- FindStat

Identifier

Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000649: Permutations ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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search for individual values

searching the database for the individual values of this statistic

Description

The number of 3-excedences of a permutation.
This is the number of positions $1\leq i\leq n$ such that $\sigma(i)=i+3$.

Map

to 321-avoiding permutation (Billey-Jockusch-Stanley)

Description

The Billey-Jockusch-Stanley bijection to 321-avoiding permutations.

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.