- FindStat

Identifier

Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000620: Integer partitions ⟶ ℤ

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

>>> Load all 28094 entries. <<<

search for individual values

searching the database for the individual values of this statistic

Description

The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is odd.
The case of an even minimum is St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even..

Map

Simion-Schmidt map

Description

The Simion-Schmidt map sends any permutation to a $123$-avoiding permutation.
Details can be found in [1].
In particular, this is a bijection between $132$-avoiding permutations and $123$-avoiding permutations, see [1, Proposition 19].

Map

cycle type

Description

The cycle type of a permutation as a partition.

Map

first row removal

Description

Removes the first entry of an integer partition