- FindStat

Your data matches 156 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000720
(load all 5 compositions to match this statistic)

Mapping

St000720: 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)]
 => 1
[(1,3),(2,4)]
 => 2
[(1,4),(2,3)]
 => 2
[(1,2),(3,4),(5,6)]
 => 1
[(1,3),(2,4),(5,6)]
 => 2
[(1,4),(2,3),(5,6)]
 => 2
[(1,5),(2,3),(4,6)]
 => 2
[(1,6),(2,3),(4,5)]
 => 2
[(1,6),(2,4),(3,5)]
 => 3
[(1,5),(2,4),(3,6)]
 => 3
[(1,4),(2,5),(3,6)]
 => 3
[(1,3),(2,5),(4,6)]
 => 2
[(1,2),(3,5),(4,6)]
 => 2
[(1,2),(3,6),(4,5)]
 => 2
[(1,3),(2,6),(4,5)]
 => 2
[(1,4),(2,6),(3,5)]
 => 3
[(1,5),(2,6),(3,4)]
 => 3
[(1,6),(2,5),(3,4)]
 => 3
[(1,2),(3,4),(5,6),(7,8)]
 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => 2
[(1,4),(2,3),(5,6),(7,8)]
 => 2
[(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)]
 => 2
[(1,8),(2,3),(4,5),(6,7)]
 => 2
[(1,8),(2,4),(3,5),(6,7)]
 => 3
[(1,7),(2,4),(3,5),(6,8)]
 => 3
[(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)]
 => 2
[(1,2),(3,6),(4,5),(7,8)]
 => 2
[(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)]
 => 3
[(1,8),(2,5),(3,4),(6,7)]
 => 3
[(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)]
 => 3
[(1,4),(2,7),(3,5),(6,8)]
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => 2
[(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)]
 => 2
[(1,4),(2,8),(3,5),(6,7)]
 => 3

Description

The size of the largest partition in the oscillating tableau corresponding to the perfect matching. Equivalently, this is the maximal number of crosses in the corresponding triangular rook filling that can be covered by a rectangle.

Matching statistic: St000013
(load all 14 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
St000013: 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]
 => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => 2
[(1,4),(2,3)]
 => [1,1,0,0]
 => 2
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => 1
[(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]
 => 2
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => 2
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => 3
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => 3
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => 3
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => 2
[(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]
 => 2
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => 3
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => 3
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => 2
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => 2
[(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]
 => 2
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => 2
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 3
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 3
[(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]
 => 2
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => 2
[(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]
 => 3
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 3
[(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]
 => 3
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => 2
[(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]
 => 2
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => 3

Description

The height of a Dyck path. The height of a Dyck path

$D$ of semilength

$n$ is defined as the maximal height of a peak of

$D$ . The height of

$D$ at position

$i$ is the number of up-steps minus the number of down-steps before position

$i$ .

Matching statistic: St000062
(load all 8 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St000062: 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]
 => [2,1] => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,2] => 2
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,2] => 2
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [3,2,1] => 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [3,1,2] => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [2,1,3] => 2
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [2,1,3] => 2
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [2,1,3] => 2
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [2,3,1] => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [2,1,3] => 2
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,2,3] => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [4,3,2,1] => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [4,3,1,2] => 2
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [4,2,3,1] => 2
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [4,2,1,3] => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [4,1,2,3] => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [2,1,3,4] => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [3,2,4,1] => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [3,2,1,4] => 2
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,2,4] => 3

Description

The length of the longest increasing subsequence of the permutation.

Matching statistic: St000166
(load all 3 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000166: Ordered trees ⟶ 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]
 => [[],[]]
 => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => [[[]]]
 => 2
[(1,4),(2,3)]
 => [1,1,0,0]
 => [[[]]]
 => 2
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => 1
[(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]
 => [[[],[]]]
 => 2
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 2
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 2
[(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]
 => [[[],[]]]
 => 2
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 2
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 2
[(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]
 => [[[],[],[]]]
 => 2
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 2
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 3
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 3
[(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]
 => [[],[[]],[]]
 => 2
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 2
[(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]
 => [[[[]],[]]]
 => 3
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 3
[(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]
 => [[[[]],[]]]
 => 3
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 2
[(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]
 => [[[],[],[]]]
 => 2
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 3

Description

The depth minus 1 of an ordered tree. The ordered trees of size

$n$ are bijection with the Dyck paths of size

$n-1$ , and this statistic then corresponds to [[St000013]].

