Your data matches 117 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000746
(load all 3 compositions to match this statistic)
Mapping
St000746: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => 1
[(1,2),(3,4)]
 => 2
[(1,3),(2,4)]
 => 1
[(1,4),(2,3)]
 => 1
[(1,2),(3,4),(5,6)]
 => 3
[(1,3),(2,4),(5,6)]
 => 2
[(1,4),(2,3),(5,6)]
 => 2
[(1,5),(2,3),(4,6)]
 => 1
[(1,6),(2,3),(4,5)]
 => 1
[(1,6),(2,4),(3,5)]
 => 2
[(1,5),(2,4),(3,6)]
 => 2
[(1,4),(2,5),(3,6)]
 => 2
[(1,3),(2,5),(4,6)]
 => 1
[(1,2),(3,5),(4,6)]
 => 2
[(1,2),(3,6),(4,5)]
 => 2
[(1,3),(2,6),(4,5)]
 => 1
[(1,4),(2,6),(3,5)]
 => 2
[(1,5),(2,6),(3,4)]
 => 2
[(1,6),(2,5),(3,4)]
 => 2
[(1,2),(3,4),(5,6),(7,8)]
 => 4
[(1,3),(2,4),(5,6),(7,8)]
 => 3
[(1,4),(2,3),(5,6),(7,8)]
 => 3
[(1,5),(2,3),(4,6),(7,8)]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => 2
[(1,7),(2,4),(3,5),(6,8)]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => 3
[(1,2),(3,6),(4,5),(7,8)]
 => 3
[(1,3),(2,6),(4,5),(7,8)]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => 2
[(1,8),(2,5),(3,4),(6,7)]
 => 2
[(1,8),(2,6),(3,4),(5,7)]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => 2
[(1,3),(2,7),(4,5),(6,8)]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => 2
Description
The number of pairs with odd minimum in a perfect matching.
Matching statistic: St000676
(load all 14 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => 2
[(1,3),(2,4)]
 => [1,1,0,0]
 => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => 3
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => 2
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => 2
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => 2
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => 2
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => 2
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => 2
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => 3
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => 3
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => 3
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => 3
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 2
Description
The number of odd rises of a Dyck path. This is the number of ones at an odd position, with the initial position equal to 1. The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St000031
(load all 3 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000031: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => [1] => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,2] => 2
[(1,3),(2,4)]
 => [1,1,0,0]
 => [2,1] => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [2,1] => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
Description
The number of cycles in the cycle decomposition of a permutation.
Matching statistic: St000329
(load all 4 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => [1,1,0,0]
 => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 3
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000443
(load all 6 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000443: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => [1,0]
 => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
Description
The number of long tunnels of a Dyck path. A long tunnel of a Dyck path is a longest sequence of consecutive usual tunnels, i.e., a longest sequence of tunnels where the end point of one is the starting point of the next. See [1] for the definition of tunnels.
Matching statistic: St001007
(load all 6 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001007: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => [1,0]
 => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
Description
Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001187
(load all 6 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001187: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => [1,0]
 => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
Description
The number of simple modules with grade at least one in the corresponding Nakayama algebra.
Matching statistic: St001224
(load all 6 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001224: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => [1,0]
 => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,1,0,0]
 => 2
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 3
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 3
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 3
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
Description
Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. Then the statistic gives the vector space dimension of the first Ext-group between X and the regular module.
