- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000740: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[[5],[]] => [] => [] => [1] => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 1259 entries. <<<

search for individual values

searching the database for the individual values of this statistic

/ search for generating function

searching the database for statistics with the same generating function

click to show known generating functions

Description

The last entry of a permutation.
This statistic is undefined for the empty permutation.

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.

Map

Ringel

Description

The Ringel permutation of the LNakayama algebra corresponding to a Dyck path.