- FindStat

Identifier

Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00236: Permutations —Clarke-Steingrimsson-Zeng inverse⟶ Permutations
St000868: Permutations ⟶ ℤ

Values

[1] => [1,0] => [1] => [1] => 0
[2] => [1,0,1,0] => [1,2] => [1,2] => 0
[1,1] => [1,1,0,0] => [2,1] => [2,1] => 1
[3] => [1,0,1,0,1,0] => [1,2,3] => [1,2,3] => 0
[2,1] => [1,0,1,1,0,0] => [1,3,2] => [1,3,2] => 1
[1,1,1] => [1,1,0,1,0,0] => [2,3,1] => [3,2,1] => 2
[4] => [1,0,1,0,1,0,1,0] => [1,2,3,4] => [1,2,3,4] => 0
[3,1] => [1,0,1,0,1,1,0,0] => [1,2,4,3] => [1,2,4,3] => 1
[2,2] => [1,1,1,0,0,0] => [3,2,1] => [2,3,1] => 1
[2,1,1] => [1,0,1,1,0,1,0,0] => [1,3,4,2] => [1,4,3,2] => 2
[1,1,1,1] => [1,1,0,1,0,1,0,0] => [2,3,4,1] => [4,3,2,1] => 3
[5] => [1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5] => [1,2,3,4,5] => 0
[4,1] => [1,0,1,0,1,0,1,1,0,0] => [1,2,3,5,4] => [1,2,3,5,4] => 1
[3,2] => [1,0,1,1,1,0,0,0] => [1,4,3,2] => [1,3,4,2] => 1
[3,1,1] => [1,0,1,0,1,1,0,1,0,0] => [1,2,4,5,3] => [1,2,5,4,3] => 2
[2,2,1] => [1,1,1,0,0,1,0,0] => [3,2,4,1] => [2,4,3,1] => 2
[2,1,1,1] => [1,0,1,1,0,1,0,1,0,0] => [1,3,4,5,2] => [1,5,4,3,2] => 3
[1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,0] => [2,3,4,5,1] => [5,4,3,2,1] => 4
[6] => [1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6] => [1,2,3,4,5,6] => 0
[5,1] => [1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,6,5] => [1,2,3,4,6,5] => 1
[4,2] => [1,0,1,0,1,1,1,0,0,0] => [1,2,5,4,3] => [1,2,4,5,3] => 1
[4,1,1] => [1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,5,6,4] => [1,2,3,6,5,4] => 2
[3,3] => [1,1,1,0,1,0,0,0] => [4,2,3,1] => [2,3,4,1] => 1
[3,2,1] => [1,0,1,1,1,0,0,1,0,0] => [1,4,3,5,2] => [1,3,5,4,2] => 2
[3,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,4,5,6,3] => [1,2,6,5,4,3] => 3
[2,2,2] => [1,1,1,1,0,0,0,0] => [4,3,2,1] => [3,2,4,1] => 3
[2,2,1,1] => [1,1,1,0,0,1,0,1,0,0] => [3,2,4,5,1] => [2,5,4,3,1] => 3
[2,1,1,1,1] => [1,0,1,1,0,1,0,1,0,1,0,0] => [1,3,4,5,6,2] => [1,6,5,4,3,2] => 4
[1,1,1,1,1,1] => [1,1,0,1,0,1,0,1,0,1,0,0] => [2,3,4,5,6,1] => [6,5,4,3,2,1] => 5
[7] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0] => [1,2,3,4,5,6,7] => [1,2,3,4,5,6,7] => 0
[6,1] => [1,0,1,0,1,0,1,0,1,0,1,1,0,0] => [1,2,3,4,5,7,6] => [1,2,3,4,5,7,6] => 1
[5,2] => [1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,6,5,4] => [1,2,3,5,6,4] => 1
[5,1,1] => [1,0,1,0,1,0,1,0,1,1,0,1,0,0] => [1,2,3,4,6,7,5] => [1,2,3,4,7,6,5] => 2
[4,3] => [1,0,1,1,1,0,1,0,0,0] => [1,5,3,4,2] => [1,3,4,5,2] => 1
[4,2,1] => [1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,5,4,6,3] => [1,2,4,6,5,3] => 2
[4,1,1,1] => [1,0,1,0,1,0,1,1,0,1,0,1,0,0] => [1,2,3,5,6,7,4] => [1,2,3,7,6,5,4] => 3
[3,3,1] => [1,1,1,0,1,0,0,1,0,0] => [4,2,3,5,1] => [2,3,5,4,1] => 2
[3,2,2] => [1,0,1,1,1,1,0,0,0,0] => [1,5,4,3,2] => [1,4,3,5,2] => 3
[3,2,1,1] => [1,0,1,1,1,0,0,1,0,1,0,0] => [1,4,3,5,6,2] => [1,3,6,5,4,2] => 3
[3,1,1,1,1] => [1,0,1,0,1,1,0,1,0,1,0,1,0,0] => [1,2,4,5,6,7,3] => [1,2,7,6,5,4,3] => 4
[2,2,2,1] => [1,1,1,1,0,0,0,1,0,0] => [4,3,2,5,1] => [3,2,5,4,1] => 4
[2,2,1,1,1] => [1,1,1,0,0,1,0,1,0,1,0,0] => [3,2,4,5,6,1] => [2,6,5,4,3,1] => 4
[6,2] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0] => [1,2,3,4,7,6,5] => [1,2,3,4,6,7,5] => 1
[5,3] => [1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,6,4,5,3] => [1,2,4,5,6,3] => 1
[5,2,1] => [1,0,1,0,1,0,1,1,1,0,0,1,0,0] => [1,2,3,6,5,7,4] => [1,2,3,5,7,6,4] => 2
[4,4] => [1,1,1,0,1,0,1,0,0,0] => [5,2,3,4,1] => [2,3,4,5,1] => 1
[4,3,1] => [1,0,1,1,1,0,1,0,0,1,0,0] => [1,5,3,4,6,2] => [1,3,4,6,5,2] => 2
[4,2,2] => [1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,6,5,4,3] => [1,2,5,4,6,3] => 3
[4,2,1,1] => [1,0,1,0,1,1,1,0,0,1,0,1,0,0] => [1,2,5,4,6,7,3] => [1,2,4,7,6,5,3] => 3
[3,3,2] => [1,1,1,0,1,1,0,0,0,0] => [5,2,4,3,1] => [2,4,3,5,1] => 3
[3,3,1,1] => [1,1,1,0,1,0,0,1,0,1,0,0] => [4,2,3,5,6,1] => [2,3,6,5,4,1] => 3
[3,2,2,1] => [1,0,1,1,1,1,0,0,0,1,0,0] => [1,5,4,3,6,2] => [1,4,3,6,5,2] => 4
[3,2,1,1,1] => [1,0,1,1,1,0,0,1,0,1,0,1,0,0] => [1,4,3,5,6,7,2] => [1,3,7,6,5,4,2] => 4
[2,2,2,2] => [1,1,1,1,0,1,0,0,0,0] => [5,3,4,2,1] => [4,3,2,5,1] => 5
[2,2,2,1,1] => [1,1,1,1,0,0,0,1,0,1,0,0] => [4,3,2,5,6,1] => [3,2,6,5,4,1] => 5
[6,3] => [1,0,1,0,1,0,1,1,1,0,1,0,0,0] => [1,2,3,7,5,6,4] => [1,2,3,5,6,7,4] => 1
[5,4] => [1,0,1,1,1,0,1,0,1,0,0,0] => [1,6,3,4,5,2] => [1,3,4,5,6,2] => 1
[5,3,1] => [1,0,1,0,1,1,1,0,1,0,0,1,0,0] => [1,2,6,4,5,7,3] => [1,2,4,5,7,6,3] => 2
[5,2,2] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0] => [1,2,3,7,6,5,4] => [1,2,3,6,5,7,4] => 3
[4,4,1] => [1,1,1,0,1,0,1,0,0,1,0,0] => [5,2,3,4,6,1] => [2,3,4,6,5,1] => 2
[4,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0] => [1,6,3,5,4,2] => [1,3,5,4,6,2] => 3
[4,3,1,1] => [1,0,1,1,1,0,1,0,0,1,0,1,0,0] => [1,5,3,4,6,7,2] => [1,3,4,7,6,5,2] => 3
[4,2,2,1] => [1,0,1,0,1,1,1,1,0,0,0,1,0,0] => [1,2,6,5,4,7,3] => [1,2,5,4,7,6,3] => 4
[3,3,3] => [1,1,1,1,1,0,0,0,0,0] => [5,4,3,2,1] => [3,4,2,5,1] => 4
[3,3,2,1] => [1,1,1,0,1,1,0,0,0,1,0,0] => [5,2,4,3,6,1] => [2,4,3,6,5,1] => 4
[3,2,2,2] => [1,0,1,1,1,1,0,1,0,0,0,0] => [1,6,4,5,3,2] => [1,5,4,3,6,2] => 5
[3,2,2,1,1] => [1,0,1,1,1,1,0,0,0,1,0,1,0,0] => [1,5,4,3,6,7,2] => [1,4,3,7,6,5,2] => 5
[2,2,2,2,1] => [1,1,1,1,0,1,0,0,0,1,0,0] => [5,3,4,2,6,1] => [4,3,2,6,5,1] => 6
[6,4] => [1,0,1,0,1,1,1,0,1,0,1,0,0,0] => [1,2,7,4,5,6,3] => [1,2,4,5,6,7,3] => 1
[5,5] => [1,1,1,0,1,0,1,0,1,0,0,0] => [6,2,3,4,5,1] => [2,3,4,5,6,1] => 1
[5,4,1] => [1,0,1,1,1,0,1,0,1,0,0,1,0,0] => [1,6,3,4,5,7,2] => [1,3,4,5,7,6,2] => 2
[5,3,2] => [1,0,1,0,1,1,1,0,1,1,0,0,0,0] => [1,2,7,4,6,5,3] => [1,2,4,6,5,7,3] => 3
[4,4,2] => [1,1,1,0,1,0,1,1,0,0,0,0] => [6,2,3,5,4,1] => [2,3,5,4,6,1] => 3
[4,3,3] => [1,0,1,1,1,1,1,0,0,0,0,0] => [1,6,5,4,3,2] => [1,4,5,3,6,2] => 4
[4,3,2,1] => [1,0,1,1,1,0,1,1,0,0,0,1,0,0] => [1,6,3,5,4,7,2] => [1,3,5,4,7,6,2] => 4
[4,2,2,2] => [1,0,1,0,1,1,1,1,0,1,0,0,0,0] => [1,2,7,5,6,4,3] => [1,2,6,5,4,7,3] => 5
[3,3,3,1] => [1,1,1,1,1,0,0,0,0,1,0,0] => [5,4,3,2,6,1] => [3,4,2,6,5,1] => 5
[3,3,2,2] => [1,1,1,0,1,1,0,1,0,0,0,0] => [6,2,4,5,3,1] => [2,5,4,3,6,1] => 5
[2,2,2,2,2] => [1,1,1,1,0,1,0,1,0,0,0,0] => [6,3,4,5,2,1] => [5,4,3,2,6,1] => 7
[6,5] => [1,0,1,1,1,0,1,0,1,0,1,0,0,0] => [1,7,3,4,5,6,2] => [1,3,4,5,6,7,2] => 1
[5,4,2] => [1,0,1,1,1,0,1,0,1,1,0,0,0,0] => [1,7,3,4,6,5,2] => [1,3,4,6,5,7,2] => 3
[5,3,3] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0] => [1,2,7,6,5,4,3] => [1,2,5,6,4,7,3] => 4
[4,4,3] => [1,1,1,0,1,1,1,0,0,0,0,0] => [6,2,5,4,3,1] => [2,4,5,3,6,1] => 4
[4,3,3,1] => [1,0,1,1,1,1,1,0,0,0,0,1,0,0] => [1,6,5,4,3,7,2] => [1,4,5,3,7,6,2] => 5
[4,3,2,2] => [1,0,1,1,1,0,1,1,0,1,0,0,0,0] => [1,7,3,5,6,4,2] => [1,3,6,5,4,7,2] => 5
[3,3,3,2] => [1,1,1,1,1,0,0,1,0,0,0,0] => [6,4,3,5,2,1] => [3,5,4,2,6,1] => 6
[5,4,3] => [1,0,1,1,1,0,1,1,1,0,0,0,0,0] => [1,7,3,6,5,4,2] => [1,3,5,6,4,7,2] => 4
[4,4,4] => [1,1,1,1,1,0,1,0,0,0,0,0] => [6,5,3,4,2,1] => [3,4,5,2,6,1] => 5
[4,3,3,2] => [1,0,1,1,1,1,1,0,0,1,0,0,0,0] => [1,7,5,4,6,3,2] => [1,4,6,5,3,7,2] => 6
[3,3,3,3] => [1,1,1,1,1,1,0,0,0,0,0,0] => [6,5,4,3,2,1] => [4,3,5,2,6,1] => 7
[5,4,4] => [1,0,1,1,1,1,1,0,1,0,0,0,0,0] => [1,7,6,4,5,3,2] => [1,4,5,6,3,7,2] => 5

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$F_{1} = 1$

$F_{2} = 1 + q$

$F_{3} = 1 + q + q^{2}$

$F_{4} = 1 + 2\ q + q^{2} + q^{3}$

$F_{5} = 1 + 2\ q + 2\ q^{2} + q^{3} + q^{4}$

$F_{6} = 1 + 3\ q + 2\ q^{2} + 3\ q^{3} + q^{4} + q^{5}$

Description

The aid statistic in the sense of Shareshian-Wachs.
This is the number of admissible inversions St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. plus the number of descents St000021The number of descents of a permutation.. This statistic was introduced by John Shareshian and Michelle L. Wachs in [1]. Theorem 4.1 states that the aid statistic together with the descent statistic is Euler-Mahonian.

Map

to non-crossing permutation

Description

Sends a Dyck path

$D$ with valley at positions

$\{(i_1,j_1),\ldots,(i_k,j_k)\}$ to the unique non-crossing permutation

$\pi$ having descents

$\{i_1,\ldots,i_k\}$ and whose inverse has descents

$\{j_1,\ldots,j_k\}$ .
It sends the area St000012The area of a Dyck path. to the number of inversions St000018The number of inversions of a permutation. and the major index St000027The major index of a Dyck path. to

$n(n-1)$ minus the sum of the major index St000004The major index of a permutation. and the inverse major index St000305The inverse major index of a permutation..

Map

parallelogram polyomino

Description

Return the Dyck path corresponding to the partition interpreted as a parallogram polyomino.
The Ferrers diagram of an integer partition can be interpreted as a parallogram polyomino, such that each part corresponds to a column.
This map returns the corresponding Dyck path.

Map

Clarke-Steingrimsson-Zeng inverse

Description

The inverse of the Clarke-Steingrimsson-Zeng map, sending excedances to descents.
This is the inverse of the map

$\Phi$ in [1, sec.3].