- FindStat

Identifier

Mp00056: Parking functions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤ (values match St000320The dinv adjustment of an integer partition.)

Values

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.

Map

cycle type

Description

The cycle type of a permutation as a partition.

Map

to Dyck path

Description

Sends a parking function to its support Dyck word.

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.).