Matching statistic: St000442
(load all 13 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000442: 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]
 => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,1,1,0,0,0]
 => 2
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[(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]
 => 2
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[(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]
 => 2
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[(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]
 => 1
[(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]
 => 2
[(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]
 => 2
[(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]
 => 2
[(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]
 => 2
[(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]
 => 3
[(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]
 => 3
[(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]
 => 2
[(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]
 => 2
[(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]
 => 3
[(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]
 => 3
[(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]
 => 3
[(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]
 => 3
[(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]
 => 2
[(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]
 => 2
[(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]
 => 3

Description

The maximal area to the right of an up step of a Dyck path.

Matching statistic: St000451
(load all 27 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
St000451: 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] => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => [2,1] => 2
[(1,4),(2,3)]
 => [1,1,0,0]
 => [2,1] => 2
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,2,3] => 1
[(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] => 2
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [2,3,1] => 2
[(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] => 2
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [3,1,2] => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,2,3,4] => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [2,1,3,4] => 2
[(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] => 2
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 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] => 2
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,3,2,4] => 2
[(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]
 => [3,1,2,4] => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [3,1,2,4] => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [3,4,1,2] => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [2,3,4,1] => 2
[(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] => 2
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [3,1,4,2] => 3

Description

The length of the longest pattern of the form k 1 2...(k-1).

Matching statistic: St000527
(load all 4 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St000527: Posets ⟶ 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]
 => ([(0,1)],2)
 => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => ([],2)
 => 2
[(1,4),(2,3)]
 => [1,1,0,0]
 => ([],2)
 => 2
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => ([(0,2),(2,1)],3)
 => 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => ([(0,1),(0,2)],3)
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => ([],3)
 => 3
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => ([],3)
 => 3
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => ([],3)
 => 3
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => ([(0,2),(1,2)],3)
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => ([(1,2)],3)
 => 2
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => ([],3)
 => 3
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => ([],3)
 => 3
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => ([],3)
 => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => ([(0,3),(2,1),(3,2)],4)
 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 2
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => ([(0,3),(3,1),(3,2)],4)
 => 2
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => ([(0,2),(0,3),(3,1)],4)
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => ([(0,1),(0,2),(0,3)],4)
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => ([(2,3)],4)
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => ([(0,3),(1,2),(2,3)],4)
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => ([(0,3),(1,2),(1,3)],4)
 => 2
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => ([(1,2),(1,3)],4)
 => 3

Description

The width of the poset. This is the size of the poset's longest antichain, also called Dilworth number.

Matching statistic: St001203
(load all 4 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001203: 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]
 => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[(1,4),(2,3)]
 => [1,1,0,0]
 => [1,0,1,0]
 => 2
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [1,1,0,0,1,0]
 => 2
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 2
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 2
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 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]
 => 3
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,0,1,0,1,0]
 => 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]
 => 3
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 3

Description

We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n-1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a Dyck path as follows: In the list

$L$ delete the first entry

$c_0$ and substract from all other entries

$n-1$ and then append the last element 1 (this was suggested by Christian Stump). The result is a Kupisch series of an LNakayama algebra. Example: [5,6,6,6,6] goes into [2,2,2,2,1]. Now associate to the CNakayama algebra with the above properties the Dyck path corresponding to the Kupisch series of the LNakayama algebra. The statistic return the global dimension of the CNakayama algebra divided by 2.

Matching statistic: St001589
(load all 3 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St001589: Perfect matchings ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[(1,2)]
 => [1,0]
 => [(1,2)]
 => 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [(1,2),(3,4)]
 => 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => 2
[(1,4),(2,3)]
 => [1,1,0,0]
 => [(1,4),(2,3)]
 => 2
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6)]
 => 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [(1,4),(2,3),(5,6)]
 => 2
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [(1,2),(3,6),(4,5)]
 => 2
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [(1,6),(2,3),(4,5)]
 => 2
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [(1,6),(2,5),(3,4)]
 => 3
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [(1,2),(3,4),(5,6),(7,8)]
 => 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 2
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [(1,4),(2,3),(5,6),(7,8)]
 => 2
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 2
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 2
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 2
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [(1,2),(3,6),(4,5),(7,8)]
 => 2
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [(1,6),(2,3),(4,5),(7,8)]
 => 2
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [(1,6),(2,5),(3,4),(7,8)]
 => 3
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 3
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 3
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [(1,8),(2,7),(3,4),(5,6)]
 => 3
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 2
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 2
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [(1,2),(3,8),(4,5),(6,7)]
 => 2
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [(1,8),(2,3),(4,5),(6,7)]
 => 2
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [(1,8),(2,5),(3,4),(6,7)]
 => 3

Description

The nesting number of a perfect matching. This is the maximal number of chords in the standard representation of a perfect matching that mutually nest.