Matching statistic: St001461
(load all 2 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001461: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => [1] => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,2] => 2
[(1,3),(2,4)]
 => [1,1,0,0]
 => [2,1] => 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [2,1] => 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,2,3] => 3
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [2,1,3] => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,3,2] => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [3,2,1] => 2
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 4
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 3
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 3
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [2,3,1,4] => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,2,1,4] => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [4,2,3,1] => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,3,4,2] => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,2,4,1] => 2
Description
The number of topologically connected components of the chord diagram of a permutation. The chord diagram of a permutation $\pi\in\mathfrak S_n$ is obtained by placing labels $1,\dots,n$ in cyclic order on a cycle and drawing a (straight) arc from $i$ to $\pi(i)$ for every label $i$. This statistic records the number of topologically connected components in the chord diagram. In particular, if two arcs cross, all four labels connected by the two arcs are in the same component. The permutation $\pi\in\mathfrak S_n$ stabilizes an interval $I=\{a,a+1,\dots,b\}$ if $\pi(I)=I$. It is stabilized-interval-free, if the only interval $\pi$ stablizes is $\{1,\dots,n\}$. Thus, this statistic is $1$ if $\pi$ is stabilized-interval-free.
Matching statistic: St000024
(load all 6 compositions to match this statistic)
Mapping
Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000024: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[(1,2)]
 => [1,0]
 => [1,0]
 => 0 = 1 - 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0 = 1 - 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2 = 3 - 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0 = 1 - 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 1 = 2 - 1
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3 = 4 - 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,0,1,0,1,1,0,0]
 => 1 = 2 - 1
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1 = 2 - 1
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0 = 1 - 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 1 = 2 - 1
Description
The number of double up and double down steps of a Dyck path. In other words, this is the number of double rises (and, equivalently, the number of double falls) of a Dyck path.
The following 107 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000010The length of the partition. St000015The number of peaks of a Dyck path. St000053The number of valleys of the Dyck path. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000155The number of exceedances (also excedences) of a permutation. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000216The absolute length of a permutation. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000809The reduced reflection length of the permutation. St000991The number of right-to-left minima of a permutation. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000021The number of descents of a permutation. St000120The number of left tunnels of a Dyck path. St000168The number of internal nodes of an ordered tree. St000211The rank of the set partition. St000238The number of indices that are not small weak excedances. St000245The number of ascents of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000332The positive inversions of an alternating sign matrix. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000703The number of deficiencies of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001298The number of repeated entries in the Lehmer code of a permutation. St001489The maximum of the number of descents and the number of inverse descents. St000702The number of weak deficiencies of a permutation. St000083The number of left oriented leafs of a binary tree except the first one. St000925The number of topologically connected components of a set partition. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000829The Ulam distance of a permutation to the identity permutation. St001060The distinguishing index of a graph. 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 $eA$. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St000390The number of runs of ones in a binary word. St000292The number of ascents of a binary word. St001280The number of parts of an integer partition that are at least two. St000291The number of descents of a binary word. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001092The number of distinct even parts of a partition. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001494The Alon-Tarsi number of a graph. St000891The number of distinct diagonal sums of a permutation matrix. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000445The number of rises of length 1 of a Dyck path. St000454The largest eigenvalue of a graph if it is integral. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000035The number of left outer peaks of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St000097The order of the largest clique of the graph. St001394The genus of a permutation. St000822The Hadwiger number of the graph. St000996The number of exclusive left-to-right maxima of a permutation. St000534The number of 2-rises of a permutation. St001115The number of even descents of a permutation. St000352The Elizalde-Pak rank of a permutation. St000741The Colin de Verdière graph invariant. St000834The number of right outer peaks of a permutation. St000730The maximal arc length of a set partition. St000251The number of nonsingleton blocks of a set partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001716The 1-improper chromatic number of a graph. St001644The dimension of a graph. St000647The number of big descents of a permutation. St000731The number of double exceedences of a permutation. St000871The number of very big ascents of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000201The number of leaf nodes in a binary tree. St000527The width of the poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000862The number of parts of the shifted shape of a permutation. St000259The diameter of a connected graph. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000648The number of 2-excedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000153The number of adjacent cycles of a permutation. St000098The chromatic number of a graph. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001330The hat guessing number of a graph. St000568The hook number of a binary tree. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001052The length of the exterior of a permutation. St000353The number of inner valleys of a permutation. St000022The number of fixed points of a permutation. St001096The size of the overlap set of a permutation. St000237The number of small exceedances. St000732The number of double deficiencies of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation.