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Your data matches 12 different statistics following compositions of up to 3 maps.
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Matching statistic: St001239
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(load all 3 compositions to match this statistic)
St001239: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 2
[1,1,0,0]
=> 1
[1,0,1,0,1,0]
=> 2
[1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> 3
[1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> 3
[1,1,1,0,1,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> 3
Description
The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra.
Matching statistic: St001192
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St001192: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> 0 = 1 - 1
[1,0,1,0]
=> [1,0,1,0]
=> 1 = 2 - 1
[1,1,0,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2 = 3 - 1
Description
The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$.
Matching statistic: St000147
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> 1
[1,0,1,0]
=> [2,1] => [2,1] => [2]
=> 2
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => [2,1]
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [2,3,1] => [3]
=> 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => [2,1,1]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [2,1,1]
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => [3,1]
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => [2,1,1]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => [2,2]
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [3,1]
=> 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [2,3,1,4] => [3,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [3,1]
=> 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [2,3,4,1] => [4]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [2,1,1,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [2,1,1,1]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => [3,1,1]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [2,1,1,1]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => [3,1,1]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,1,1]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [3,2]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => [4,1]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [2,1,1,1]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [2,1,1,1]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => [3,1,1]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [3,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,3,1,5] => [3,1,1]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [3,2]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => [3,2]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => [4,1]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [3,1,1]
=> 3
Description
The largest part of an integer partition.
Matching statistic: St001235
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
St001235: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 1
[1,0,1,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1] => 2
[1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [2] => 1
[1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [2,1] => 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,2] => 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2,1] => 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1] => 3
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> [3] => 1
[1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [3,1] => 2
[1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [2,2] => 2
[1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => 2
[1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => 3
[1,0,1,1,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [3,1] => 2
[1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => 2
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => 2
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1,1] => 3
[1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => 3
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => 4
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [2,2,1] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [3,2] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> [2,2,1] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,1,2] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => 3
Description
The global dimension of the corresponding Comp-Nakayama algebra.
We identify the composition [n1-1,n2-1,...,nr-1] with the Nakayama algebra with Kupisch series [n1,n1-1,...,2,n2,n2-1,...,2,...,nr,nr-1,...,3,2,1]. We call such Nakayama algebras with Kupisch series corresponding to a integer composition "Comp-Nakayama algebra".
Matching statistic: St000485
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
St000485: Permutations ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => 2
[1,1,0,0]
=> [1,2] => [1,2] => 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,2,1] => 2
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 2
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,2,3,1] => 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,2,1,3] => 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,3,2] => 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [4,1,3,2] => 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,2,3,1,4] => 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [5,2,1,4,3] => 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,2,1,3,5] => 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,2,1,3,4] => 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,3,2,4] => 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,1,3,4,2] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [4,1,3,2,5] => 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,5,1,2,4] => 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [5,1,3,2,4] => 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,4,3] => 2
Description
The length of the longest cycle of a permutation.
Matching statistic: St000444
Mp00028: Dyck paths —reverse⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St000444: Dyck paths ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [2,1] => [1,1,0,0]
=> 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,2] => [1,0,1,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0]
=> [1,1,0,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 3
[1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> 4
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 2
Description
The length of the maximal rise of a Dyck path.
