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Your data matches 145 different statistics following compositions of up to 3 maps.
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Matching statistic: St000904
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
St000904: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 1 = 2 - 1
[1,1] => 2 = 3 - 1
[2] => 1 = 2 - 1
[1,1,1] => 3 = 4 - 1
[1,2] => 1 = 2 - 1
[2,1] => 1 = 2 - 1
[3] => 1 = 2 - 1
[1,1,1,1] => 4 = 5 - 1
[1,1,2] => 2 = 3 - 1
[1,2,1] => 2 = 3 - 1
[1,3] => 1 = 2 - 1
[2,1,1] => 2 = 3 - 1
[2,2] => 2 = 3 - 1
[3,1] => 1 = 2 - 1
[4] => 1 = 2 - 1
[1,2,2] => 2 = 3 - 1
[2,1,2] => 2 = 3 - 1
[2,2,1] => 2 = 3 - 1
[2,3] => 1 = 2 - 1
[3,2] => 1 = 2 - 1
[2,2,2] => 3 = 4 - 1
[3,3] => 2 = 3 - 1
Description
The maximal number of repetitions of an integer composition.
Matching statistic: St001933
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 1 = 2 - 1
[1,1] => [1,1]
=> 2 = 3 - 1
[2] => [2]
=> 1 = 2 - 1
[1,1,1] => [1,1,1]
=> 3 = 4 - 1
[1,2] => [2,1]
=> 1 = 2 - 1
[2,1] => [2,1]
=> 1 = 2 - 1
[3] => [3]
=> 1 = 2 - 1
[1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,1,2] => [2,1,1]
=> 2 = 3 - 1
[1,2,1] => [2,1,1]
=> 2 = 3 - 1
[1,3] => [3,1]
=> 1 = 2 - 1
[2,1,1] => [2,1,1]
=> 2 = 3 - 1
[2,2] => [2,2]
=> 2 = 3 - 1
[3,1] => [3,1]
=> 1 = 2 - 1
[4] => [4]
=> 1 = 2 - 1
[1,2,2] => [2,2,1]
=> 2 = 3 - 1
[2,1,2] => [2,2,1]
=> 2 = 3 - 1
[2,2,1] => [2,2,1]
=> 2 = 3 - 1
[2,3] => [3,2]
=> 1 = 2 - 1
[3,2] => [3,2]
=> 1 = 2 - 1
[2,2,2] => [2,2,2]
=> 3 = 4 - 1
[3,3] => [3,3]
=> 2 = 3 - 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St001091
Mp00040: Integer compositions —to partition⟶ Integer partitions
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001091: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 0 = 2 - 2
[1,1] => [1,1]
=> 1 = 3 - 2
[2] => [2]
=> 0 = 2 - 2
[1,1,1] => [1,1,1]
=> 2 = 4 - 2
[1,2] => [2,1]
=> 0 = 2 - 2
[2,1] => [2,1]
=> 0 = 2 - 2
[3] => [3]
=> 0 = 2 - 2
[1,1,1,1] => [1,1,1,1]
=> 3 = 5 - 2
[1,1,2] => [2,1,1]
=> 1 = 3 - 2
[1,2,1] => [2,1,1]
=> 1 = 3 - 2
[1,3] => [3,1]
=> 0 = 2 - 2
[2,1,1] => [2,1,1]
=> 1 = 3 - 2
[2,2] => [2,2]
=> 1 = 3 - 2
[3,1] => [3,1]
=> 0 = 2 - 2
[4] => [4]
=> 0 = 2 - 2
[1,2,2] => [2,2,1]
=> 1 = 3 - 2
[2,1,2] => [2,2,1]
=> 1 = 3 - 2
[2,2,1] => [2,2,1]
=> 1 = 3 - 2
[2,3] => [3,2]
=> 0 = 2 - 2
[3,2] => [3,2]
=> 0 = 2 - 2
[2,2,2] => [2,2,2]
=> 2 = 4 - 2
[3,3] => [3,3]
=> 1 = 3 - 2
Description
The number of parts in an integer partition whose next smaller part has the same size.
In other words, this is the number of distinct parts subtracted from the number of all parts.
