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Your data matches 159 different statistics following compositions of up to 3 maps.
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Matching statistic: St001254
(load all 33 compositions to match this statistic)
(load all 33 compositions to match this statistic)
St001254: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1 = 4 - 3
[1,0,1,0]
=> 2 = 5 - 3
[1,1,0,0]
=> 3 = 6 - 3
[1,0,1,0,1,0]
=> 3 = 6 - 3
[1,0,1,1,0,0]
=> 4 = 7 - 3
[1,1,0,0,1,0]
=> 4 = 7 - 3
[1,1,0,1,0,0]
=> 4 = 7 - 3
[1,1,1,0,0,0]
=> 6 = 9 - 3
[1,0,1,0,1,0,1,0]
=> 4 = 7 - 3
[1,0,1,0,1,1,0,0]
=> 5 = 8 - 3
[1,0,1,1,0,0,1,0]
=> 5 = 8 - 3
[1,0,1,1,0,1,0,0]
=> 5 = 8 - 3
[1,0,1,1,1,0,0,0]
=> 7 = 10 - 3
[1,1,0,0,1,0,1,0]
=> 5 = 8 - 3
[1,1,0,0,1,1,0,0]
=> 6 = 9 - 3
[1,1,0,1,0,0,1,0]
=> 5 = 8 - 3
[1,1,0,1,0,1,0,0]
=> 5 = 8 - 3
[1,1,0,1,1,0,0,0]
=> 6 = 9 - 3
[1,1,1,0,0,0,1,0]
=> 7 = 10 - 3
[1,1,1,0,0,1,0,0]
=> 7 = 10 - 3
[1,1,1,0,1,0,0,0]
=> 7 = 10 - 3
[1,1,1,1,0,0,0,0]
=> 10 = 13 - 3
Description
The vector space dimension of the first extension-group between A/soc(A) and J when A is the corresponding Nakayama algebra with Jacobson radical J.
Matching statistic: St000984
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000984: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000984: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 1 = 4 - 3
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 5 - 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 6 - 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 6 - 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4 = 7 - 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 7 - 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 7 - 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6 = 9 - 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 7 - 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 8 - 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5 = 8 - 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 5 = 8 - 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7 = 10 - 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 8 - 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6 = 9 - 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 5 = 8 - 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 5 = 8 - 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 7 = 10 - 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7 = 10 - 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 7 = 10 - 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6 = 9 - 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10 = 13 - 3
Description
The number of boxes below precisely one peak.
Imagine that each peak of the Dyck path, drawn with north and east steps, casts a shadow onto the triangular region between it and the diagonal. This statistic is the number of cells which are in the shade of precisely one peak.
Matching statistic: St001003
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001003: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00142: Dyck paths —promotion⟶ Dyck paths
St001003: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 4
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 5
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,1,0,0]
=> 6
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 6
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 7
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 7
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 7
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 9
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 7
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 8
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 8
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 8
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 10
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 8
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 9
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 8
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 8
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 9
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 10
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 10
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 10
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 13
Description
The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001019
(load all 13 compositions to match this statistic)
(load all 13 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001019: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
St001019: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> 2 = 4 - 2
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 4 = 6 - 2
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 3 = 5 - 2
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 7 = 9 - 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> 5 = 7 - 2
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 5 = 7 - 2
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> 5 = 7 - 2
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 4 = 6 - 2
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 11 = 13 - 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 8 = 10 - 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 7 = 9 - 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 8 = 10 - 2
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 6 = 8 - 2
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 8 = 10 - 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 6 = 8 - 2
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 7 = 9 - 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 8 = 10 - 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 6 = 8 - 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 6 = 8 - 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 6 = 8 - 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 7 - 2
Description
Sum of the projective dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001228
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001228: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> 1 = 4 - 3
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 2 = 5 - 3
[1,1,0,0]
=> [2] => [1,1,0,0]
=> 3 = 6 - 3
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 3 = 6 - 3