Matching statistic: St001062
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001062: Set partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1,0]
=> {{1}}
=> ? = 1
[1,0,1,0]
=> [2,1] => [1,1,0,0]
=> {{1,2}}
=> 2
[1,1,0,0]
=> [1,2] => [1,0,1,0]
=> {{1},{2}}
=> 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,1,0,1,0,0]
=> {{1,3},{2}}
=> 2
[1,0,1,1,0,0]
=> [2,1,3] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2
[1,1,0,0,1,0]
=> [1,3,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 2
[1,1,0,1,0,0]
=> [3,1,2] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 3
[1,1,1,0,0,0]
=> [1,2,3] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> {{1,4},{2},{3}}
=> 2
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> {{1,3},{2},{4}}
=> 2
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> {{1,3,4},{2}}
=> 3
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> {{1},{2,4},{3}}
=> 2
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> {{1,4},{2,3}}
=> 2
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> {{1,2,4},{3}}
=> 3
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 3
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 3
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 4
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> {{1,5},{2},{3},{4}}
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> {{1,4},{2},{3},{5}}
=> 2
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> {{1,3},{2},{4,5}}
=> 2
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,1,0,1,0,1,1,0,0,0]
=> {{1,4,5},{2},{3}}
=> 3
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> {{1,3},{2},{4},{5}}
=> 2
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> {{1,2},{3,5},{4}}
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,1,0,1,1,0,0,1,0,0]
=> {{1,5},{2},{3,4}}
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> {{1,3,5},{2},{4}}
=> 3
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [1,1,0,1,1,0,0,0,1,0]
=> {{1,3,4},{2},{5}}
=> 3
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 3
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> {{1,3,4,5},{2}}
=> 4
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 2
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> {{1},{2,5},{3},{4}}
=> 2
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> {{1},{2,4},{3},{5}}
=> 2
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> {{1},{2,4,5},{3}}
=> 3
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 2
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [1,1,1,0,0,1,0,1,0,0]
=> {{1,5},{2,3},{4}}
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [1,1,1,0,0,1,0,0,1,0]
=> {{1,4},{2,3},{5}}
=> 2
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,1,1,0,1,0,0,1,0,0]
=> {{1,5},{2,4},{3}}
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,1,1,0,1,0,1,0,0,0]
=> {{1,2,5},{3},{4}}
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [1,1,1,0,1,0,0,0,1,0]
=> {{1,2,4},{3},{5}}
=> 3
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 3
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [1,1,1,0,0,1,1,0,0,0]
=> {{1,4,5},{2,3}}
=> 3
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> {{1,2,4,5},{3}}
=> 4
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 3
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> {{1},{2},{3,5},{4}}
=> 2
Description
The maximal size of a block of a set partition.
Matching statistic: St000392
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00130: Permutations —descent tops⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => => ? = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [3,1,2] => 01 => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 10 => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 01 => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,2,1] => 11 => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 00 => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [4,1,2,3] => 001 => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [3,1,2,4] => 010 => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 101 => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [4,3,1,2] => 011 => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 100 => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,4,2,3] => 001 => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 010 => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [4,2,1,3] => 101 => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [3,1,4,2] => 011 => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,2,1,4] => 110 => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 001 => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,3,2] => 011 => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,3,2,1] => 111 => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 000 => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [5,1,2,3,4] => 0001 => 1 = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [4,1,2,3,5] => 0010 => 1 = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [3,1,2,5,4] => 0101 => 1 = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [5,4,1,2,3] => 0011 => 2 = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [3,1,2,4,5] => 0100 => 1 = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,5,3,4] => 1001 => 1 = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => 1010 => 1 = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [5,3,1,2,4] => 0101 => 1 = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [4,1,2,5,3] => 0011 => 2 = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [4,3,1,2,5] => 0110 => 2 = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => 1001 => 1 = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,4,3] => 1011 => 2 = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [5,4,3,1,2] => 0111 => 3 = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => 1000 => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,5,2,3,4] => 0001 => 1 = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,4,2,3,5] => 0010 => 1 = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => 0101 => 1 = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,5,4,2,3] => 0011 => 2 = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => 0100 => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [5,2,1,3,4] => 1001 => 1 = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [4,2,1,3,5] => 1010 => 1 = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [3,1,5,2,4] => 0101 => 1 = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [5,2,4,1,3] => 0011 => 2 = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [3,1,4,2,5] => 0110 => 2 = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,2,1,5,4] => 1101 => 2 = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [5,4,2,1,3] => 1011 => 2 = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,1,5,4,2] => 0111 => 3 = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,2,1,4,5] => 1100 => 2 = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,5,3,4] => 0001 => 1 = 2 - 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000374
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Mp00058: Perfect matchings —to permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
St000374: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [(1,2)]
=> [2,1] => [1,2] => 0 = 1 - 1
[1,0,1,0]
=> [(1,2),(3,4)]
=> [2,1,4,3] => [3,2,1,4] => 1 = 2 - 1
[1,1,0,0]
=> [(1,4),(2,3)]
=> [4,3,2,1] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> [2,1,4,3,6,5] => [3,2,5,4,1,6] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> [2,1,6,5,4,3] => [3,2,1,4,5,6] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> [4,3,2,1,6,5] => [5,4,3,2,1,6] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> [6,3,2,5,4,1] => [1,5,4,2,3,6] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> [6,5,4,3,2,1] => [1,2,3,4,5,6] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8)]
=> [2,1,4,3,6,5,8,7] => [3,2,5,4,7,6,1,8] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> [2,1,4,3,8,7,6,5] => [3,2,5,4,1,6,7,8] => ? = 2 - 1
