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Your data matches 266 different statistics following compositions of up to 3 maps.
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Matching statistic: St000475
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St000475: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 0
[1,1]
=> 2
[3]
=> 0
[2,1]
=> 1
[1,1,1]
=> 3
[4]
=> 0
[3,1]
=> 1
[2,2]
=> 0
[2,1,1]
=> 2
[1,1,1,1]
=> 4
[5]
=> 0
[4,1]
=> 1
[3,2]
=> 0
[3,1,1]
=> 2
[2,2,1]
=> 1
[2,1,1,1]
=> 3
[1,1,1,1,1]
=> 5
[6]
=> 0
[5,1]
=> 1
[4,2]
=> 0
[4,1,1]
=> 2
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 3
[2,2,2]
=> 0
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 4
[1,1,1,1,1,1]
=> 6
Description
The number of parts equal to 1 in a partition.
Matching statistic: St000674
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000674: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> 2
[1,1]
=> [1,1,0,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
Description
The number of hills of a Dyck path.
A hill is a peak with up step starting and down step ending at height zero.
Matching statistic: St000022
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000022: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => 2
[1,1]
=> [1,1,0,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
Description
The number of fixed points of a permutation.
Matching statistic: St000117
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,0]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 0
Description
The number of centered tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.
Matching statistic: St000221
(load all 52 compositions to match this statistic)
(load all 52 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000221: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => 2
[1,1]
=> [1,1,0,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
Description
The number of strong fixed points of a permutation.
$i$ is called a strong fixed point of $\pi$ if
1. $j < i$ implies $\pi_j < \pi_i$, and
2. $j > i$ implies $\pi_j > \pi_i$
This can be described as an occurrence of the mesh pattern ([1], {(0,1),(1,0)}), i.e., the upper left and the lower right quadrants are shaded, see [3].
The generating function for the joint-distribution (RLmin, LRmax, strong fixed points) has a continued fraction expression as given in [4, Lemma 3.2], for LRmax see [[St000314]].
Matching statistic: St000241
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000241: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => 0
[1,1]
=> [1,1,0,0]
=> [2,1] => 2
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 3
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,1,2] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 2
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 4
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,2,3] => 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,4,2] => 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 3
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 5
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [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
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [3,4,1,2] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 3
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,1,2,3] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 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
Description
The number of cyclical small excedances.
A cyclical small excedance is an index $i$ such that $\pi_i = i+1$ considered cyclically.
Matching statistic: St000247
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00284: Standard tableaux —rows⟶ Set partitions
St000247: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [[1,2]]
=> {{1,2}}
=> 0
[1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2
[3]
=> [[1,2,3]]
=> {{1,2,3}}
=> 0
[2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 3
[4]
=> [[1,2,3,4]]
=> {{1,2,3,4}}
=> 0
[3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0
[2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 2
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 4
[5]
=> [[1,2,3,4,5]]
=> {{1,2,3,4,5}}
=> 0
[4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 1
[3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 0
[3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 2
[2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 3
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 5
[6]
=> [[1,2,3,4,5,6]]
=> {{1,2,3,4,5,6}}
=> 0
[5,1]
=> [[1,3,4,5,6],[2]]
=> {{1,3,4,5,6},{2}}
=> 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> {{1,2,5,6},{3,4}}
=> 0
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> {{1,4,5,6},{2},{3}}
=> 2
[3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> {{1,3,6},{2,5},{4}}
=> 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> {{1,5,6},{2},{3},{4}}
=> 3
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 2
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 4
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 6
Description
The number of singleton blocks of a set partition.
Matching statistic: St000873
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000873: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [1,2] => 2
[1,1]
=> [1,1,0,0]
=> [2,1] => 0
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => 3
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => 0
Description
The aix statistic of a permutation.
According to [1], this statistic on finite strings $\pi$ of integers is given as follows: let $m$ be the leftmost occurrence of the minimal entry and let $\pi = \alpha\ m\ \beta$. Then
$$
\operatorname{aix}\pi = \begin{cases}
\operatorname{aix}\alpha & \text{ if } \alpha,\beta \neq \emptyset \\
1 + \operatorname{aix}\beta & \text{ if } \alpha = \emptyset \\
0 & \text{ if } \beta = \emptyset
\end{cases}\ .
$$
Matching statistic: St000877
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00105: Binary words —complement⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00105: Binary words —complement⟶ Binary words
St000877: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> 0 => 1 => 0
[1,1]
=> 11 => 00 => 2
[3]
=> 1 => 0 => 1
[2,1]
=> 01 => 10 => 0
[1,1,1]
=> 111 => 000 => 3
[4]
=> 0 => 1 => 0
[3,1]
=> 11 => 00 => 2
[2,2]
=> 00 => 11 => 0
[2,1,1]
=> 011 => 100 => 1
[1,1,1,1]
=> 1111 => 0000 => 4
[5]
=> 1 => 0 => 1
[4,1]
=> 01 => 10 => 0
[3,2]
=> 10 => 01 => 1
[3,1,1]
=> 111 => 000 => 3
[2,2,1]
=> 001 => 110 => 0
[2,1,1,1]
=> 0111 => 1000 => 2
[1,1,1,1,1]
=> 11111 => 00000 => 5
[6]
=> 0 => 1 => 0
[5,1]
=> 11 => 00 => 2
[4,2]
=> 00 => 11 => 0
[4,1,1]
=> 011 => 100 => 1
[3,3]
=> 11 => 00 => 2
[3,2,1]
=> 101 => 010 => 1
[3,1,1,1]
=> 1111 => 0000 => 4
[2,2,2]
=> 000 => 111 => 0
[2,2,1,1]
=> 0011 => 1100 => 0
[2,1,1,1,1]
=> 01111 => 10000 => 3
[1,1,1,1,1,1]
=> 111111 => 000000 => 6
Description
The depth of the binary word interpreted as a path.
This is the maximal value of the number of zeros minus the number of ones occurring in a prefix of the binary word, see [1, sec.9.1.2].
The number of binary words of length $n$ with depth $k$ is $\binom{n}{\lfloor\frac{(n+1) - (-1)^{n-k}(k+1)}{2}\rfloor}$, see [2].
Matching statistic: St000895
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00035: Dyck paths —to alternating sign matrix⟶ Alternating sign matrices
St000895: Alternating sign matrices ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2]
=> [1,0,1,0]
=> [[1,0],[0,1]]
=> 2
[1,1]
=> [1,1,0,0]
=> [[0,1],[1,0]]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [[1,0,0],[0,1,0],[0,0,1]]
=> 3
[2,1]
=> [1,0,1,1,0,0]
=> [[1,0,0],[0,0,1],[0,1,0]]
=> 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [[0,1,0],[1,-1,1],[0,1,0]]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
=> 4
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
=> 2
[2,2]
=> [1,1,1,0,0,0]
=> [[0,0,1],[1,0,0],[0,1,0]]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,0,0,0],[0,0,1,0],[0,1,-1,1],[0,0,1,0]]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [[0,1,0,0],[1,-1,1,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
=> 5
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
=> 3
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
=> 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[0,0,1,0],[1,0,0,0],[0,1,-1,1],[0,0,1,0]]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0],[0,0,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0],[1,-1,1,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,1,0],[0,0,0,0,0,1]]
=> 6
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,1,0,0],[0,0,0,0,0,1],[0,0,0,0,1,0]]
=> 4
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
=> 2
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,1,0,0,0],[0,0,0,0,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 3
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [[0,0,1,0],[1,0,-1,1],[0,1,0,0],[0,0,1,0]]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,1,0,0,0,0],[0,0,0,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 2
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
=> 0
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[0,0,1,0,0],[1,0,0,0,0],[0,1,-1,1,0],[0,0,1,-1,1],[0,0,0,1,0]]
=> 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [[1,0,0,0,0,0],[0,0,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [[0,1,0,0,0,0],[1,-1,1,0,0,0],[0,1,-1,1,0,0],[0,0,1,-1,1,0],[0,0,0,1,-1,1],[0,0,0,0,1,0]]
=> 0
Description
The number of ones on the main diagonal of an alternating sign matrix.
