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Your data matches 261 different statistics following compositions of up to 3 maps.
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Matching statistic: St000998
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000998: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 2
[1,1] => [1,0,1,0]
=> 3
[2] => [1,1,0,0]
=> 3
[1,1,1] => [1,0,1,0,1,0]
=> 4
[1,2] => [1,0,1,1,0,0]
=> 4
[2,1] => [1,1,0,0,1,0]
=> 4
[3] => [1,1,1,0,0,0]
=> 4
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
[1,3] => [1,0,1,1,1,0,0,0]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> 5
[3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4] => [1,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 6
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 6
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 6
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 6
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 6
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 6
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 6
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 7
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 7
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 7
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 7
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 7
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 7
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 7
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 7
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 7
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 7
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
Description
Number of indecomposable projective modules with injective dimension smaller than or equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001240
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001240: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001240: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 2
[1,1] => [1,0,1,0]
=> 3
[2] => [1,1,0,0]
=> 3
[1,1,1] => [1,0,1,0,1,0]
=> 4
[1,2] => [1,0,1,1,0,0]
=> 4
[2,1] => [1,1,0,0,1,0]
=> 4
[3] => [1,1,1,0,0,0]
=> 4
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
[1,3] => [1,0,1,1,1,0,0,0]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> 5
[3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4] => [1,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 6
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 6
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 6
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 6
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 6
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 6
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 6
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 7
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 7
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 7
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 7
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 7
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 7
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 7
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 7
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 7
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 7
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
Description
The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra
Matching statistic: St001650
(load all 31 compositions to match this statistic)
(load all 31 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001650: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001650: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 2
[1,1] => [1,0,1,0]
=> 3
[2] => [1,1,0,0]
=> 3
[1,1,1] => [1,0,1,0,1,0]
=> 4
[1,2] => [1,0,1,1,0,0]
=> 4
[2,1] => [1,1,0,0,1,0]
=> 4
[3] => [1,1,1,0,0,0]
=> 4
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 5
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 5
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 5
[1,3] => [1,0,1,1,1,0,0,0]
=> 5
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 5
[2,2] => [1,1,0,0,1,1,0,0]
=> 5
[3,1] => [1,1,1,0,0,0,1,0]
=> 5
[4] => [1,1,1,1,0,0,0,0]
=> 5
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 6
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 6
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 6
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 6
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 6
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 6
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 6
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 6
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 6
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 6
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 6
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 6
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 6
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 6
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 6
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 7
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 7
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 7
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 7
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 7
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 7
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 7
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 7
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 7
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 7
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 7
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 7
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 7
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 7
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 7
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 7
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 7
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 7
Description
The order of Ringel's homological bijection associated to the linear Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000144
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000144: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 2 - 1
[1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2] => [1,1,0,0]
=> 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2] => [1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1] => [1,1,0,0,1,0]
=> 3 = 4 - 1
[3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5 = 6 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6 = 7 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6 = 7 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 6 = 7 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6 = 7 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6 = 7 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 6 = 7 - 1
Description
The pyramid weight of the Dyck path.
The pyramid weight of a Dyck path is the sum of the lengths of the maximal pyramids (maximal sequences of the form $1^h0^h$) in the path.
Maximal pyramids are called lower interactions by Le Borgne [2], see [[St000331]] and [[St000335]] for related statistics.