[1,0,1,1,0,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8)]
=> [2,1,6,5,4,3,8,7] => [3,2,7,6,5,4,1,8] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> [2,1,8,5,4,7,6,3] => [3,2,1,7,6,4,5,8] => ? = 3 - 1
[1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> [2,1,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8)]
=> [4,3,2,1,6,5,8,7] => [5,4,3,2,7,6,1,8] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> [4,3,2,1,8,7,6,5] => [5,4,3,2,1,6,7,8] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> [6,3,2,5,4,1,8,7] => [7,4,3,6,5,2,1,8] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> [8,3,2,5,4,7,6,1] => [1,5,4,7,6,2,3,8] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [(1,8),(2,3),(4,7),(5,6)]
=> [8,3,2,7,6,5,4,1] => [1,5,4,2,3,6,7,8] => ? = 3 - 1
[1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> [6,5,4,3,2,1,8,7] => [7,6,5,4,3,2,1,8] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> [8,5,4,3,2,7,6,1] => [1,7,6,5,4,2,3,8] => ? = 3 - 1
[1,1,1,0,1,0,0,0]
=> [(1,8),(2,7),(3,4),(5,6)]
=> [8,7,4,3,6,5,2,1] => [1,2,7,6,3,4,5,8] => ? = 4 - 1
[1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> [8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10)]
=> [2,1,4,3,6,5,8,7,10,9] => [3,2,5,4,7,6,9,8,1,10] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> [2,1,4,3,6,5,10,9,8,7] => [3,2,5,4,7,6,1,8,9,10] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10)]
=> [2,1,4,3,8,7,6,5,10,9] => [3,2,5,4,9,8,7,6,1,10] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> [2,1,4,3,10,7,6,9,8,5] => [3,2,5,4,1,9,8,6,7,10] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8)]
=> [2,1,4,3,10,9,8,7,6,5] => [3,2,5,4,1,6,7,8,9,10] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,6),(4,5),(7,8),(9,10)]
=> [2,1,6,5,4,3,8,7,10,9] => [3,2,7,6,5,4,9,8,1,10] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> [2,1,6,5,4,3,10,9,8,7] => [3,2,7,6,5,4,1,8,9,10] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,8),(4,5),(6,7),(9,10)]
=> [2,1,8,5,4,7,6,3,10,9] => [3,2,9,6,5,8,7,4,1,10] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [(1,2),(3,10),(4,5),(6,7),(8,9)]
=> [2,1,10,5,4,7,6,9,8,3] => [3,2,1,7,6,9,8,4,5,10] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [(1,2),(3,10),(4,5),(6,9),(7,8)]
=> [2,1,10,5,4,9,8,7,6,3] => [3,2,1,7,6,4,5,8,9,10] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10)]
=> [2,1,8,7,6,5,4,3,10,9] => [3,2,9,8,7,6,5,4,1,10] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [(1,2),(3,10),(4,7),(5,6),(8,9)]
=> [2,1,10,7,6,5,4,9,8,3] => [3,2,1,9,8,7,6,4,5,10] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [(1,2),(3,10),(4,9),(5,6),(7,8)]
=> [2,1,10,9,6,5,8,7,4,3] => [3,2,1,4,9,8,5,6,7,10] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> [2,1,10,9,8,7,6,5,4,3] => [3,2,1,4,5,6,7,8,9,10] => 1 = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [(1,4),(2,3),(5,6),(7,8),(9,10)]
=> [4,3,2,1,6,5,8,7,10,9] => [5,4,3,2,7,6,9,8,1,10] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> [4,3,2,1,6,5,10,9,8,7] => [5,4,3,2,7,6,1,8,9,10] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> [4,3,2,1,8,7,6,5,10,9] => [5,4,3,2,9,8,7,6,1,10] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [(1,4),(2,3),(5,10),(6,7),(8,9)]
=> [4,3,2,1,10,7,6,9,8,5] => [5,4,3,2,1,9,8,6,7,10] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [(1,4),(2,3),(5,10),(6,9),(7,8)]
=> [4,3,2,1,10,9,8,7,6,5] => [5,4,3,2,1,6,7,8,9,10] => 1 = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8),(9,10)]
=> [6,3,2,5,4,1,8,7,10,9] => [7,4,3,6,5,2,9,8,1,10] => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [(1,6),(2,3),(4,5),(7,10),(8,9)]
=> [6,3,2,5,4,1,10,9,8,7] => [7,4,3,6,5,2,1,8,9,10] => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [(1,8),(2,3),(4,5),(6,7),(9,10)]
=> [8,3,2,5,4,7,6,1,10,9] => [9,4,3,6,5,8,7,2,1,10] => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [(1,10),(2,3),(4,5),(6,7),(8,9)]
=> [10,3,2,5,4,7,6,9,8,1] => [1,5,4,7,6,9,8,2,3,10] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [(1,10),(2,3),(4,5),(6,9),(7,8)]
=> [10,3,2,5,4,9,8,7,6,1] => [1,5,4,7,6,2,3,8,9,10] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [(1,8),(2,3),(4,7),(5,6),(9,10)]
=> [8,3,2,7,6,5,4,1,10,9] => [9,4,3,8,7,6,5,2,1,10] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [(1,10),(2,3),(4,7),(5,6),(8,9)]
=> [10,3,2,7,6,5,4,9,8,1] => [1,5,4,9,8,7,6,2,3,10] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [(1,10),(2,3),(4,9),(5,6),(7,8)]
=> [10,3,2,9,6,5,8,7,4,1] => [1,5,4,2,9,8,3,6,7,10] => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [(1,10),(2,3),(4,9),(5,8),(6,7)]
=> [10,3,2,9,8,7,6,5,4,1] => [1,5,4,2,3,6,7,8,9,10] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10)]
=> [6,5,4,3,2,1,8,7,10,9] => [7,6,5,4,3,2,9,8,1,10] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [(1,6),(2,5),(3,4),(7,10),(8,9)]
=> [6,5,4,3,2,1,10,9,8,7] => [7,6,5,4,3,2,1,8,9,10] => 1 = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [(1,8),(2,5),(3,4),(6,7),(9,10)]
=> [8,5,4,3,2,7,6,1,10,9] => [9,6,5,4,3,8,7,2,1,10] => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [(1,10),(2,5),(3,4),(6,7),(8,9)]
=> [10,5,4,3,2,7,6,9,8,1] => [1,7,6,5,4,9,8,2,3,10] => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [(1,10),(2,5),(3,4),(6,9),(7,8)]
=> [10,5,4,3,2,9,8,7,6,1] => [1,7,6,5,4,2,3,8,9,10] => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [(1,8),(2,7),(3,4),(5,6),(9,10)]
=> [8,7,4,3,6,5,2,1,10,9] => [9,8,5,4,7,6,3,2,1,10] => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [(1,10),(2,7),(3,4),(5,6),(8,9)]
=> [10,7,4,3,6,5,2,9,8,1] => [1,9,6,5,8,7,4,2,3,10] => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [(1,10),(2,9),(3,4),(5,6),(7,8)]
=> [10,9,4,3,6,5,8,7,2,1] => [1,2,7,6,9,8,3,4,5,10] => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [(1,10),(2,9),(3,4),(5,8),(6,7)]
=> [10,9,4,3,8,7,6,5,2,1] => [1,2,7,6,3,4,5,8,9,10] => ? = 4 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> [8,7,6,5,4,3,2,1,10,9] => [9,8,7,6,5,4,3,2,1,10] => 1 = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [(1,10),(2,7),(3,6),(4,5),(8,9)]
=> [10,7,6,5,4,3,2,9,8,1] => [1,9,8,7,6,5,4,2,3,10] => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [(1,10),(2,9),(3,6),(4,5),(7,8)]
=> [10,9,6,5,4,3,8,7,2,1] => [1,2,9,8,7,6,3,4,5,10] => ? = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,5),(6,7)]
=> [10,9,8,5,4,7,6,3,2,1] => [1,2,3,9,8,4,5,6,7,10] => ? = 5 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> [10,9,8,7,6,5,4,3,2,1] => [1,2,3,4,5,6,7,8,9,10] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> [2,1,4,3,6,5,8,7,10,9,12,11] => [3,2,5,4,7,6,9,8,11,10,1,12] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,12),(10,11)]
=> [2,1,4,3,6,5,8,7,12,11,10,9] => [3,2,5,4,7,6,9,8,1,10,11,12] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9),(11,12)]
=> [2,1,4,3,6,5,10,9,8,7,12,11] => [3,2,5,4,7,6,11,10,9,8,1,12] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,9),(10,11)]
=> [2,1,4,3,6,5,12,9,8,11,10,7] => [3,2,5,4,7,6,1,11,10,8,9,12] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> [2,1,4,3,6,5,12,11,10,9,8,7] => [3,2,5,4,7,6,1,8,9,10,11,12] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,10),(11,12)]
=> [2,1,4,3,8,7,6,5,10,9,12,11] => [3,2,5,4,9,8,7,6,11,10,1,12] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7),(9,12),(10,11)]
=> [2,1,4,3,8,7,6,5,12,11,10,9] => [3,2,5,4,9,8,7,6,1,10,11,12] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9),(11,12)]
=> [2,1,4,3,10,7,6,9,8,5,12,11] => [3,2,5,4,11,8,7,10,9,6,1,12] => ? = 2 - 1
Description
The number of exclusive right-to-left minima of a permutation.
This is the number of right-to-left minima that are not left-to-right maxima.
This is also the number of non weak exceedences of a permutation that are also not mid-points of a decreasing subsequence of length 3.
Given a permutation $\pi = [\pi_1,\ldots,\pi_n]$, this statistic counts the number of position $j$ such that $\pi_j < j$ and there do not exist indices $i,k$ with $i < j < k$ and $\pi_i > \pi_j > \pi_k$.
See also [[St000213]] and [[St000119]].
