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Your data matches 222 different statistics following compositions of up to 3 maps.
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Matching statistic: St001198
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
Description
The 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 $eA$.
Matching statistic: St001206
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 3
Description
The 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 $eA$.
Matching statistic: St001530
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001530: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001530: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 3
Description
The depth of a Dyck path. That is the depth of the corresponding Nakayama algebra with a linear quiver.
Matching statistic: St001294
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001294: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1 = 2 - 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 2 = 3 - 1
Description
The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra.
See [[http://www.findstat.org/DyckPaths/NakayamaAlgebras]].
The number of algebras where the statistic returns a value less than or equal to 1 might be given by the Motzkin numbers https://oeis.org/A001006.
Matching statistic: St001717
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
Mp00047: Ordered trees —to poset⟶ Posets
St001717: Posets ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 100%
Values
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [[[[[]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 2 + 3
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [[],[[[[]]]]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [[[],[[[]]]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [[[[],[[]]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [[[[[]]]],[]]
=> ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
=> 5 = 2 + 3
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [[[[[]]],[]]]
=> ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
=> 5 = 2 + 3
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [[[[[]],[]]]]
=> ([(0,5),(1,3),(3,5),(4,2),(5,4)],6)
=> 5 = 2 + 3
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5 = 2 + 3
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5 = 2 + 3
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5 = 2 + 3
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5 = 2 + 3
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5 = 2 + 3
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [[[[[],[]]]]]
=> ([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> 5 = 2 + 3
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 3 + 3
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [[[[[[]]]]]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 6 = 3 + 3
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[],[[],[[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[],[[],[[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[],[[],[[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[],[[],[[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[],[[],[[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [[],[[],[[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,5,3,2,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,5,3,4,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,5,4,3,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [[],[[[[]]],[]]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,3,4,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,3,4,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,3,5,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,3,5,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,4,2,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,4,2,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,4,3,2,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,4,3,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,4,5,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,4,5,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,5,2,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,5,2,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,5,3,2,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,5,3,4,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,5,4,2,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[1,6,5,4,3,2] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[],[[[[[]]]]]]
=> ([(0,6),(1,5),(2,6),(3,4),(4,2),(5,3)],7)
=> ? = 3 + 3
[2,1,6,3,4,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2 + 3
[2,1,6,3,5,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2 + 3
[2,1,6,4,3,5] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2 + 3
[2,1,6,4,5,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2 + 3
[2,1,6,5,3,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2 + 3
[2,1,6,5,4,3] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [[[]],[[[[]]]]]
=> ([(0,5),(1,3),(2,6),(3,6),(4,2),(5,4)],7)
=> ? = 2 + 3
[2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2 + 3
[2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2 + 3
[2,4,6,3,1,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2 + 3
[2,4,6,3,5,1] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2 + 3
[2,4,6,5,1,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2 + 3
[2,4,6,5,3,1] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [[[],[[],[[]]]]]
=> ([(0,6),(1,5),(2,3),(3,6),(5,4),(6,5)],7)
=> ? = 2 + 3
[2,5,1,3,4,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[[],[[[]]]],[]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
[2,5,1,4,3,6] => [1,1,0,1,1,1,0,0,0,0,1,0]
=> [[[],[[[]]]],[]]
=> ([(0,6),(1,5),(2,3),(3,4),(4,5),(5,6)],7)
=> ? = 2 + 3
Description
The largest size of an interval in a poset.
Matching statistic: St000703
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000703: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 3
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 3
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,3,2,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,3,5,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,2,6,4,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,3,2,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,3,2,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,3,4,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,3,6,4,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,4,3,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,4,6,3,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,3,2,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,3,4,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,4,3,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
Description
The number of deficiencies of a permutation.
This is defined as
$$\operatorname{dec}(\sigma)=\#\{i:\sigma(i) < i\}.$$
The number of exceedances is [[St000155]].
Matching statistic: St000994
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000994: Permutations ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 100%
Values
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,5,4,3,1] => 2
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => 2
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => 2
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => 2
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => 2
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => 2
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 3
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 3
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,1,0,0,0,0,0]
=> [2,3,7,6,5,4,1] => ? = 2
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,1,0,1,1,1,0,0,0,0,0]
=> [2,7,3,6,5,4,1] => ? = 2
[1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,3,2,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,3,5,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,1,1,0,0,0,0,0]
=> [2,7,4,6,5,3,1] => ? = 2
[1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,2,6,4,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,3,2,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,3,2,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,3,4,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,3,6,4,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,1,1,1,0,0,0,0,1,0,0]
=> [2,6,5,4,3,7,1] => ? = 2
[1,5,4,3,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,1,1,1,0,0,0,1,0,0,0]
=> [2,7,5,4,3,6,1] => ? = 2
[1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,4,6,3,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,1,1,1,0,0,1,0,0,0,0]
=> [2,7,5,4,6,3,1] => ? = 2
[1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,3,2,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,3,4,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,5,6,4,3,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,6,4,5,3,1] => ? = 2
[1,6,2,3,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,3,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,4,3,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,4,5,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,5,3,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,2,5,4,3] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,3,2,4,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
[1,6,3,2,5,4] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> [2,7,6,5,4,3,1] => ? = 3
Description
The number of cycle peaks and the number of cycle valleys of a permutation.
A '''cycle peak''' of a permutation $\pi$ is an index $i$ such that $\pi^{-1}(i) < i > \pi(i)$. Analogously, a '''cycle valley''' is an index $i$ such that $\pi^{-1}(i) > i < \pi(i)$.
Clearly, every cycle of $\pi$ contains as many peaks as valleys.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000264: Graphs ⟶ ℤResult quality: 27% ●values known / values provided: 27%●distinct values known / distinct values provided: 50%
Values
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? = 2 + 1
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? = 2 + 1
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? = 2 + 1
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? = 2 + 1
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? = 2 + 1
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> ? = 2 + 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? = 2 + 1
[2,5,1,3,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,5,3,1,4] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 + 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,5,2,3] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,1,5,3,2] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,5,1,3] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,2,5,3,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,5,1,2] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,3,5,2,1] => [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,5,1,2,3] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,5,1,3,2] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,5,2,1,3] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,5,2,3,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,5,3,1,2] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[4,5,3,2,1] => [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ? = 2 + 1
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ? = 3 + 1
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> ? = 3 + 1
[1,2,6,3,4,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,6,3,5,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,6,4,3,5] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,6,4,5,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,6,5,3,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,2,6,5,4,3] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,4] => ([(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,2,4,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,2,5,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,4,2,5] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,4,5,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,5,2,4] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,3,6,5,4,2] => [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,2,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,6,2,3,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,6,2,5,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,6,3,2,5] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,6,3,5,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,6,5,2,3] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,4,6,5,3,2] => [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,3,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,3,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,4,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,4,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,6,3,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,2,6,4,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,2,4,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,2,6,4] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,4,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,4,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,6,2,4] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,3,6,4,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,2,3,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,2,6,3] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,3,2,6] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,3,6,2] => [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,6,2,3] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,4,6,3,2] => [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,6,2,3,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,6,2,4,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,6,3,2,4] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,6,3,4,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,6,4,2,3] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,5,6,4,3,2] => [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,3,6,1,4,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,3,6,1,5,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,3,6,4,1,5] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,3,6,4,5,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,3,6,5,1,4] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,3,6,5,4,1] => [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,1,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,4,6,1,3,5] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,4,6,1,5,3] => [1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000259
Mp00064: Permutations —reverse⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[4,1,2,3] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2
[4,1,3,2] => [2,3,1,4] => [4] => ([],4)
=> ? = 2
[4,2,1,3] => [3,1,2,4] => [4] => ([],4)
=> ? = 2
[4,2,3,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? = 2
[4,3,1,2] => [2,1,3,4] => [4] => ([],4)
=> ? = 2
[4,3,2,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 2
[1,5,2,3,4] => [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[1,5,2,4,3] => [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,5,3,2,4] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,5,3,4,2] => [2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[1,5,4,2,3] => [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[1,5,4,3,2] => [2,3,4,5,1] => [5] => ([],5)
=> ? = 2
[2,5,1,3,4] => [4,3,1,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,1,4,3] => [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,5,3,1,4] => [4,1,3,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[2,5,3,4,1] => [1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[2,5,4,1,3] => [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[2,5,4,3,1] => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? = 2
[3,5,1,2,4] => [4,2,1,5,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,5,1,4,2] => [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[3,5,2,1,4] => [4,1,2,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[3,5,4,1,2] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[3,5,4,2,1] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,1,2,3,5] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,1,2,5,3] => [3,5,2,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,1,3,2,5] => [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,1,3,5,2] => [2,5,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,1,5,2,3] => [3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,1,5,3,2] => [2,3,5,1,4] => [5] => ([],5)
=> ? = 2
[4,2,1,3,5] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2
[4,2,1,5,3] => [3,5,1,2,4] => [5] => ([],5)
=> ? = 2
[4,2,3,1,5] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,2,3,5,1] => [1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,2,5,1,3] => [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,2,5,3,1] => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2
[4,3,1,2,5] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,3,1,5,2] => [2,5,1,3,4] => [5] => ([],5)
=> ? = 2
[4,3,2,1,5] => [5,1,2,3,4] => [5] => ([],5)
=> ? = 2
[4,3,2,5,1] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? = 2
[4,3,5,1,2] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,3,5,2,1] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2
[4,5,1,2,3] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,5,1,3,2] => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,5,2,1,3] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,5,2,3,1] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2
[4,5,3,1,2] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[4,5,3,2,1] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2
[5,1,2,3,4] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3
[5,1,2,4,3] => [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3
[1,2,6,3,4,5] => [5,4,3,6,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,3,5,4] => [4,5,3,6,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,4,3,5] => [5,3,4,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,4,5,3] => [3,5,4,6,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,5,3,4] => [4,3,5,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,2,6,5,4,3] => [3,4,5,6,2,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,3,6,2,4,5] => [5,4,2,6,3,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,2,5,4] => [4,5,2,6,3,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,4,2,5] => [5,2,4,6,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,4,5,2] => [2,5,4,6,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,5,2,4] => [4,2,5,6,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,3,6,5,4,2] => [2,4,5,6,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,4,6,2,3,5] => [5,3,2,6,4,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,6,2,5,3] => [3,5,2,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,6,3,2,5] => [5,2,3,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,6,3,5,2] => [2,5,3,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,6,5,2,3] => [3,2,5,6,4,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,4,6,5,3,2] => [2,3,5,6,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,2,3,4,6] => [6,4,3,2,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,2,3,6,4] => [4,6,3,2,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,2,4,3,6] => [6,3,4,2,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,2,4,6,3] => [3,6,4,2,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,2,6,3,4] => [4,3,6,2,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,2,6,4,3] => [3,4,6,2,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,3,2,4,6] => [6,4,2,3,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,2,6,4] => [4,6,2,3,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,3,4,2,6] => [6,2,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,4,6,2] => [2,6,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,6,2,4] => [4,2,6,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,3,6,4,2] => [2,4,6,3,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,4,2,3,6] => [6,3,2,4,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,4,2,6,3] => [3,6,2,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,4,3,2,6] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,4,3,6,2] => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,4,6,2,3] => [3,2,6,4,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,4,6,3,2] => [2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,5,6,2,3,4] => [4,3,2,6,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,2,4,3] => [3,4,2,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,3,2,4] => [4,2,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,3,4,2] => [2,4,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,4,2,3] => [3,2,4,6,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[1,5,6,4,3,2] => [2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[2,3,6,1,4,5] => [5,4,1,6,3,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,1,5,4] => [4,5,1,6,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,4,1,5] => [5,1,4,6,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,4,5,1] => [1,5,4,6,3,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,5,1,4] => [4,1,5,6,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,3,6,5,4,1] => [1,4,5,6,3,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,6,1,3,5] => [5,3,1,6,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[2,4,6,1,5,3] => [3,5,1,6,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mp00064: Permutations —reverse⟶ Permutations
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[4,1,2,3] => [3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 2 - 1
[4,1,3,2] => [2,3,1,4] => [4] => ([],4)
=> ? = 2 - 1
[4,2,1,3] => [3,1,2,4] => [4] => ([],4)
=> ? = 2 - 1
[4,2,3,1] => [1,3,2,4] => [1,3] => ([(2,3)],4)
=> ? = 2 - 1
[4,3,1,2] => [2,1,3,4] => [4] => ([],4)
=> ? = 2 - 1
[4,3,2,1] => [1,2,3,4] => [4] => ([],4)
=> ? = 2 - 1
[1,5,2,3,4] => [4,3,2,5,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,2,4,3] => [3,4,2,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,3,2,4] => [4,2,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,3,4,2] => [2,4,3,5,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,4,2,3] => [3,2,4,5,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[1,5,4,3,2] => [2,3,4,5,1] => [5] => ([],5)
=> ? = 2 - 1
[2,5,1,3,4] => [4,3,1,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,1,4,3] => [3,4,1,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,3,1,4] => [4,1,3,5,2] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,3,4,1] => [1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,4,1,3] => [3,1,4,5,2] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[2,5,4,3,1] => [1,3,4,5,2] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[3,5,1,2,4] => [4,2,1,5,3] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,1,4,2] => [2,4,1,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,2,1,4] => [4,1,2,5,3] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,2,4,1] => [1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,4,1,2] => [2,1,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[3,5,4,2,1] => [1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,2,3,5] => [5,3,2,1,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,2,5,3] => [3,5,2,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,3,2,5] => [5,2,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,3,5,2] => [2,5,3,1,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,5,2,3] => [3,2,5,1,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,1,5,3,2] => [2,3,5,1,4] => [5] => ([],5)
=> ? = 2 - 1
[4,2,1,3,5] => [5,3,1,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[4,2,1,5,3] => [3,5,1,2,4] => [5] => ([],5)
=> ? = 2 - 1
[4,2,3,1,5] => [5,1,3,2,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,3,5,1] => [1,5,3,2,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,5,1,3] => [3,1,5,2,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,2,5,3,1] => [1,3,5,2,4] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[4,3,1,2,5] => [5,2,1,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,1,5,2] => [2,5,1,3,4] => [5] => ([],5)
=> ? = 2 - 1
[4,3,2,1,5] => [5,1,2,3,4] => [5] => ([],5)
=> ? = 2 - 1
[4,3,2,5,1] => [1,5,2,3,4] => [1,4] => ([(3,4)],5)
=> ? = 2 - 1
[4,3,5,1,2] => [2,1,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,3,5,2,1] => [1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,1,2,3] => [3,2,1,5,4] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,1,3,2] => [2,3,1,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,2,1,3] => [3,1,2,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,2,3,1] => [1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,3,1,2] => [2,1,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[4,5,3,2,1] => [1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 2 - 1
[5,1,2,3,4] => [4,3,2,1,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[5,1,2,4,3] => [3,4,2,1,5] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 3 - 1
[1,2,6,3,4,5] => [5,4,3,6,2,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,6,3,5,4] => [4,5,3,6,2,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,6,4,3,5] => [5,3,4,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,6,4,5,3] => [3,5,4,6,2,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,6,5,3,4] => [4,3,5,6,2,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,2,6,5,4,3] => [3,4,5,6,2,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,2,4,5] => [5,4,2,6,3,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,2,5,4] => [4,5,2,6,3,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,4,2,5] => [5,2,4,6,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,4,5,2] => [2,5,4,6,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,5,2,4] => [4,2,5,6,3,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,3,6,5,4,2] => [2,4,5,6,3,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,2,3,5] => [5,3,2,6,4,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,2,5,3] => [3,5,2,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,3,2,5] => [5,2,3,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,3,5,2] => [2,5,3,6,4,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,5,2,3] => [3,2,5,6,4,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,4,6,5,3,2] => [2,3,5,6,4,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,2,3,4,6] => [6,4,3,2,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,2,3,6,4] => [4,6,3,2,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,2,4,3,6] => [6,3,4,2,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,2,4,6,3] => [3,6,4,2,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,2,6,3,4] => [4,3,6,2,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,2,6,4,3] => [3,4,6,2,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,3,2,4,6] => [6,4,2,3,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,3,2,6,4] => [4,6,2,3,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,3,4,2,6] => [6,2,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,3,4,6,2] => [2,6,4,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,3,6,2,4] => [4,2,6,3,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,3,6,4,2] => [2,4,6,3,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,4,2,3,6] => [6,3,2,4,5,1] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,4,2,6,3] => [3,6,2,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,4,3,2,6] => [6,2,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,4,3,6,2] => [2,6,3,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,4,6,2,3] => [3,2,6,4,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,4,6,3,2] => [2,3,6,4,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,2,3,4] => [4,3,2,6,5,1] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,2,4,3] => [3,4,2,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,3,2,4] => [4,2,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,3,4,2] => [2,4,3,6,5,1] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,4,2,3] => [3,2,4,6,5,1] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[1,5,6,4,3,2] => [2,3,4,6,5,1] => [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,1,4,5] => [5,4,1,6,3,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,1,5,4] => [4,5,1,6,3,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,4,1,5] => [5,1,4,6,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,4,5,1] => [1,5,4,6,3,2] => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,5,1,4] => [4,1,5,6,3,2] => [2,3,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,3,6,5,4,1] => [1,4,5,6,3,2] => [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,4,6,1,3,5] => [5,3,1,6,4,2] => [1,2,2,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
[2,4,6,1,5,3] => [3,5,1,6,4,2] => [3,2,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 1 = 2 - 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 212 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000297The number of leading ones in a binary word. St000288The number of ones in a binary word. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000223The number of nestings in the permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001095The number of non-isomorphic posets with precisely one further covering relation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001597The Frobenius rank of a skew partition. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001801Half the number of preimage-image pairs of different parity in a permutation. St000039The number of crossings of a permutation. St000155The number of exceedances (also excedences) of a permutation. St000213The number of weak exceedances (also weak excedences) of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001520The number of strict 3-descents. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001589The nesting number of a perfect matching. St000443The number of long tunnels of a Dyck path. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001015Number of indecomposable injective modules with codominant dimension equal to one in the Nakayama algebra corresponding to the Dyck path. St001016Number of indecomposable injective modules with codominant dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001297The number of indecomposable non-injective projective modules minus the number of indecomposable non-injective projective modules that have reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St000031The number of cycles in the cycle decomposition of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001948The number of augmented double ascents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000120The number of left tunnels of a Dyck path. St000181The number of connected components of the Hasse diagram for the poset. St000242The number of indices that are not cyclical small weak excedances. St000317The cycle descent number of a permutation. St000327The number of cover relations in a poset. St000331The number of upper interactions of a Dyck path. St000358The number of occurrences of the pattern 31-2. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000487The length of the shortest cycle of a permutation. St000662The staircase size of the code of a permutation. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001298The number of repeated entries in the Lehmer code of a permutation. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001668The number of points of the poset minus the width of the poset. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000015The number of peaks of a Dyck path. St000021The number of descents of a permutation. St000062The length of the longest increasing subsequence of the permutation. St000080The rank of the poset. St000083The number of left oriented leafs of a binary tree except the first one. St000168The number of internal nodes of an ordered tree. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St000210Minimum over maximum difference of elements in cycles. St000238The number of indices that are not small weak excedances. St000314The number of left-to-right-maxima of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000332The positive inversions of an alternating sign matrix. St000354The number of recoils of a permutation. St000619The number of cyclic descents of a permutation. St000649The number of 3-excedences of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000702The number of weak deficiencies of a permutation. St000740The last entry of a permutation. St000829The Ulam distance of a permutation to the identity permutation. St000863The length of the first row of the shifted shape of a permutation. St000923The minimal number with no two order isomorphic substrings of this length in a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000956The maximal displacement of a permutation. St000991The number of right-to-left minima 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. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001061The number of indices that are both descents and recoils of a permutation. St001130The number of two successive successions in a permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001246The maximal difference between two consecutive entries of a permutation. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001480The number of simple summands of the module J^2/J^3. St001489The maximum of the number of descents and the number of inverse descents. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001569The maximal modular displacement of a permutation. St001590The crossing number of a perfect matching. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001684The reduced word complexity of a permutation. St001726The number of visible inversions of a permutation. St001964The interval resolution global dimension of a poset. St000024The number of double up and double down steps of a Dyck path. St000144The pyramid weight of the Dyck path. St000166The depth minus 1 of an ordered tree. St000325The width of the tree associated to a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000470The number of runs in a permutation. St000720The size of the largest partition in the oscillating tableau corresponding to the perfect matching. St000982The length of the longest constant subword. St001180Number of indecomposable injective modules with projective dimension at most 1. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001065Number of indecomposable reflexive modules in the corresponding Nakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001237The number of simple modules with injective dimension at most one or dominant dimension at least one. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001211The number of simple modules in the corresponding Nakayama algebra that have vanishing second Ext-group with the regular module. St001492The number of simple modules that do not appear in the socle of the regular module or have no nontrivial selfextensions with the regular module in the corresponding Nakayama algebra. St001966Half the global dimension of the stable Auslander algebra of a sincere Nakayama algebra (with associated Dyck path). St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St000454The largest eigenvalue of a graph if it is integral. St000889The number of alternating sign matrices with the same antidiagonal sums. St000422The energy of a graph, if it is integral. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000628The balance of a binary word. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001498The normalised height of a Nakayama algebra with magnitude 1. St000392The length of the longest run of ones in a binary word. St000479The Ramsey number of a graph. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St001861The number of Bruhat lower covers of a permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001896The number of right descents of a signed permutations. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001434The number of negative sum pairs of a signed permutation. St001769The reflection length of a signed permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001864The number of excedances of a signed permutation. St001866The nesting alignments of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001895The oddness of a signed permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001817The number of flag weak exceedances of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001618The cardinality of the Frattini sublattice of a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000635The number of strictly order preserving maps of a poset into itself. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001490The number of connected components of a skew partition. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001820The size of the image of the pop stack sorting operator. St001881The number of factors of a lattice as a Cartesian product of lattices. St001890The maximum magnitude of the Möbius function of a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St001545The second Elser number of a connected graph. St001845The number of join irreducibles minus the rank of a lattice. St001846The number of elements which do not have a complement in the lattice. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000189The number of elements in the poset. St001616The number of neutral elements in a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001619The number of non-isomorphic sublattices of a lattice. St001666The number of non-isomorphic subposets of a lattice which are lattices. St001833The number of linear intervals in a lattice. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001620The number of sublattices of a lattice. St001679The number of subsets of a lattice whose meet is the bottom element. St001868The number of alignments of type NE of a signed permutation.
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