- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000018: Permutations ⟶ ℤ

Values

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

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

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

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

[[3],[]] => [] => [] => [] => 0

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

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

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

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

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

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

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

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

[[4],[]] => [] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[5],[]] => [] => [] => [] => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The number of inversions of a permutation.
This equals the minimal number of simple transpositions $(i,i+1)$ needed to write $\pi$. Thus, it is also the Coxeter length of $\pi$.

Map

to 132-avoiding permutation

Description

Sends a Dyck path to a 132-avoiding permutation.
This bijection is defined in [1, Section 2].

Map

to Dyck path

Description

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

Map

inner shape

Description

The inner shape of a skew partition.