Matching statistic: St001207
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Mp00254: Permutations —Inverse fireworks map⟶ Permutations
St001207: Permutations ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 67%
Values
[1,0]
=> [1] => [1] => ? = 1 - 1
[1,0,1,0]
=> [2,1] => [2,1] => 1 = 2 - 1
[1,1,0,0]
=> [1,2] => [1,2] => 0 = 1 - 1
[1,0,1,0,1,0]
=> [2,3,1] => [1,3,2] => 1 = 2 - 1
[1,0,1,1,0,0]
=> [2,1,3] => [2,1,3] => 1 = 2 - 1
[1,1,0,0,1,0]
=> [1,3,2] => [1,3,2] => 1 = 2 - 1
[1,1,0,1,0,0]
=> [3,1,2] => [3,1,2] => 2 = 3 - 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [2,3,4,1] => [1,2,4,3] => 1 = 2 - 1
[1,0,1,0,1,1,0,0]
=> [2,3,1,4] => [1,3,2,4] => 1 = 2 - 1
[1,0,1,1,0,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => 1 = 2 - 1
[1,0,1,1,0,1,0,0]
=> [2,4,1,3] => [2,4,1,3] => 2 = 3 - 1
[1,0,1,1,1,0,0,0]
=> [2,1,3,4] => [2,1,3,4] => 1 = 2 - 1
[1,1,0,0,1,0,1,0]
=> [1,3,4,2] => [1,2,4,3] => 1 = 2 - 1
[1,1,0,0,1,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => 1 = 2 - 1
[1,1,0,1,0,0,1,0]
=> [3,1,4,2] => [2,1,4,3] => 1 = 2 - 1
[1,1,0,1,0,1,0,0]
=> [3,4,1,2] => [2,4,1,3] => 2 = 3 - 1
[1,1,0,1,1,0,0,0]
=> [3,1,2,4] => [3,1,2,4] => 2 = 3 - 1
[1,1,1,0,0,0,1,0]
=> [1,2,4,3] => [1,2,4,3] => 1 = 2 - 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,4,2,3] => 2 = 3 - 1
[1,1,1,0,1,0,0,0]
=> [4,1,2,3] => [4,1,2,3] => 3 = 4 - 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => [1,2,3,5,4] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,1,5] => [1,2,4,3,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,3,1,5,4] => [1,3,2,5,4] => ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => [1,3,5,2,4] => ? = 3 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => [1,3,2,4,5] => ? = 2 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,4,5,3] => [2,1,3,5,4] => ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => [1,3,2,5,4] => ? = 2 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => [1,3,5,2,4] => ? = 3 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => [2,4,1,3,5] => ? = 3 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => [2,1,5,3,4] => ? = 3 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => [2,5,1,3,4] => ? = 4 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => ? = 2 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => [1,2,3,5,4] => ? = 2 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => [1,2,4,3,5] => ? = 2 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => ? = 2 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => [1,3,5,2,4] => ? = 3 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => ? = 2 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => [2,1,3,5,4] => ? = 2 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => [2,1,4,3,5] => ? = 2 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => [1,3,2,5,4] => ? = 2 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,4,5,1,2] => [1,3,5,2,4] => ? = 3 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,4,1,3,5] => ? = 3 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => [3,1,2,5,4] => ? = 3 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,5,2,4] => [3,1,5,2,4] => ? = 3 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,5,1,2,4] => [3,5,1,2,4] => ? = 4 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [3,1,2,4,5] => [3,1,2,4,5] => ? = 3 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => [1,2,3,5,4] => ? = 2 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => [1,2,4,3,5] => ? = 2 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => [1,3,2,5,4] => ? = 2 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => [1,3,5,2,4] => ? = 3 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,4,2,3,5] => [1,4,2,3,5] => ? = 3 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => [3,1,2,5,4] => ? = 3 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => [3,1,5,2,4] => ? = 3 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => [3,5,1,2,4] => ? = 4 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [4,1,2,3,5] => [4,1,2,3,5] => ? = 4 - 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => [1,2,3,5,4] => ? = 2 - 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => [1,2,5,3,4] => ? = 3 - 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => [1,5,2,3,4] => ? = 4 - 1
[1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => [5,1,2,3,4] => ? = 5 - 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,2,3,4,5] => ? = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,6,1] => [1,2,3,4,6,5] => ? = 2 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [2,3,4,5,1,6] => [1,2,3,5,4,6] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [2,3,4,1,6,5] => [1,2,4,3,6,5] => ? = 2 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [2,3,4,6,1,5] => [1,2,4,6,3,5] => ? = 3 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [2,3,4,1,5,6] => [1,2,4,3,5,6] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [2,3,1,5,6,4] => [1,3,2,4,6,5] => ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [2,3,1,5,4,6] => [1,3,2,5,4,6] => ? = 2 - 1
Description
The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$.
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