Matching statistic: St000228
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000228: 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]
=> 2 = 3 - 1
[1,1,1] => [1,1,1]
=> 3 = 4 - 1
[1,2] => [2,1]
=> 3 = 4 - 1
[2,1] => [2,1]
=> 3 = 4 - 1
[3] => [3]
=> 3 = 4 - 1
[1,1,1,1] => [1,1,1,1]
=> 4 = 5 - 1
[1,1,2] => [2,1,1]
=> 4 = 5 - 1
[1,2,1] => [2,1,1]
=> 4 = 5 - 1
[1,3] => [3,1]
=> 4 = 5 - 1
[2,1,1] => [2,1,1]
=> 4 = 5 - 1
[2,2] => [2,2]
=> 4 = 5 - 1
[3,1] => [3,1]
=> 4 = 5 - 1
[4] => [4]
=> 4 = 5 - 1
[1,1,1,1,1] => [1,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,2] => [2,1,1,1]
=> 5 = 6 - 1
[1,1,2,1] => [2,1,1,1]
=> 5 = 6 - 1
[1,1,3] => [3,1,1]
=> 5 = 6 - 1
[1,2,1,1] => [2,1,1,1]
=> 5 = 6 - 1
[1,2,2] => [2,2,1]
=> 5 = 6 - 1
[1,3,1] => [3,1,1]
=> 5 = 6 - 1
[1,4] => [4,1]
=> 5 = 6 - 1
[2,1,1,1] => [2,1,1,1]
=> 5 = 6 - 1
[2,1,2] => [2,2,1]
=> 5 = 6 - 1
[2,2,1] => [2,2,1]
=> 5 = 6 - 1
[2,3] => [3,2]
=> 5 = 6 - 1
[3,1,1] => [3,1,1]
=> 5 = 6 - 1
[3,2] => [3,2]
=> 5 = 6 - 1
[4,1] => [4,1]
=> 5 = 6 - 1
[5] => [5]
=> 5 = 6 - 1
[1,1,1,1,1,1] => [1,1,1,1,1,1]
=> 6 = 7 - 1
[1,1,1,1,2] => [2,1,1,1,1]
=> 6 = 7 - 1
[1,1,1,2,1] => [2,1,1,1,1]
=> 6 = 7 - 1
[1,1,1,3] => [3,1,1,1]
=> 6 = 7 - 1
[1,1,2,1,1] => [2,1,1,1,1]
=> 6 = 7 - 1
[1,1,2,2] => [2,2,1,1]
=> 6 = 7 - 1
[1,1,3,1] => [3,1,1,1]
=> 6 = 7 - 1
[1,1,4] => [4,1,1]
=> 6 = 7 - 1
[1,2,1,1,1] => [2,1,1,1,1]
=> 6 = 7 - 1
[1,2,1,2] => [2,2,1,1]
=> 6 = 7 - 1
[1,2,2,1] => [2,2,1,1]
=> 6 = 7 - 1
[1,2,3] => [3,2,1]
=> 6 = 7 - 1
[1,3,1,1] => [3,1,1,1]
=> 6 = 7 - 1
[1,3,2] => [3,2,1]
=> 6 = 7 - 1
[1,4,1] => [4,1,1]
=> 6 = 7 - 1
[1,5] => [5,1]
=> 6 = 7 - 1
[2,1,1,1,1] => [2,1,1,1,1]
=> 6 = 7 - 1
[2,1,1,2] => [2,2,1,1]
=> 6 = 7 - 1
[2,1,2,1] => [2,2,1,1]
=> 6 = 7 - 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000395
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 2 - 1
[1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2] => [1,1,0,0]
=> 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2] => [1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1] => [1,1,0,0,1,0]
=> 3 = 4 - 1
[3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5 = 6 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6 = 7 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6 = 7 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 6 = 7 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6 = 7 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6 = 7 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 6 = 7 - 1
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St000967
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000967: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000967: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 3 = 2 + 1
[1,1] => [1,0,1,0]
=> 4 = 3 + 1
[2] => [1,1,0,0]
=> 4 = 3 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2] => [1,0,1,1,0,0]
=> 5 = 4 + 1
[2,1] => [1,1,0,0,1,0]
=> 5 = 4 + 1
[3] => [1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 7 = 6 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 7 = 6 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 8 = 7 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 8 = 7 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 8 = 7 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 8 = 7 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 8 = 7 + 1
Description
The value p(1) for the Coxeterpolynomial p of the corresponding LNakayama algebra.
Matching statistic: St001018
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001018: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001018: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 2 - 1
[1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2] => [1,1,0,0]
=> 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2] => [1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1] => [1,1,0,0,1,0]
=> 3 = 4 - 1
[3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5 = 6 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6 = 7 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6 = 7 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 6 = 7 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6 = 7 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6 = 7 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 6 = 7 - 1
Description
Sum of projective dimension of the indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001020
(load all 53 compositions to match this statistic)
(load all 53 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001020: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 1 = 2 - 1
[1,1] => [1,0,1,0]
=> 2 = 3 - 1
[2] => [1,1,0,0]
=> 2 = 3 - 1
[1,1,1] => [1,0,1,0,1,0]
=> 3 = 4 - 1
[1,2] => [1,0,1,1,0,0]
=> 3 = 4 - 1
[2,1] => [1,1,0,0,1,0]
=> 3 = 4 - 1
[3] => [1,1,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 4 = 5 - 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 4 = 5 - 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 4 = 5 - 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 4 = 5 - 1
[4] => [1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 5 = 6 - 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 5 = 6 - 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 5 = 6 - 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 5 = 6 - 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 5 = 6 - 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 5 = 6 - 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 5 = 6 - 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 5 = 6 - 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 5 = 6 - 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 5 = 6 - 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 6 = 7 - 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 6 = 7 - 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 6 = 7 - 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 6 = 7 - 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 6 = 7 - 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 6 = 7 - 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 6 = 7 - 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 6 = 7 - 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 6 = 7 - 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 6 = 7 - 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 6 = 7 - 1
Description
Sum of the codominant dimensions of the non-projective indecomposable injective modules of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001218
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001218: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001218: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1,0]
=> 3 = 2 + 1
[1,1] => [1,0,1,0]
=> 4 = 3 + 1
[2] => [1,1,0,0]
=> 4 = 3 + 1
[1,1,1] => [1,0,1,0,1,0]
=> 5 = 4 + 1
[1,2] => [1,0,1,1,0,0]
=> 5 = 4 + 1
[2,1] => [1,1,0,0,1,0]
=> 5 = 4 + 1
[3] => [1,1,1,0,0,0]
=> 5 = 4 + 1
[1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 6 = 5 + 1
[1,1,2] => [1,0,1,0,1,1,0,0]
=> 6 = 5 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> 6 = 5 + 1
[1,3] => [1,0,1,1,1,0,0,0]
=> 6 = 5 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> 6 = 5 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> 6 = 5 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> 6 = 5 + 1
[4] => [1,1,1,1,0,0,0,0]
=> 6 = 5 + 1
[1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 7 = 6 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 7 = 6 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 7 = 6 + 1
[1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 7 = 6 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 7 = 6 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 7 = 6 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 7 = 6 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 7 = 6 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 7 = 6 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 7 = 6 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 7 = 6 + 1
[5] => [1,1,1,1,1,0,0,0,0,0]
=> 7 = 6 + 1
[1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,1,1,2] => [1,0,1,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> 8 = 7 + 1
[1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> 8 = 7 + 1
[1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> 8 = 7 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> 8 = 7 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> 8 = 7 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 8 = 7 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 8 = 7 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 8 = 7 + 1
[1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 8 = 7 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> 8 = 7 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> 8 = 7 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> 8 = 7 + 1
Description
Smallest index k greater than or equal to one such that the Coxeter matrix C of the corresponding Nakayama algebra has C^k=1.
It returns zero in case there is no such k.
The following 251 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000026The position of the first return of a Dyck path. St000058The order of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001012Number of simple modules with projective dimension at most 2 in the Nakayama algebra corresponding to the Dyck path. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St001179Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra. St001190Number of simple modules with projective dimension at most 4 in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St000189The number of elements in the poset. St000229Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition. St000288The number of ones in a binary word. St000293The number of inversions of a binary word. St000336The leg major index of a standard tableau. 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. St000460The hook length of the last cell along the main diagonal of an integer partition. St000636The hull number of a graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001034The area of the parallelogram polyomino associated with the Dyck path. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001342The number of vertices in the center of a graph. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St000167The number of leaves of an ordered tree. St000451The length of the longest pattern of the form k 1 2. St000505The biggest entry in the block containing the 1. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001430The number of positive entries in a signed permutation. St001492The 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. St000018The number of inversions of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000050The depth or height of a binary tree. St000081The number of edges of a graph. St000141The maximum drop size of a permutation. St000246The number of non-inversions of a permutation. St000290The major index of a binary word. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000503The maximal difference between two elements in a common block. St000553The number of blocks of a graph. St000703The number of deficiencies of a permutation. St000733The row containing the largest entry of a standard tableau. St000863The length of the first row of the shifted shape of a permutation. St000867The sum of the hook lengths in the first row of an integer partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001348The bounce of the parallelogram polyomino associated with the Dyck path. St001437The flex of a binary word. St001479The number of bridges of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St001523The degree of symmetry of a Dyck path. St001554The number of distinct nonempty subtrees of a binary tree. St001643The Frobenius dimension of the Nakayama algebra corresponding to the Dyck path. St001672The restrained domination number of a graph. St000921The number of internal inversions of a binary word. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001176The size of a partition minus its first part. St001245The cyclic maximal difference between two consecutive entries of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St001958The degree of the polynomial interpolating the values of a permutation. St000806The semiperimeter of the associated bargraph. St001746The coalition number of a graph. St000171The degree of the graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000294The number of distinct factors of a binary word. St000518The number of distinct subsequences in a binary word. St000625The sum of the minimal distances to a greater element. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St001074The number of inversions of the cyclic embedding of a permutation. St001759The Rajchgot index of a permutation. St000060The greater neighbor of the maximum. St000296The length of the symmetric border of a binary word. St000385The number of vertices with out-degree 1 in a binary tree. St000393The number of strictly increasing runs in a binary word. St000414The binary logarithm of the number of binary trees with the same underlying unordered tree. St000619The number of cyclic descents of a permutation. St000627The exponent of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000922The minimal number such that all substrings of this length are unique. St000982The length of the longest constant subword. St001246The maximal difference between two consecutive entries of a permutation. St001267The length of the Lyndon factorization of the binary word. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000519The largest length of a factor maximising the subword complexity. St001974The rank of the alternating sign matrix. St001925The minimal number of zeros in a row of an alternating sign matrix. St000890The number of nonzero entries in an alternating sign matrix. St001917The order of toric promotion on the set of labellings of a graph. St000719The number of alignments in a perfect matching. St000528The height of a poset. St000054The first entry of the permutation. St000019The cardinality of the support of a permutation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000906The length of the shortest maximal chain in a poset. St000080The rank of the poset. St001622The number of join-irreducible elements of a lattice. St000672The number of minimal elements in Bruhat order not less than the permutation. St001725The harmonious chromatic number of a graph. St000924The number of topologically connected components of a perfect matching. St000883The number of longest increasing subsequences of a permutation. St001268The size of the largest ordinal summand in the poset. St000907The number of maximal antichains of minimal length in a poset. St000911The number of maximal antichains of maximal size in a poset. St000912The number of maximal antichains in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001645The pebbling number of a connected graph. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000740The last entry of a permutation. St001497The position of the largest weak excedence of a permutation. St001778The largest greatest common divisor of an element and its image in a permutation. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000673The number of non-fixed points of a permutation. St000844The size of the largest block in the direct sum decomposition of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000029The depth of a permutation. St000197The number of entries equal to positive one in the alternating sign matrix. St000209Maximum difference of elements in cycles. St000210Minimum over maximum difference of elements in cycles. St000216The absolute length of a permutation. St000809The reduced reflection length of the permutation. St000957The number of Bruhat lower covers of a permutation. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001468The smallest fixpoint of a permutation. St001480The number of simple summands of the module J^2/J^3. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St000235The number of indices that are not cyclical small weak excedances. St000240The number of indices that are not small excedances. 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. St000004The major index of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000067The inversion number of the alternating sign matrix. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000653The last descent of a permutation. St000702The number of weak deficiencies of a permutation. St000795The mad of a permutation. St000831The number of indices that are either descents or recoils. St000956The maximal displacement of a permutation. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of 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. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001428The number of B-inversions of a signed permutation. St001726The number of visible inversions of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St001727The number of invisible inversions of a permutation. St000327The number of cover relations in a poset. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001875The number of simple modules with projective dimension at most 1. St000044The number of vertices of the unicellular map given by a perfect matching. St000744The length of the path to the largest entry in a standard Young tableau. St000820The number of compositions obtained by rotating the composition. St000896The number of zeros on the main diagonal of an alternating sign matrix. St001045The number of leaves in the subtree not containing one in the decreasing labelled binary unordered tree associated with the perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001555The order of a signed permutation. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000199The column of the unique '1' in the last row of the alternating sign matrix. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000017The number of inversions of a standard tableau. St001965The number of decreasable positions in the corner sum matrix of an alternating sign matrix. St001927Sparre Andersen's number of positives of a signed permutation. St000245The number of ascents of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000211The rank of the set partition. St000213The number of weak exceedances (also weak excedences) of a permutation. St000251The number of nonsingleton blocks of a set partition. St000308The height of the tree associated to a permutation. St000314The number of left-to-right-maxima of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000354The number of recoils of a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000654The first descent of a permutation. St000794The mak of a permutation. St000798The makl of a permutation. St000833The comajor index of a permutation. St000925The number of topologically connected components of a set partition. St000991The number of right-to-left minima of a permutation. St001114The number of odd descents of a permutation. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001220The width of a permutation. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001461The number of topologically connected components of the chord diagram of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St001517The length of a longest pair of twins in a permutation. St001665The number of pure excedances of a permutation. St001667The maximal size of a pair of weak twins for a permutation. St001729The number of visible descents of a permutation. St001769The reflection length of a signed permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St001861The number of Bruhat lower covers of a permutation. St001874Lusztig's a-function for the symmetric group. St001894The depth of a signed permutation. St001928The number of non-overlapping descents in a permutation. St000133The "bounce" of a permutation. St000168The number of internal nodes of an ordered tree. St000338The number of pixed points of a permutation. St000358The number of occurrences of the pattern 31-2. St000624The normalized sum of the minimal distances to a greater element. St000989The number of final rises of a permutation. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001926Sparre Andersen's position of the maximum of a signed permutation.
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