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Your data matches 259 different statistics following compositions of up to 3 maps.
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Matching statistic: St001063
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001063: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001063: 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]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[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]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 1
Description
Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra.
Matching statistic: St001066
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001066: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> 1
Description
The number of simple reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St001087
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00241: Permutations —invert Laguerre heap⟶ Permutations
St001087: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1] => 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,2] => 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [2,1] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,3,1] => [3,1,2] => 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1,3] => 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,1,2] => [2,3,1] => 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,3,4,1] => [4,1,2,3] => 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,3,1,4] => [3,1,2,4] => 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [2,4,1,3] => [3,4,1,2] => 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [3,4,2,1] => [4,2,1,3] => 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [3,2,4,1] => [4,1,3,2] => 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [3,1,4,2] => 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [3,2,1,4] => 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [4,2,1,3] => [3,4,2,1] => 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,1,3,2] => [3,2,4,1] => 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,3,4,5,1] => [5,1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,3,4,1,5] => [4,1,2,3,5] => 0 = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [2,3,5,1,4] => [4,5,1,2,3] => 0 = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [3,4,5,2,1] => [5,2,1,3,4] => 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,3,1,4,5] => [3,1,2,4,5] => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,4,2,5,1] => [5,1,4,2,3] => 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [2,4,1,3,5] => [3,4,1,2,5] => 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [2,5,1,3,4] => [3,4,5,1,2] => 1 = 2 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,5,2,4,1] => [4,1,5,2,3] => 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [3,4,2,1,5] => [4,2,1,3,5] => 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,5,2,1,4] => [4,5,2,1,3] => 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [3,5,1,4,2] => [4,2,5,1,3] => 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [4,5,3,2,1] => [5,3,2,1,4] => 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [3,2,4,5,1] => [5,1,3,2,4] => 0 = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [3,2,4,1,5] => [4,1,3,2,5] => 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2,5,1,4] => [4,5,1,3,2] => 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [4,3,5,2,1] => [5,2,1,4,3] => 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [4,2,3,5,1] => [5,1,3,4,2] => 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [4,1,3,5,2] => 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [3,1,4,2,5] => 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [5,2,3,1,4] => [3,1,4,5,2] => 1 = 2 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [5,2,1,4,3] => [4,3,5,2,1] => 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,3,4,2,1] => [4,2,1,5,3] => 0 = 1 - 1
Description
The number of occurrences of the vincular pattern |12-3 in a permutation.
This is the number of occurrences of the pattern $123$, where the first matched entry is the first entry of the permutation and the other two matched entries are consecutive.
In other words, this is the number of ascents whose bottom value is strictly larger than the first entry of the permutation.
Matching statistic: St001163
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001163: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001163: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
Description
The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra.
Matching statistic: St001167
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001167: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra.
The top of a module is the cokernel of the inclusion of the radical of the module into the module.
For Nakayama algebras with at most 8 simple modules, the statistic also coincides with the number of simple modules with projective dimension at least 3 in the corresponding Nakayama algebra.
Matching statistic: St001253
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001253: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 0 = 1 - 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,0]
=> [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0]
=> [3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> 0 = 1 - 1
Description
The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra.
Matching statistic: St000993
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,2}
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,2}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [3,2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [3,2,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [2,2,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [2,2,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,1,1]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,1,1]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,1]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [4,3,2,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> [4,3,2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,2,1]
=> [4,3,2]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [4,3,2]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [4,3,2]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [4,2,2,1]
=> [4,3,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [4,2,2,1]
=> [4,3,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [4,3,1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [4,3,1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [3,2,2,1]
=> [4,3,1]
=> 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> [1]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001038
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 82%●distinct values known / distinct values provided: 80%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,2}
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,2}
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,2}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,2}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,3}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 2
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,1,1,2,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [3,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0]
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,2,2,2,3,3,3,5}
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001199
(load all 105 compositions to match this statistic)
(load all 105 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> [1,3,2] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {1,1}
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {1,1}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1}
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1}
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1}
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> 4
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 3
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,4,1,5] => [1,1,1,0,0,1,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,4,5,1] => [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [3,4,2,1,5] => [1,1,1,0,1,0,0,0,1,0]
=> 2
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [3,4,2,5,1] => [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,4,5,2,1] => [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => [4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,3,2,1] => [4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,2,2,2}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,2,3,4,5,6] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> 5
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [5,1,2,3,4,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [5,6,1,2,3,4] => [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => [6,5,1,2,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,2,4,6,5,3] => [6,4,5,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => [6,5,4,1,2,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,6,5,4] => [6,3,5,1,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [1,3,4,6,5,2] => [6,3,4,5,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => [6,5,3,4,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => [6,4,5,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => [6,5,4,3,1,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => [6,2,5,1,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,0,1,1,0,1,1,0,0,0]
=> [2,1,4,6,5,3] => [6,2,4,5,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => [6,5,2,4,1,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [2,3,1,6,5,4] => [6,2,3,5,1,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,5,1] => [6,2,3,4,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,5,4,1] => [6,5,2,3,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => [6,4,5,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => [6,5,4,2,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,2,6,5,4,1] => [6,3,5,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,5,2,1] => [6,3,4,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [3,6,5,4,2,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,5,3,2,1] => [6,4,5,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001432
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 79%●distinct values known / distinct values provided: 40%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001432: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 79%●distinct values known / distinct values provided: 40%
Values
[1,0]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,2}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,2}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,2}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,2}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,2,2,3}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,3,3,4}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> []
=> ? ∊ {1,2,3,3,3,3,3,3,3,3,3,3,3,4,4,5}
Description
The order dimension of the partition.