The following 256 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000164The number of short pairs. St000215The number of adjacencies of a permutation, zero appended. St000248The number of anti-singletons of a set partition. St000297The number of leading ones in a binary word. St000315The number of isolated vertices of a graph. St000338The number of pixed points of a permutation. St000461The rix statistic of a permutation. St000974The length of the trunk of an ordered tree. St001126Number of simple module that are 1-regular in the corresponding Nakayama algebra. St001459The number of zero columns in the nullspace of a graph. St001691The number of kings in a graph. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000382The first part of an integer composition. St000765The number of weak records in an integer composition. St000899The maximal number of repetitions of an integer composition. St000237The number of small exceedances. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St000894The trace of an alternating sign matrix. St000148The number of odd parts of a partition. St000377The dinv defect of an integer partition. St000214The number of adjacencies of a permutation. St000369The dinv deficit of a Dyck path. St000835The minimal difference in size when partitioning the integer partition into two subpartitions. St000992The alternating sum of the parts of an integer partition. St001055The Grundy value for the game of removing cells of a row in an integer partition. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001274The number of indecomposable injective modules with projective dimension equal to two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000239The number of small weak excedances. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St000335The difference of lower and upper interactions. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St000454The largest eigenvalue of a graph if it is integral. St000693The modular (standard) major index of a standard tableau. St000376The bounce deficit of a Dyck path. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000567The sum of the products of all pairs of parts. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001176The size of a partition minus its first part. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001961The sum of the greatest common divisors of all pairs of parts. St000010The length of the partition. St000012The area of a Dyck path. St000160The multiplicity of the smallest part of a partition. St000185The weighted size of a partition. St000228The size of a partition. St000288The number of ones in a binary word. St000295The length of the border of a binary word. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000380Half of the maximal perimeter of a rectangle fitting into the diagram of an integer partition. St000384The maximal part of the shifted composition of an integer partition. St000392The length of the longest run of ones in a binary word. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000459The hook length of the base cell of a partition. St000519The largest length of a factor maximising the subword complexity. St000548The number of different non-empty partial sums of an integer partition. St000644The number of graphs with given frequency partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000784The maximum of the length and the largest part of the integer partition. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000867The sum of the hook lengths in the first row of an integer partition. St000869The sum of the hook lengths of an integer partition. St001017Number of indecomposable injective modules with projective dimension equal to the codominant dimension in the Nakayama algebra corresponding to the Dyck path. St001091The number of parts in an integer partition whose next smaller part has the same size. St001127The sum of the squares of the parts of a partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001279The sum of the parts of an integer partition that are at least two. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001498The normalised height of a Nakayama algebra with magnitude 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001541The Gini index of an integer partition. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001561The value of the elementary symmetric function evaluated at 1. St001697The shifted natural comajor index of a standard Young tableau. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001903The number of fixed points of a parking function. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001930The weak major index of a binary word. St001956The comajor index for set-valued two-row standard Young tableaux. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000938The number of zeros of the symmetric group character corresponding to the partition. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001267The length of the Lyndon factorization of the binary word. St000219The number of occurrences of the pattern 231 in a permutation. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000259The diameter of a connected graph. St000153The number of adjacent cycles of a permutation. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St000993The multiplicity of the largest part of an integer partition. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000678The number of up steps after the last double rise of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001139The number of occurrences of hills of size 2 in a Dyck path. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001248Sum of the even parts of a partition. St001280The number of parts of an integer partition that are at least two. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St001587Half of the largest even part of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001657The number of twos in an integer partition. St001808The box weight or horizontal decoration of a Dyck path. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000455The second largest eigenvalue of a graph if it is integral. St000460The hook length of the last cell along the main diagonal of an integer partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000939The number of characters of the symmetric group whose value on the partition is positive. St000944The 3-degree of an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001525The number of symmetric hooks on the diagonal of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000260The radius of a connected graph. St000456The monochromatic index of a connected graph. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000706The product of the factorials of the multiplicities of an integer partition. St000707The product of the factorials of the parts. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000933The number of multipartitions of sizes given by an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001568The smallest positive integer that does not appear twice in the partition. St001651The Frankl number of a lattice. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001645The pebbling number of a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001060The distinguishing index of a graph. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000284The Plancherel distribution on integer partitions. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000664The number of right ropes of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St001128The exponens consonantiae of a partition. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001556The number of inversions of the third entry of a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St000090The variation of a composition. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St001823The Stasinski-Voll length of a signed permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000990The first ascent of a permutation. St001058The breadth of the ordered tree. St001209The pmaj statistic of a parking function. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001893The flag descent of a signed permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St000472The sum of the ascent bottoms of a permutation. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001074The number of inversions of the cyclic embedding of a permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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