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 4 = 7 - 3
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 4 = 7 - 3
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> 4 = 7 - 3
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> 6 = 9 - 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 7 - 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 5 = 8 - 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5 = 8 - 3
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 5 = 8 - 3
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 7 = 10 - 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5 = 8 - 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 6 = 9 - 3
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5 = 8 - 3
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 5 = 8 - 3
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 6 = 9 - 3
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 7 = 10 - 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 7 = 10 - 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 7 = 10 - 3
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> 10 = 13 - 3
Description
The vector space dimension of the space of module homomorphisms between J and itself when J denotes the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000012
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [1,1,0,0]
=> 1 = 4 - 3
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 5 - 3
[1,1,0,0]
=> [2] => [1,1,0,0]
=> [1,1,1,0,0,0]
=> 3 = 6 - 3
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 6 - 3
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4 = 7 - 3
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 7 - 3
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4 = 7 - 3
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 6 = 9 - 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 7 - 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 5 = 8 - 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5 = 8 - 3
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 5 = 8 - 3
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 7 = 10 - 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 8 - 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6 = 9 - 3
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 8 - 3
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5 = 8 - 3
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 6 = 9 - 3
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7 = 10 - 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7 = 10 - 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7 = 10 - 3
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 10 = 13 - 3
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000224
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000224: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000224: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> [2,1] => 1 = 4 - 3
[1,0,1,0]
=> [1,1] => [1,0,1,0]
=> [3,1,2] => 3 = 6 - 3
[1,1,0,0]
=> [2] => [1,1,0,0]
=> [2,3,1] => 2 = 5 - 3
[1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> [4,1,2,3] => 6 = 9 - 3
[1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> [3,1,4,2] => 4 = 7 - 3
[1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 4 = 7 - 3
[1,1,0,1,0,0]
=> [2,1] => [1,1,0,0,1,0]
=> [2,4,1,3] => 4 = 7 - 3
[1,1,1,0,0,0]
=> [3] => [1,1,1,0,0,0]
=> [2,3,4,1] => 3 = 6 - 3
[1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [5,1,2,3,4] => 10 = 13 - 3
[1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [4,1,2,5,3] => 7 = 10 - 3
[1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 6 = 9 - 3
[1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,5,2,4] => 6 = 9 - 3
[1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => 5 = 8 - 3
[1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 7 = 10 - 3
[1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 5 = 8 - 3
[1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 7 = 10 - 3
[1,1,0,1,0,1,0,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => 7 = 10 - 3
[1,1,0,1,1,0,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => 5 = 8 - 3
[1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 5 = 8 - 3
[1,1,1,0,0,1,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 5 = 8 - 3
[1,1,1,0,1,0,0,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => 5 = 8 - 3
[1,1,1,1,0,0,0,0]
=> [4] => [1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 4 = 7 - 3
Description
The sorting index of a permutation.
The sorting index counts the total distance that symbols move during a selection sort of a permutation. This sorting algorithm swaps symbol n into index n and then recursively sorts the first n-1 symbols.
Compare this to [[St000018]], the number of inversions of a permutation, which is also the total distance that elements move during a bubble sort.
Matching statistic: St000400
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000400: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00010: Binary trees —to ordered tree: left child = left brother⟶ Ordered trees
St000400: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [.,.]
=> [[]]
=> 1 = 4 - 3
[1,0,1,0]
=> [2,1] => [[.,.],.]
=> [[],[]]
=> 2 = 5 - 3
[1,1,0,0]
=> [1,2] => [.,[.,.]]
=> [[[]]]
=> 3 = 6 - 3
[1,0,1,0,1,0]
=> [3,2,1] => [[[.,.],.],.]
=> [[],[],[]]
=> 3 = 6 - 3
[1,0,1,1,0,0]
=> [2,3,1] => [[.,[.,.]],.]
=> [[[]],[]]
=> 4 = 7 - 3
[1,1,0,0,1,0]
=> [3,1,2] => [[.,.],[.,.]]
=> [[],[[]]]
=> 4 = 7 - 3
[1,1,0,1,0,0]
=> [2,1,3] => [[.,.],[.,.]]
=> [[],[[]]]
=> 4 = 7 - 3
[1,1,1,0,0,0]
=> [1,2,3] => [.,[.,[.,.]]]
=> [[[[]]]]
=> 6 = 9 - 3
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [[[[.,.],.],.],.]
=> [[],[],[],[]]
=> 4 = 7 - 3
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [[[.,[.,.]],.],.]
=> [[[]],[],[]]
=> 5 = 8 - 3
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 5 = 8 - 3
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [[[.,.],[.,.]],.]
=> [[],[[]],[]]
=> 5 = 8 - 3
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [[.,[.,[.,.]]],.]
=> [[[[]]],[]]
=> 7 = 10 - 3
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 5 = 8 - 3
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 6 = 9 - 3
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 5 = 8 - 3
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [[[.,.],.],[.,.]]
=> [[],[],[[]]]
=> 5 = 8 - 3
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [[.,[.,.]],[.,.]]
=> [[[]],[[]]]
=> 6 = 9 - 3
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 7 = 10 - 3
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 7 = 10 - 3
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [[.,.],[.,[.,.]]]
=> [[],[[[]]]]
=> 7 = 10 - 3
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [.,[.,[.,[.,.]]]]
=> [[[[[]]]]]
=> 10 = 13 - 3
Description
The path length of an ordered tree.
This is the sum of the lengths of all paths from the root to a node, see Section 2.3.4.5 of [1].
Matching statistic: St000448
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000448: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00046: Ordered trees —to graph⟶ Graphs
St000448: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [[[]]]
=> ([(0,2),(1,2)],3)
=> 1 = 4 - 3
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [[[],[]]]
=> ([(0,3),(1,3),(2,3)],4)
=> 3 = 6 - 3
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [[[[]]]]
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 5 - 3
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [[[],[],[]]]
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> 6 = 9 - 3
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [[[],[[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4 = 7 - 3
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [[[[]],[]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4 = 7 - 3
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [[[[],[]]]]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 4 = 7 - 3
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3 = 6 - 3
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [[[],[],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 10 = 13 - 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [[[],[],[[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7 = 10 - 3
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [[[],[[]],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7 = 10 - 3
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [[[],[[],[]]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 9 - 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5 = 8 - 3
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [[[[]],[],[]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7 = 10 - 3
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [[[[]],[[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 8 - 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [[[[],[]],[]]]
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 6 = 9 - 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [[[[],[],[]]]]
=> ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 7 = 10 - 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 8 - 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5 = 8 - 3
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 5 = 8 - 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
=> 5 = 8 - 3
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 4 = 7 - 3
Description
The number of pairs of vertices of a graph with distance 2.
This is the coefficient of the quadratic term of the Wiener polynomial.
Matching statistic: St000566
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000566: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => [1,2] => [2,1]
=> 1 = 4 - 3
[1,0,1,0]
=> 1010 => [1,2,2] => [2,2,1]
=> 2 = 5 - 3
[1,1,0,0]
=> 1100 => [1,1,3] => [3,1,1]
=> 3 = 6 - 3
[1,0,1,0,1,0]
=> 101010 => [1,2,2,2] => [2,2,2,1]
=> 3 = 6 - 3
[1,0,1,1,0,0]
=> 101100 => [1,2,1,3] => [3,2,1,1]
=> 4 = 7 - 3
[1,1,0,0,1,0]
=> 110010 => [1,1,3,2] => [3,2,1,1]
=> 4 = 7 - 3
[1,1,0,1,0,0]
=> 110100 => [1,1,2,3] => [3,2,1,1]
=> 4 = 7 - 3
[1,1,1,0,0,0]
=> 111000 => [1,1,1,4] => [4,1,1,1]
=> 6 = 9 - 3
[1,0,1,0,1,0,1,0]
=> 10101010 => [1,2,2,2,2] => [2,2,2,2,1]
=> 4 = 7 - 3
[1,0,1,0,1,1,0,0]
=> 10101100 => [1,2,2,1,3] => [3,2,2,1,1]
=> 5 = 8 - 3
[1,0,1,1,0,0,1,0]
=> 10110010 => [1,2,1,3,2] => [3,2,2,1,1]
=> 5 = 8 - 3
[1,0,1,1,0,1,0,0]
=> 10110100 => [1,2,1,2,3] => [3,2,2,1,1]
=> 5 = 8 - 3
[1,0,1,1,1,0,0,0]
=> 10111000 => [1,2,1,1,4] => [4,2,1,1,1]
=> 7 = 10 - 3
[1,1,0,0,1,0,1,0]
=> 11001010 => [1,1,3,2,2] => [3,2,2,1,1]
=> 5 = 8 - 3
[1,1,0,0,1,1,0,0]
=> 11001100 => [1,1,3,1,3] => [3,3,1,1,1]
=> 6 = 9 - 3
[1,1,0,1,0,0,1,0]
=> 11010010 => [1,1,2,3,2] => [3,2,2,1,1]
=> 5 = 8 - 3
[1,1,0,1,0,1,0,0]
=> 11010100 => [1,1,2,2,3] => [3,2,2,1,1]
=> 5 = 8 - 3
[1,1,0,1,1,0,0,0]
=> 11011000 => [1,1,2,1,4] => [4,2,1,1,1]
=> 7 = 10 - 3
[1,1,1,0,0,0,1,0]
=> 11100010 => [1,1,1,4,2] => [4,2,1,1,1]
=> 7 = 10 - 3
[1,1,1,0,0,1,0,0]
=> 11100100 => [1,1,1,3,3] => [3,3,1,1,1]
=> 6 = 9 - 3
[1,1,1,0,1,0,0,0]
=> 11101000 => [1,1,1,2,4] => [4,2,1,1,1]
=> 7 = 10 - 3
[1,1,1,1,0,0,0,0]
=> 11110000 => [1,1,1,1,5] => [5,1,1,1,1]
=> 10 = 13 - 3
Description
The number of ways to select a row of a Ferrers shape and two cells in this row. Equivalently, if $\lambda = (\lambda_0\geq\lambda_1 \geq \dots\geq\lambda_m)$ is an integer partition, then the statistic is
$$\frac{1}{2} \sum_{i=0}^m \lambda_i(\lambda_i -1).$$
The following 149 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St001308The number of induced paths on three vertices in a graph. St001874Lusztig's a-function for the symmetric group. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000055The inversion sum of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000558The number of occurrences of the pattern {{1,2}} in a set partition. 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. St000809The reduced reflection length of the permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001821The sorting index of a signed permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001388The number of non-attacking neighbors of a permutation. St001645The pebbling number of a connected graph. St001812The biclique partition number of a graph. St001626The number of maximal proper sublattices of a lattice. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. 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. St000077The number of boxed and circled entries. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001960The number of descents of a permutation minus one if its first entry is not one. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000863The length of the first row of the shifted shape of a permutation. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000021The number of descents of a permutation. St000060The greater neighbor of the maximum. St000216The absolute length of a permutation. St000238The number of indices that are not small weak excedances. St000240The number of indices that are not small excedances. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000354The number of recoils of a permutation. St000619The number of cyclic descents of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000831The number of indices that are either descents or recoils. St000864The number of circled entries of the shifted recording tableau of a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001489The maximum of the number of descents and the number of inverse descents. St000028The number of stack-sorts needed to sort a permutation. St000235The number of indices that are not cyclical small weak excedances. St000242The number of indices that are not cyclical small weak excedances. St000393The number of strictly increasing runs in a binary word. St000505The biggest entry in the block containing the 1. St000579The number of occurrences of the pattern {{1},{2}} such that 2 is a maximal element. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000740The last entry of a permutation. St000796The stat' of a permutation. St000971The smallest closer of a set partition. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001298The number of repeated entries in the Lehmer code of a permutation. St001391The disjunction number of a graph. St001424The number of distinct squares in a binary word. St001488The number of corners of a skew partition. St001497The position of the largest weak excedence of a permutation. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001722The number of minimal chains with small intervals between a binary word and the top element. St001760The number of prefix or suffix reversals needed to sort a permutation. St001894The depth of a signed permutation. St001959The product of the heights of the peaks of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000120The number of left tunnels of a Dyck path. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000173The segment statistic of a semistandard tableau. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000223The number of nestings in the permutation. St000331The number of upper interactions of a Dyck path. St000360The number of occurrences of the pattern 32-1. St000454The largest eigenvalue of a graph if it is integral. St000491The number of inversions of a set partition. St000503The maximal difference between two elements in a common block. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000624The normalized sum of the minimal distances to a greater element. St000672The number of minimal elements in Bruhat order not less than the permutation. St000736The last entry in the first row of a semistandard tableau. St000747A variant of the major index of a set partition. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St001030Half the number of non-boundary horizontal edges in the fully packed loop corresponding to the alternating sign matrix. 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 $K[x]/(x^n)$. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001684The reduced word complexity of a permutation. St001727The number of invisible inversions of a permutation. St001742The difference of the maximal and the minimal degree in a graph. St001769The reflection length of a signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001823The Stasinski-Voll length of a signed permutation. St001843The Z-index of a set partition. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001869The maximum cut size of a graph. St001896The number of right descents of a signed permutations. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation. St001946The number of descents in a parking function. St000112The sum of the entries reduced by the index of their row in a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001209The pmaj statistic of a parking function. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew 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. St001557The number of inversions of the second entry of a permutation. St001569The maximal modular displacement of a permutation. St001668The number of points of the poset minus the width of the poset. St001811The Castelnuovo-Mumford regularity of a permutation. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St001862The number of crossings of a signed permutation. St001948The number of augmented double ascents of a permutation.
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