Given a partition $\lambda$, let $I(\lambda)$ be the principal order ideal in the Young lattice generated by $\lambda$. The order dimension of a partition is defined as the order dimension of the poset $I(\lambda)$.
The following 249 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001722The number of minimal chains with small intervals between a binary word and the top element. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001933The largest multiplicity of a part in an integer partition. St000026The position of the first return of a Dyck path. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000120The number of left tunnels of a Dyck path. St000291The number of descents of a binary word. St000297The number of leading ones in a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000335The difference of lower and upper interactions. St000390The number of runs of ones in a binary word. St000392The length of the longest run of ones in a binary word. St000531The leading coefficient of the rook polynomial of an integer partition. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000733The row containing the largest entry of a standard tableau. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000876The number of factors in the Catalan decomposition of a binary word. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. 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. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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 $eA$. 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. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in 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. St001313The number of Dyck paths above the lattice path given by a binary word. St001372The length of a longest cyclic run of ones of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. 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. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001659The number of ways to place as many non-attacking rooks as possible on a Ferrers board. St001732The number of peaks visible from the left. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001128The exponens consonantiae of a partition. St001389The number of partitions of the same length below the given integer partition. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000899The maximal number of repetitions of an integer composition. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000990The first ascent of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001964The interval resolution global dimension of a poset. St001162The minimum jump of a permutation. St001875The number of simple modules with projective dimension at most 1. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St000937The number of positive values of the symmetric group character corresponding to the partition. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St000570The Edelman-Greene number of a permutation. St000706The product of the factorials of the multiplicities of an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St000028The number of stack-sorts needed to sort a permutation. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000665The number of rafts of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St000781The number of proper colouring schemes of a Ferrers diagram. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000100The number of linear extensions of a poset. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000678The number of up steps after the last double rise of a Dyck path. St000744The length of the path to the largest entry in a standard Young tableau. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001890The maximum magnitude of the Möbius function of a poset. St000546The number of global descents of a permutation. St000668The least common multiple of the parts of the partition. St000708The product of the parts of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001597The Frobenius rank of a skew partition. St000729The minimal arc length of a set partition. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001330The hat guessing number of a graph. St000654The first descent of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000022The number of fixed points of a permutation. St000534The number of 2-rises of a permutation. St000056The decomposition (or block) number of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St000260The radius of a connected graph. St001810The number of fixed points of a permutation smaller than its largest moved point. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000456The monochromatic index of a connected graph. St000731The number of double exceedences of a permutation. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000487The length of the shortest cycle of a permutation. St000527The width of the poset. St001344The neighbouring number of a permutation. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001527The cyclic permutation representation number of an integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001118The acyclic chromatic index of a graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000908The length of the shortest maximal antichain in a poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. 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 $eA$. 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 $eA$. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St000145The Dyson rank of a partition. St000181The number of connected components of the Hasse diagram for the poset. St000284The Plancherel distribution on integer partitions. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000618The number of self-evacuating tableaux of given shape. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001982The number of orbits of the action of a permutation of given cycle type on the set of edges of the complete graph. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001846The number of elements which do not have a complement in the lattice. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000650The number of 3-rises of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000214The number of adjacencies of a permutation. St000058The order of a permutation. St001820The size of the image of the pop stack sorting operator. St000455The second largest eigenvalue of a graph if it is integral. St000454The largest eigenvalue of a graph if it is integral. St001095The number of non-isomorphic posets with precisely one further covering relation. St001868The number of alignments of type NE of a signed permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000807The sum of the heights of the valleys of the associated bargraph. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000741The Colin de Verdière graph invariant. St001060The distinguishing index of a graph. St001867The number of alignments of type EN of a signed permutation. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St001130The number of two successive successions in a permutation. St000732The number of double deficiencies of a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St000669The number of permutations obtained by switching ascents or descents of size 2. St000237The number of small exceedances. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000884The number of isolated descents of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001394The genus of a permutation. St001556The number of inversions of the third entry of a permutation. St001715The number of non-records in a permutation. St001730The number of times the path corresponding to a binary word crosses the base line. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000764The number of strong records in an integer composition. St000883The number of longest increasing subsequences of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000352The Elizalde-Pak rank of a permutation.
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