Matching statistic: St000094
(load all 3 compositions to match this statistic)

Mapping

Mp00150: Perfect matchings —to Dyck path⟶ Dyck paths
Mp00026: Dyck paths —to ordered tree⟶ Ordered trees
St000094: Ordered trees ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

[(1,2)]
 => [1,0]
 => [[]]
 => 2 = 1 + 1
[(1,2),(3,4)]
 => [1,0,1,0]
 => [[],[]]
 => 2 = 1 + 1
[(1,3),(2,4)]
 => [1,1,0,0]
 => [[[]]]
 => 3 = 2 + 1
[(1,4),(2,3)]
 => [1,1,0,0]
 => [[[]]]
 => 3 = 2 + 1
[(1,2),(3,4),(5,6)]
 => [1,0,1,0,1,0]
 => [[],[],[]]
 => 2 = 1 + 1
[(1,3),(2,4),(5,6)]
 => [1,1,0,0,1,0]
 => [[[]],[]]
 => 3 = 2 + 1
[(1,4),(2,3),(5,6)]
 => [1,1,0,0,1,0]
 => [[[]],[]]
 => 3 = 2 + 1
[(1,5),(2,3),(4,6)]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 3 = 2 + 1
[(1,6),(2,3),(4,5)]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 3 = 2 + 1
[(1,6),(2,4),(3,5)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[(1,5),(2,4),(3,6)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[(1,4),(2,5),(3,6)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[(1,3),(2,5),(4,6)]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 3 = 2 + 1
[(1,2),(3,5),(4,6)]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => 3 = 2 + 1
[(1,2),(3,6),(4,5)]
 => [1,0,1,1,0,0]
 => [[],[[]]]
 => 3 = 2 + 1
[(1,3),(2,6),(4,5)]
 => [1,1,0,1,0,0]
 => [[[],[]]]
 => 3 = 2 + 1
[(1,4),(2,6),(3,5)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[(1,5),(2,6),(3,4)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[(1,6),(2,5),(3,4)]
 => [1,1,1,0,0,0]
 => [[[[]]]]
 => 4 = 3 + 1
[(1,2),(3,4),(5,6),(7,8)]
 => [1,0,1,0,1,0,1,0]
 => [[],[],[],[]]
 => 2 = 1 + 1
[(1,3),(2,4),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 3 = 2 + 1
[(1,4),(2,3),(5,6),(7,8)]
 => [1,1,0,0,1,0,1,0]
 => [[[]],[],[]]
 => 3 = 2 + 1
[(1,5),(2,3),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 3 = 2 + 1
[(1,6),(2,3),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 3 = 2 + 1
[(1,7),(2,3),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 3 = 2 + 1
[(1,8),(2,3),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 3 = 2 + 1
[(1,8),(2,4),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[(1,7),(2,4),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[(1,6),(2,4),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[(1,5),(2,4),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[(1,4),(2,5),(3,6),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[(1,3),(2,5),(4,6),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 3 = 2 + 1
[(1,2),(3,5),(4,6),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3 = 2 + 1
[(1,2),(3,6),(4,5),(7,8)]
 => [1,0,1,1,0,0,1,0]
 => [[],[[]],[]]
 => 3 = 2 + 1
[(1,3),(2,6),(4,5),(7,8)]
 => [1,1,0,1,0,0,1,0]
 => [[[],[]],[]]
 => 3 = 2 + 1
[(1,4),(2,6),(3,5),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[(1,5),(2,6),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[(1,6),(2,5),(3,4),(7,8)]
 => [1,1,1,0,0,0,1,0]
 => [[[[]]],[]]
 => 4 = 3 + 1
[(1,7),(2,5),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[(1,8),(2,5),(3,4),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[(1,8),(2,6),(3,4),(5,7)]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 4 = 3 + 1
[(1,7),(2,6),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 4 = 3 + 1
[(1,6),(2,7),(3,4),(5,8)]
 => [1,1,1,0,1,0,0,0]
 => [[[[],[]]]]
 => 4 = 3 + 1
[(1,5),(2,7),(3,4),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[(1,4),(2,7),(3,5),(6,8)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1
[(1,3),(2,7),(4,5),(6,8)]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 3 = 2 + 1
[(1,2),(3,7),(4,5),(6,8)]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 3 = 2 + 1
[(1,2),(3,8),(4,5),(6,7)]
 => [1,0,1,1,0,1,0,0]
 => [[],[[],[]]]
 => 3 = 2 + 1
[(1,3),(2,8),(4,5),(6,7)]
 => [1,1,0,1,0,1,0,0]
 => [[[],[],[]]]
 => 3 = 2 + 1
[(1,4),(2,8),(3,5),(6,7)]
 => [1,1,1,0,0,1,0,0]
 => [[[[]],[]]]
 => 4 = 3 + 1

Description

The depth of an ordered tree.

The following 146 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

St000141The maximum drop size of a permutation. St000306The bounce count of a Dyck path. St000662The staircase size of the code of a permutation. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001197The global dimension of

$eAe$ for 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

$eA$ . St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000080The rank of the poset. St000093The cardinality of a maximal independent set of vertices of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000147The largest part of an integer partition. St000172The Grundy number of a graph. St000308The height of the tree associated to 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. St000528The height of a poset. St000542The number of left-to-right-minima of a permutation. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000822The Hadwiger number of the graph. St000877The depth of the binary word interpreted as a path. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series

$L=[c_0,c_1,...,c_{n−1}]$ such that

$n=c_0 < c_i$ for all

$i > 0$ a special CNakayama algebra. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001343The dimension of the reduced incidence algebra of a poset. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001494The Alon-Tarsi number of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001590The crossing number of a perfect matching. St001717The largest size of an interval in a poset. St001963The tree-depth of a graph. St000021The number of descents of a permutation. St000028The number of stack-sorts needed to sort a permutation. St000053The number of valleys of the Dyck path. St000245The number of ascents of a permutation. St000272The treewidth of a graph. St000317The cycle descent number of a permutation. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000536The pathwidth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001047The maximal number of arcs crossing a given arc of a perfect matching. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by

$\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001358The largest degree of a regular subgraph of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000888The maximal sum of entries on a diagonal of an alternating sign matrix. St000892The maximal number of nonzero entries on a diagonal of an alternating sign matrix. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000730The maximal arc length of a set partition. St001330The hat guessing number of a graph. St000259The diameter of a connected graph. St000454The largest eigenvalue of a graph if it is integral. St000392The length of the longest run of ones in a binary word. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000264The girth of a graph, which is not a tree. St000260The radius of a connected graph. St001060The distinguishing index of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000628The balance of a binary word. St001484The number of singletons of an integer partition. St000444The length of the maximal rise of a Dyck path. St000983The length of the longest alternating subword. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001090The number of pop-stack-sorts needed to sort a permutation. St001732The number of peaks visible from the left. St000445The number of rises of length 1 of a Dyck path. St000932The number of occurrences of the pattern UDU in a Dyck path. St000422The energy of a graph, if it is integral. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000201The number of leaf nodes in a binary tree. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000674The number of hills of a Dyck path. St000931The number of occurrences of the pattern UUU in a 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

$eA$ . St000374The number of exclusive right-to-left minima of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000035The number of left outer peaks of a permutation. St000884The number of isolated descents of a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000834The number of right outer peaks of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001394The genus of a permutation. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000386The number of factors DDU in a Dyck path. St000568The hook number of a binary tree. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000871The number of very big ascents of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000352The Elizalde-Pak rank of a permutation. St001581The achromatic number of a graph. St000455The second largest eigenvalue of a graph if it is integral. St000647The number of big descents of a permutation. St001812The biclique partition number of a graph. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001115The number of even descents of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000742The number of big ascents of a permutation after prepending zero. St000022The number of fixed points of a permutation. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St000731The number of double exceedences of a permutation. St001669The number of single rises in a Dyck path. St001729The number of visible descents of a permutation. St000624The normalized sum of the minimal distances to a greater element. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001153The number of blocks with even minimum in a set partition. St001083The number of boxed occurrences of 132 in a permutation. St001427The number of descents of a signed permutation. St000366The number of double descents of a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length

$3$ . St001665The number of pure excedances of a permutation. St001801Half the number of preimage-image pairs of different parity in a permutation. St000237The number of small exceedances. St000247The number of singleton blocks of a set partition. St000356The number of occurrences of the pattern 13-2. St001469The holeyness of a permutation. St000068The number of minimal elements in a poset. St000071The number of maximal chains in a poset. St000153The number of adjacent cycles of a permutation. St001907The number of Bastidas - Hohlweg - Saliola excedances of a signed permutation. St000167The number of leaves of an ordered tree. St001737The number of descents of type 2 in a permutation. St000632The jump number of the poset. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition.