0,1,1,1,0,1,0,0,0,1,0] => [1,2,5,6,4,3,7] => 2

[4,3,3,2,1,1,1,1] => [6,4,2,2,1,1] => [1,0,1,1,0,1,1,0,0,1,0,0,1,0] => [1,3,5,4,6,2,7] => 2

[3,3,3,2,1,1,1,1,1] => [6,4,3,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0,1,0] => [1,4,3,5,6,2,7] => 1

[10,4,1,1,1] => [5,5,2,2,2,1] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0] => [1,2,5,4,3,7,6] => 5

[10,3,2,1,1] => [5,5,3,2,1,1] => [1,0,1,1,0,1,0,1,0,0,1,1,0,0] => [1,3,4,5,2,7,6] => 5

[10,2,1,1,1,1,1] => [5,5,4,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0] => [1,4,3,2,5,7,6] => 5

[8,6,1,1,1] => [4,4,3,3,2,1] => [1,0,1,0,1,0,1,1,0,1,1,0,0,0] => [1,2,3,5,7,6,4] => 4

[8,5,2,1,1] => [5,4,4,2,1,1] => [1,0,1,1,0,1,0,0,1,1,0,1,0,0] => [1,3,4,2,6,7,5] => 4

[8,4,1,1,1,1,1] => [4,4,4,2,2,1] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0] => [1,2,4,3,7,6,5] => 4

[8,3,3,2,1] => [6,4,4,1,1,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0] => [1,4,3,2,6,5,7] => 4

[8,3,2,1,1,1,1] => [4,4,4,3,1,1] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0] => [1,3,2,4,7,6,5] => 4

[6,5,3,1,1,1] => [5,3,3,3,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => 3

[6,5,2,1,1,1,1] => [5,4,3,3,1,1] => [1,0,1,1,0,0,1,1,0,1,0,1,0,0] => [1,3,2,5,6,7,4] => 3

[6,4,3,3,1] => [6,3,3,2,2,1] => [1,0,1,0,1,1,0,1,1,0,0,0,1,0] => [1,2,4,6,5,3,7] => 3

[6,3,3,3,2] => [6,3,3,3,1,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0] => [1,3,2,6,5,4,7] => 3

[5,4,3,3,2] => [6,5,2,2,1,1] => [1,0,1,1,0,1,1,0,0,0,1,0,1,0] => [1,3,5,4,2,6,7] => 2

[5,4,3,1,1,1,1,1] => [5,4,3,2,2,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => 2

[5,3,3,3,2,1] => [6,5,3,1,1,1] => [1,0,1,1,1,0,0,1,0,0,1,0,1,0] => [1,4,3,5,2,6,7] => 1

[4,3,3,1,1,1,1,1,1,1] => [6,4,2,2,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0,1,0] => [1,2,5,4,6,3,7] => 2

[3,3,3,2,1,1,1,1,1,1] => [6,4,3,2,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0,1,0] => [1,3,4,5,6,2,7] => 1

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 2 + q$

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

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

Description

The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation.

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.).

Map

Glaisher-Franklin

Description

The Glaisher-Franklin bijection on integer partitions.
This map sends the set of even part sizes, each divided by two, to the set of repeated part sizes, see [1, 3.3.1].

Map

to Dyck path

Description

Sends a partition to the shortest Dyck path tracing the shape of its Ferrers diagram.