Matching statistic: St001180
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001180: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001180: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2
[1,1] => [2] => [1,1,0,0]
=> 3
[2] => [1] => [1,0]
=> 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 4
[1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> 2
[3] => [1] => [1,0]
=> 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3] => [1,1] => [1,0,1,0]
=> 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[2,2] => [2] => [1,1,0,0]
=> 3
[3,1] => [1,1] => [1,0,1,0]
=> 2
[4] => [1] => [1,0]
=> 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
[2,3] => [1,1] => [1,0,1,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 4
[3,3] => [2] => [1,1,0,0]
=> 3
Description
Number of indecomposable injective modules with projective dimension at most 1.
Matching statistic: St001211
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001211: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2
[1,1] => [2] => [1,1,0,0]
=> 3
[2] => [1] => [1,0]
=> 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 4
[1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> 2
[3] => [1] => [1,0]
=> 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3] => [1,1] => [1,0,1,0]
=> 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[2,2] => [2] => [1,1,0,0]
=> 3
[3,1] => [1,1] => [1,0,1,0]
=> 2
[4] => [1] => [1,0]
=> 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
[2,3] => [1,1] => [1,0,1,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 4
[3,3] => [2] => [1,1,0,0]
=> 3
Description
The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module.
Matching statistic: St001492
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001492: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1] => [1,0]
=> 2
[1,1] => [2] => [1,1,0,0]
=> 3
[2] => [1] => [1,0]
=> 2
[1,1,1] => [3] => [1,1,1,0,0,0]
=> 4
[1,2] => [1,1] => [1,0,1,0]
=> 2
[2,1] => [1,1] => [1,0,1,0]
=> 2
[3] => [1] => [1,0]
=> 2
[1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> 5
[1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 3
[1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[1,3] => [1,1] => [1,0,1,0]
=> 2
[2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 3
[2,2] => [2] => [1,1,0,0]
=> 3
[3,1] => [1,1] => [1,0,1,0]
=> 2
[4] => [1] => [1,0]
=> 2
[1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 3
[2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 3
[2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 3
[2,3] => [1,1] => [1,0,1,0]
=> 2
[3,2] => [1,1] => [1,0,1,0]
=> 2
[2,2,2] => [3] => [1,1,1,0,0,0]
=> 4
[3,3] => [2] => [1,1,0,0]
=> 3
Description
The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra.
Matching statistic: St000392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 1 = 2 - 1
[1,1] => [1,1]
=> 110 => 2 = 3 - 1
[2] => [2]
=> 100 => 1 = 2 - 1
[1,1,1] => [1,1,1]
=> 1110 => 3 = 4 - 1
[1,2] => [2,1]
=> 1010 => 1 = 2 - 1
[2,1] => [2,1]
=> 1010 => 1 = 2 - 1
[3] => [3]
=> 1000 => 1 = 2 - 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => 4 = 5 - 1
[1,1,2] => [2,1,1]
=> 10110 => 2 = 3 - 1
[1,2,1] => [2,1,1]
=> 10110 => 2 = 3 - 1
[1,3] => [3,1]
=> 10010 => 1 = 2 - 1
[2,1,1] => [2,1,1]
=> 10110 => 2 = 3 - 1
[2,2] => [2,2]
=> 1100 => 2 = 3 - 1
[3,1] => [3,1]
=> 10010 => 1 = 2 - 1
[4] => [4]
=> 10000 => 1 = 2 - 1
[1,2,2] => [2,2,1]
=> 11010 => 2 = 3 - 1
[2,1,2] => [2,2,1]
=> 11010 => 2 = 3 - 1
[2,2,1] => [2,2,1]
=> 11010 => 2 = 3 - 1
[2,3] => [3,2]
=> 10100 => 1 = 2 - 1
[3,2] => [3,2]
=> 10100 => 1 = 2 - 1
[2,2,2] => [2,2,2]
=> 11100 => 3 = 4 - 1
[3,3] => [3,3]
=> 11000 => 2 = 3 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St001000
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001000: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2] => [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
Description
Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001184
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001184: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1] => [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
[2] => [2]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,1] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[1,2] => [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,1,1] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,2,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[2,1,1] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[2,2] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,1] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[4] => [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,2,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,2] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,2,1] => [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,3] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[3,2] => [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,2,2] => [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[3,3] => [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
Description
Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra.
Matching statistic: St001372
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 1 = 2 - 1
[1,1] => [1,1]
=> 110 => 2 = 3 - 1
[2] => [2]
=> 100 => 1 = 2 - 1
[1,1,1] => [1,1,1]
=> 1110 => 3 = 4 - 1
[1,2] => [2,1]
=> 1010 => 1 = 2 - 1
[2,1] => [2,1]
=> 1010 => 1 = 2 - 1
[3] => [3]
=> 1000 => 1 = 2 - 1
[1,1,1,1] => [1,1,1,1]
=> 11110 => 4 = 5 - 1
[1,1,2] => [2,1,1]
=> 10110 => 2 = 3 - 1
[1,2,1] => [2,1,1]
=> 10110 => 2 = 3 - 1
[1,3] => [3,1]
=> 10010 => 1 = 2 - 1
[2,1,1] => [2,1,1]
=> 10110 => 2 = 3 - 1
[2,2] => [2,2]
=> 1100 => 2 = 3 - 1
[3,1] => [3,1]
=> 10010 => 1 = 2 - 1
[4] => [4]
=> 10000 => 1 = 2 - 1
[1,2,2] => [2,2,1]
=> 11010 => 2 = 3 - 1
[2,1,2] => [2,2,1]
=> 11010 => 2 = 3 - 1
[2,2,1] => [2,2,1]
=> 11010 => 2 = 3 - 1
[2,3] => [3,2]
=> 10100 => 1 = 2 - 1
[3,2] => [3,2]
=> 10100 => 1 = 2 - 1
[2,2,2] => [2,2,2]
=> 11100 => 3 = 4 - 1
[3,3] => [3,3]
=> 11000 => 2 = 3 - 1
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
The following 135 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001571The Cartan determinant of the integer partition. St000144The pyramid weight of the Dyck path. St000439The position of the first down step of a Dyck path. 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. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000007The number of saliances of the permutation. St000011The number of touch points (or returns) of a Dyck path. St000013The height of a Dyck path. St000015The number of peaks of a Dyck path. St000025The number of initial rises of a Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St000239The number of small weak excedances. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000393The number of strictly increasing runs in a binary word. St000443The number of long tunnels of a Dyck path. St000654The first descent of a permutation. St000676The number of odd rises of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000877The depth of the binary word interpreted as a path. St000899The maximal number of repetitions of an integer composition. St000922The minimal number such that all substrings of this length are unique. St000996The number of exclusive left-to-right maxima of a permutation. St000998Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001203We associate to 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 Dyck path as follows:
St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001498The normalised height of a Nakayama algebra with magnitude 1. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000024The number of double up and double down steps of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000053The number of valleys of the Dyck path. St000245The number of ascents of a permutation. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000338The number of pixed points of a permutation. St000356The number of occurrences of the pattern 13-2. St000672The number of minimal elements in Bruhat order not less than the permutation. St000674The number of hills of a Dyck path. St000989The number of final rises of a permutation. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001777The number of weak descents in an integer composition. St000932The number of occurrences of the pattern UDU in a Dyck path. St001399The distinguishing number of a poset. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000444The length of the maximal rise of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000442The maximal area to the right of an up step of a Dyck path. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000438The position of the last up step in a Dyck path. St000939The number of characters of the symmetric group whose value on the partition is positive. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000806The semiperimeter of the associated bargraph. St000045The number of linear extensions of a binary tree. St000907The number of maximal antichains of minimal length in a poset. St001330The hat guessing number of a graph. St000650The number of 3-rises of a permutation. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001060The distinguishing index of a graph. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001875The number of simple modules with projective dimension at most 1. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001645The pebbling number of a connected graph. St000464The Schultz index of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. 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. 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 $eA$. St001545The second Elser number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000302The determinant of the distance matrix of a connected graph. St000456The monochromatic index of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a graph. St000455The second largest eigenvalue of a graph if it is integral. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001811The Castelnuovo-Mumford regularity of a permutation.
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