searching the database
Your data matches 1380 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000011: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> []
=> []
=> 0
[1,1,0,0]
=> [1,1,0,0]
=> []
=> []
=> 0
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> []
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> []
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> 0
[]
=> []
=> []
=> []
=> 0
Description
The number of touch points (or returns) of a Dyck path.
This is the number of points, excluding the origin, where the Dyck path has height 0.
Matching statistic: St000013
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000013: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> []
=> []
=> []
=> 0
[1,1,0,0]
=> []
=> []
=> []
=> 0
[1,1,1,0,0,0]
=> []
=> []
=> []
=> 0
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 2
[1,1,1,1,0,0,0,0]
=> []
=> []
=> []
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> []
=> 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1]
=> [1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> []
=> 0
[]
=> []
=> []
=> []
=> 0
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: St000220
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00120: Dyck paths —Lalanne-Kreweras involution⟶ Dyck paths
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000220: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1] => 0
[1,1,0,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 3
[1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 2
[1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 2
[1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,2,4,3,5,6] => 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => 0
[]
=> []
=> []
=> [] => 0
Description
The number of occurrences of the pattern 132 in a permutation.
Matching statistic: St000454
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => ([],6)
=> 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => ([],7)
=> 0
[]
=> [1,0]
=> [1] => ([],1)
=> 0
Description
The largest eigenvalue of a graph if it is integral.
If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree.
This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St001270
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001270: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => ([],6)
=> 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => ([],7)
=> 0
[]
=> [1,0]
=> [1] => ([],1)
=> 0
Description
The bandwidth of a graph.
The bandwidth of a graph is the smallest number $k$ such that the vertices of the graph can be
ordered as $v_1,\dots,v_n$ with $k \cdot d(v_i,v_j) \geq |i-j|$.
We adopt the convention that the singleton graph has bandwidth $0$, consistent with the bandwith of the complete graph on $n$ vertices having bandwidth $n-1$, but in contrast to any path graph on more than one vertex having bandwidth $1$. The bandwidth of a disconnected graph is the maximum of the bandwidths of the connected components.
Matching statistic: St001279
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00201: Dyck paths —Ringel⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St001279: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [2,1] => [1,2] => [1,1]
=> 0
[1,1,0,0]
=> [2,3,1] => [1,2,3] => [1,1,1]
=> 0
[1,1,1,0,0,0]
=> [2,3,4,1] => [1,2,3,4] => [1,1,1,1]
=> 0
[1,1,0,0,1,0,1,0]
=> [2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> 3
[1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> 3
[1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> 3
[1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => [1,2,3,5,4] => [2,1,1,1]
=> 2
[1,1,1,0,0,1,0,0]
=> [2,5,4,1,3] => [1,2,5,3,4] => [2,1,1,1]
=> 2
[1,1,1,0,1,0,0,0]
=> [5,3,4,1,2] => [1,5,2,3,4] => [2,1,1,1]
=> 2
[1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [1,2,3,4,5] => [1,1,1,1,1]
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [2,5,1,3,6,4] => [1,2,5,6,4,3] => [3,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [5,3,1,2,6,4] => [1,5,6,4,2,3] => [3,1,1,1]
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [5,4,1,2,6,3] => [1,5,6,3,2,4] => [3,1,1,1]
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [2,3,5,1,6,4] => [1,2,3,5,6,4] => [2,1,1,1,1]
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [2,5,4,1,6,3] => [1,2,5,6,3,4] => [2,1,1,1,1]
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [5,3,4,1,6,2] => [1,5,6,2,3,4] => [2,1,1,1,1]
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,5,1,3,6,7,4] => [1,2,5,6,7,4,3] => [3,1,1,1,1]
=> 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [5,3,1,2,6,7,4] => [1,5,6,7,4,2,3] => [3,1,1,1,1]
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [5,4,1,2,6,7,3] => [1,5,6,7,3,2,4] => [3,1,1,1,1]
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,3,5,1,6,7,4] => [1,2,3,5,6,7,4] => [2,1,1,1,1,1]
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [2,5,4,1,6,7,3] => [1,2,5,6,7,3,4] => [2,1,1,1,1,1]
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [5,3,4,1,6,7,2] => [1,5,6,7,2,3,4] => [2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => [1,1,1,1,1,1,1]
=> 0
[]
=> [1] => [1] => [1]
=> 0
Description
The sum of the parts of an integer partition that are at least two.
Matching statistic: St001458
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001458: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001458: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => ([],6)
=> 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => ([],7)
=> 0
[]
=> [1,0]
=> [1] => ([],1)
=> 0
Description
The rank of the adjacency matrix of a graph.
Matching statistic: St001962
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001962: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001962: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 0
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 0
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 0
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 2
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 0
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 2
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => ([],6)
=> 0
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 2
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => ([],7)
=> 0
[]
=> [1,0]
=> [1] => ([],1)
=> 0
Description
The proper pathwidth of a graph.
The proper pathwidth $\operatorname{ppw}(G)$ was introduced in [1] as the minimum width of a proper-path-decomposition. Barioli et al. [2] showed that if $G$ has at least one edge, then $\operatorname{ppw}(G)$ is the minimum $k$ for which $G$ is a minor of the Cartesian product $K_k \square P$ of a complete graph on $k$ vertices with a path; and further that $\operatorname{ppw}(G)$ is the minor monotone floor $\lfloor \operatorname{Z} \rfloor(G) := \min\{\operatorname{Z}(H) \mid G \preceq H\}$ of the [[St000482|zero forcing number]] $\operatorname{Z}(G)$. It can be shown [3, Corollary 9.130] that only the spanning supergraphs need to be considered for $H$ in this definition, i.e. $\lfloor \operatorname{Z} \rfloor(G) = \min\{\operatorname{Z}(H) \mid G \le H,\; V(H) = V(G)\}$.
The minimum degree $\delta$, treewidth $\operatorname{tw}$, and pathwidth $\operatorname{pw}$ satisfy
$$\delta \le \operatorname{tw} \le \operatorname{pw} \le \operatorname{ppw} = \lfloor \operatorname{Z} \rfloor \le \operatorname{pw} + 1.$$
Note that [4] uses a different notion of proper pathwidth, which is equal to bandwidth.
Matching statistic: St000452
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000452: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000452: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [2] => ([],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [3] => ([],3)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [4] => ([],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5] => ([],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6] => ([],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [3,1,3] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [4,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [7] => ([],7)
=> 1 = 0 + 1
[]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
Description
The number of distinct eigenvalues of a graph.
Matching statistic: St001108
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001108: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> [1,2] => ([],2)
=> 1 = 0 + 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,2,3] => ([],3)
=> 1 = 0 + 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => ([],4)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,2,5,3,4] => ([(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => ([],5)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> [1,4,5,2,3,6] => ([(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> [4,1,5,2,3,6] => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [4,5,1,2,3,6] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> [1,2,5,3,4,6] => ([(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,5,2,3,4,6] => ([(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,1,0,0,0,0,0]
=> [5,1,2,3,4,6] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,2,3,4,5,6] => ([],6)
=> 1 = 0 + 1
[1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> [1,4,5,2,3,6,7] => ([(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 3 + 1
[1,1,0,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,1,1,1,0,0,0,0]
=> [4,1,5,2,3,6,7] => ([(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [4,5,1,2,3,6,7] => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 4 = 3 + 1
[1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> [1,2,5,3,4,6,7] => ([(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> [1,5,2,3,4,6,7] => ([(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> [5,1,2,3,4,6,7] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> 1 = 0 + 1
[]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
Description
The 2-dynamic chromatic number of a graph.
A $k$-dynamic coloring of a graph $G$ is a proper coloring of $G$ in such a way that each vertex $v$ sees at least $\min\{d(v), k\}$ colors in its neighborhood. The $k$-dynamic chromatic number of a graph is the smallest number of colors needed to find an $k$-dynamic coloring.
This statistic records the $2$-dynamic chromatic number of a graph.
The following 1370 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001963The tree-depth of a graph. St001504The sum of all indegrees of vertices with indegree at least two in the resolution quiver of a Nakayama algebra corresponding to the Dyck path. St000537The cutwidth of a graph. St001120The length of a longest path in a graph. St001626The number of maximal proper sublattices of a lattice. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001112The 3-weak dynamic number of a graph. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St000004The major index of a permutation. St000024The number of double up and double down steps of a Dyck path. St000235The number of indices that are not cyclical small weak excedances. St000238The number of indices that are not small weak excedances. St000424The number of occurrences of the pattern 132 or of the pattern 231 in a permutation. St000426The number of occurrences of the pattern 132 or of the pattern 312 in a permutation. St000446The disorder of a permutation. St000495The number of inversions of distance at most 2 of a permutation. St000539The number of odd inversions of a permutation. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St001459The number of zero columns in the nullspace of a graph. St001644The dimension of a graph. St001647The number of edges that can be added without increasing the clique number. St001648The number of edges that can be added without increasing the chromatic number. St000025The number of initial rises of a Dyck path. St000443The number of long tunnels of a Dyck path. St000453The number of distinct Laplacian eigenvalues of a graph. St000638The number of up-down runs of a permutation. St000722The number of different neighbourhoods in a graph. St000947The major index east count of a Dyck path. St001007Number of simple modules with projective dimension 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. 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. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001809The index of the step at the first peak of maximal height in a Dyck path. St001883The mutual visibility number of a graph. St000439The position of the first down step of a Dyck path. St000289The decimal representation of a binary word. St000391The sum of the positions of the ones in a binary word. St000435The number of occurrences of the pattern 213 or of the pattern 231 in a permutation. St000442The maximal area to the right of an up step of a Dyck path. St000462The major index minus the number of excedences of a permutation. St000499The rcb statistic of a set partition. St000556The number of occurrences of the pattern {{1},{2,3}} in a set partition. St000670The reversal length of a permutation. St000673The number of non-fixed points of a permutation. St000792The Grundy value for the game of ruler on a binary word. St000874The position of the last double rise in a Dyck path. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St000005The bounce statistic of a Dyck path. St000444The length of the maximal rise of a Dyck path. St000652The maximal difference between successive positions of a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St000240The number of indices that are not small excedances. St000451The length of the longest pattern of the form k 1 2. St000824The sum of the number of descents and the number of recoils of a permutation. St000831The number of indices that are either descents or recoils. St001489The maximum of the number of descents and the number of inverse descents. St000015The number of peaks of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We 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:
St001530The depth of a Dyck path. St000053The number of valleys of the Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000390The number of runs of ones in a binary word. St000806The semiperimeter of the associated bargraph. St000955Number of times one has $Ext^i(D(A),A)>0$ for $i>0$ for the corresponding LNakayama algebra. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001118The acyclic chromatic index of a graph. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001199The dominant 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 $eA$. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St000225Difference between largest and smallest parts in a partition. St000292The number of ascents of a binary word. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001651The Frankl number of a lattice. St000062The length of the longest increasing subsequence of the permutation. St000105The number of blocks in the set partition. St000157The number of descents of a standard tableau. St000164The number of short pairs. St000167The number of leaves of an ordered tree. St000213The number of weak exceedances (also weak excedences) of a permutation. St000236The number of cyclical small weak excedances. St000239The number of small weak excedances. 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. St000829The Ulam distance of a permutation to the identity permutation. St000925The number of topologically connected components of a set partition. 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 $eA$. 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 $eA$. 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. 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 $eA$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001235The global dimension of the corresponding Comp-Nakayama algebra. 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. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St000083The number of left oriented leafs of a binary tree except the first one. St000155The number of exceedances (also excedences) of a permutation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000331The number of upper interactions of a Dyck path. St000340The number of non-final maximal constant sub-paths of length greater than one. St000354The number of recoils of a permutation. St000522The number of 1-protected nodes of a rooted tree. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000619The number of cyclic descents of a permutation. St000630The length of the shortest palindromic decomposition of a binary word. St000706The product of the factorials of the multiplicities of an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St000886The number of permutations with the same antidiagonal sums. St000939The number of characters of the symmetric group whose value on the partition is positive. St000956The maximal displacement of a permutation. St000983The length of the longest alternating subword. St000993The multiplicity of the largest part of an integer partition. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001180Number of indecomposable injective modules with projective dimension at most 1. St001183The maximum of $projdim(S)+injdim(S)$ over all simple modules in the Nakayama algebra corresponding to the Dyck path. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001281The normalized isoperimetric number of a graph. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001568The smallest positive integer that does not appear twice in the partition. St001592The maximal number of simple paths between any two different vertices of a graph. St001608The number of coloured rooted trees such that the multiplicities of colours are given by a partition. St001610The number of coloured endofunctions such that the multiplicities of colours are given by a partition. St001726The number of visible inversions of a permutation. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001933The largest multiplicity of a part in an integer partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000436The number of occurrences of the pattern 231 or of the pattern 321 in a permutation. St000486The number of cycles of length at least 3 of a permutation. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000567The sum of the products of all pairs of parts. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000608The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal, 3 is maximal. St000615The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are maximal. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000628The balance of a binary word. St000661The number of rises of length 3 of a Dyck path. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000779The tier of a permutation. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St000929The constant term of the character polynomial of an integer partition. St000931The number of occurrences of the pattern UUU in a Dyck path. St000941The number of characters of the symmetric group whose value on the partition is even. 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. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001172The number of 1-rises at odd height of a Dyck path. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001175The size of a partition minus the hook length of the base cell. St001176The size of a partition minus its first part. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001423The number of distinct cubes in a binary word. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001587Half of the largest even part of an integer partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001657The number of twos in an integer partition. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001932The number of pairs of singleton blocks in the noncrossing set partition corresponding to a Dyck path, that can be merged to create another noncrossing set partition. St001961The sum of the greatest common divisors of all pairs of parts. St000896The number of zeros on the main diagonal of an alternating sign matrix. St000060The greater neighbor of the maximum. St000646The number of big ascents of a permutation. St001673The degree of asymmetry of an integer composition. St001488The number of corners of a skew partition. St000402Half the size of the symmetry class of a permutation. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000767The number of runs in an integer composition. St000903The number of different parts of an integer composition. St000761The number of ascents in an integer composition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000702The number of weak deficiencies of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001960The number of descents of a permutation minus one if its first entry is not one. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000711The number of big exceedences of a permutation. St000667The greatest common divisor of the parts of the partition. St001571The Cartan determinant of the integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000133The "bounce" of a permutation. St000216The absolute length of a permutation. 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. St001618The cardinality of the Frattini sublattice of a lattice. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St000259The diameter of a connected graph. St000742The number of big ascents of a permutation after prepending zero. St001823The Stasinski-Voll length of a signed permutation. St000820The number of compositions obtained by rotating the composition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001487The number of inner corners of a skew partition. St001712The number of natural descents of a standard Young tableau. St000264The girth of a graph, which is not a tree. St001875The number of simple modules with projective dimension at most 1. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St000422The energy of a graph, if it is integral. St001060The distinguishing index of a graph. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000662The staircase size of the code of a permutation. St000668The least common multiple of the parts of the partition. St000770The major index of an integer partition when read from bottom to top. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000937The number of positive values of the symmetric group character corresponding to the partition. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001510The number of self-evacuating linear extensions of a finite poset. St000137The Grundy value of an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St001902The number of potential covers of a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001472The permanent of the Coxeter matrix of the poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000618The number of self-evacuating tableaux of given shape. St000781The number of proper colouring schemes of a Ferrers diagram. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000632The jump number of the poset. St000647The number of big descents of a permutation. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000068The number of minimal elements in a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000669The number of permutations obtained by switching ascents or descents of size 2. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000909The number of maximal chains of maximal size in a poset. St001330The hat guessing number of a graph. St001581The achromatic number of a graph. St001964The interval resolution global dimension of a poset. St000002The number of occurrences of the pattern 123 in a permutation. St000022The number of fixed points of a permutation. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000218The number of occurrences of the pattern 213 in a permutation. St000223The number of nestings in the permutation. St000245The number of ascents of a permutation. St000356The number of occurrences of the pattern 13-2. St000359The number of occurrences of the pattern 23-1. St000366The number of double descents of a permutation. 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 $3$. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000431The number of occurrences of the pattern 213 or of the pattern 321 in a permutation. St000433The number of occurrences of the pattern 132 or of the pattern 321 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000441The number of successions of a permutation. St000457The number of occurrences of one of the patterns 132, 213 or 321 in a permutation. St000497The lcb statistic of a set partition. 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. St000534The number of 2-rises of a permutation. St000648The number of 2-excedences of a permutation. St000651The maximal size of a rise in a permutation. St000665The number of rafts of a permutation. St000672The number of minimal elements in Bruhat order not less than the permutation. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000731The number of double exceedences of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001115The number of even descents of a permutation. St001301The first Betti number of the order complex associated with the poset. St001394The genus of a permutation. St001396Number of triples of incomparable elements in a finite poset. St001397Number of pairs of incomparable elements in a finite poset. St000028The number of stack-sorts needed to sort a permutation. St000035The number of left outer peaks of a permutation. St000100The number of linear extensions of a poset. St000141The maximum drop size of a permutation. St000153The number of adjacent cycles of a permutation. St000237The number of small exceedances. St000284The Plancherel distribution on integer partitions. St000352The Elizalde-Pak rank of a permutation. St000456The monochromatic index of a connected graph. St000477The weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000546The number of global descents of a permutation. St000681The Grundy value of Chomp on Ferrers diagrams. St000692Babson and Steingrímsson's statistic of a permutation. St000696The number of cycles in the breakpoint graph of a permutation. 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. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000738The first entry in the last row of a standard tableau. St000815The number of semistandard Young tableaux of partition weight of given shape. St000884The number of isolated descents of a permutation. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000908The length of the shortest maximal antichain in a poset. St000933The number of multipartitions of sizes given by an integer partition. St000996The number of exclusive left-to-right maxima of a permutation. St000997The even-odd crank of an integer partition. St001128The exponens consonantiae of a partition. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000007The number of saliances of the permutation. St000054The first entry of the permutation. St000478Another weight of a partition according to Alladi. St000509The diagonal index (content) of a partition. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000725The smallest label of a leaf of the increasing binary tree associated to a permutation. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000842The breadth of a permutation. St000883The number of longest increasing subsequences of a permutation. St000891The number of distinct diagonal sums of a permutation matrix. St000927The alternating sum of the coefficients of the character polynomial of an integer partition. St000934The 2-degree of an integer partition. 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000181The number of connected components of the Hasse diagram for the poset. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000246The number of non-inversions of a permutation. St000295The length of the border of a binary word. St000347The inversion sum of a binary word. St000463The number of admissible inversions of a permutation. St000850The number of 1/2-balanced pairs in a poset. St000879The number of long braid edges in the graph of braid moves of a permutation. St000881The number of short braid edges in the graph of braid moves of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001684The reduced word complexity of a permutation. St001769The reflection length of a signed permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001845The number of join irreducibles minus the rank of a lattice. St001848The atomic length of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001862The number of crossings of 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. St001868The number of alignments of type NE of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St001892The flag excedance statistic of a signed permutation. St001893The flag descent of a signed permutation. St001894The depth of a signed permutation. St001895The oddness of a signed permutation. St001896The number of right descents of a signed permutations. St000001The number of reduced words for a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000069The number of maximal elements of a poset. St000115The single entry in the last row. St000255The number of reduced Kogan faces with the permutation as type. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000525The number of posets with the same zeta polynomial. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000847The number of standard Young tableaux whose descent set is the binary word. St000862The number of parts of the shifted shape of a permutation. St000880The number of connected components of long braid edges in the graph of braid moves of a permutation. St000914The sum of the values of the Möbius function of a poset. St001052The length of the exterior of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001096The size of the overlap set of a permutation. St001490The number of connected components of a skew partition. St001529The number of monomials in the expansion of the nabla operator applied to the power-sum symmetric function indexed by the partition. St001770The number of facets of a certain subword complex associated with the signed permutation. St001851The number of Hecke atoms of a signed permutation. St001852The size of the conjugacy class of the signed permutation. St001855The number of signed permutations less than or equal to a signed permutation in left weak order. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St001058The breadth of the ordered tree. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000348The non-inversion sum of a binary word. St000682The Grundy value of Welter's game on a binary word. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000943The number of spots the most unlucky car had to go further in a parking function. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001371The length of the longest Yamanouchi prefix of a binary word. St001524The degree of symmetry of a binary word. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001730The number of times the path corresponding to a binary word crosses the base line. St001811The Castelnuovo-Mumford regularity of a permutation. St001822The number of alignments of a signed permutation. St001856The number of edges in the reduced word graph of a permutation. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000635The number of strictly order preserving maps of a poset into itself. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001313The number of Dyck paths above the lattice path given by a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001462The number of factors of a standard tableaux under concatenation. St001569The maximal modular displacement of a permutation. St001768The number of reduced words of a signed permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001884The number of borders of a binary word. St001890The maximum magnitude of the Möbius function of a poset. St001937The size of the center of a parking function. St000006The dinv of a Dyck path. St000017The number of inversions of a standard tableau. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000052The number of valleys of a Dyck path not on the x-axis. St000065The number of entries equal to -1 in an alternating sign matrix. St000090The variation of a composition. St000091The descent variation of a composition. St000120The number of left tunnels of a Dyck path. St000136The dinv of a parking function. St000194The number of primary dinversion pairs of a labelled dyck path corresponding to a parking function. St000217The number of occurrences of the pattern 312 in a permutation. St000221The number of strong fixed points of a permutation. St000222The number of alignments in the permutation. St000227The osculating paths major index of an alternating sign matrix. St000232The number of crossings of a set partition. St000233The number of nestings of a set partition. St000242The number of indices that are not cyclical small weak excedances. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000296The length of the symmetric border of a binary word. St000297The number of leading ones in a binary word. St000317The cycle descent number of a permutation. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000353The number of inner valleys of a permutation. St000355The number of occurrences of the pattern 21-3. St000358The number of occurrences of the pattern 31-2. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000386The number of factors DDU in a Dyck path. St000403The Szeged index minus the Wiener index of a graph. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000434The number of occurrences of the pattern 213 or of the pattern 312 in a permutation. St000437The number of occurrences of the pattern 312 or of the pattern 321 in a permutation. St000448The number of pairs of vertices of a graph with distance 2. St000488The number of cycles of a permutation of length at most 2. St000496The rcs statistic of a set partition. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000516The number of stretching pairs of a permutation. St000552The number of cut vertices of a graph. St000557The number of occurrences of the pattern {{1},{2},{3}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000560The number of occurrences of the pattern {{1,2},{3,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000572The dimension exponent of a set partition. St000580The number of occurrences of the pattern {{1},{2},{3}} such that 2 is minimal, 3 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000584The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal, 3 is maximal. St000586The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal. St000587The number of occurrences of the pattern {{1},{2},{3}} such that 1 is minimal. St000588The number of occurrences of the pattern {{1},{2},{3}} such that 1,3 are minimal, 2 is maximal. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000591The number of occurrences of the pattern {{1},{2},{3}} such that 2 is maximal. St000592The number of occurrences of the pattern {{1},{2},{3}} such that 1 is maximal. St000593The number of occurrences of the pattern {{1},{2},{3}} such that 1,2 are minimal. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000597The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, (2,3) are consecutive in a block. St000598The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, 3 is maximal, (2,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000601The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal, (2,3) are consecutive in a block. St000603The number of occurrences of the pattern {{1},{2},{3}} such that 2,3 are minimal. St000604The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 2 is maximal. St000607The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000609The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000629The defect of a binary word. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000671The maximin edge-connectivity for choosing a subgraph. St000687The dimension of $Hom(I,P)$ for the LNakayama algebra of a Dyck path. St000693The modular (standard) major index of a standard tableau. St000709The number of occurrences of 14-2-3 or 14-3-2. St000719The number of alignments in a perfect matching. St000726The normalized sum of the leaf labels of the increasing binary tree associated to a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000753The Grundy value for the game of Kayles on a binary word. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000787The number of flips required to make a perfect matching noncrossing. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000840The number of closers smaller than the largest opener in a perfect matching. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000873The aix statistic of a permutation. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St000894The trace of an alternating sign matrix. St000895The number of ones on the main diagonal of an alternating sign matrix. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000989The number of final rises 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. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001078The minimal number of occurrences of (12) in a factorization of a permutation into transpositions (12) and cycles (1,. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001119The length of a shortest maximal path in a graph. St001131The number of trivial trees on the path to label one in the decreasing labelled binary unordered tree associated with the perfect matching. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three 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. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001193The dimension of $Ext_A^1(A/AeA,A)$ in the corresponding Nakayama algebra $A$ such that $eA$ is a minimal faithful projective-injective module. St001204Call 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. St001209The pmaj statistic of a parking function. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001271The competition number of a graph. St001274The number of indecomposable injective modules with projective dimension equal to two. St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001307The number of induced stars on four vertices in a graph. St001308The number of induced paths on three vertices in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001350Half of the Albertson index of a graph. St001351The Albertson index of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001374The Padmakar-Ivan index of a graph. St001377The major index minus the number of inversions of a permutation. St001381The fertility of a permutation. St001395The number of strictly unfriendly partitions of a graph. St001411The number of patterns 321 or 3412 in a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001429The number of negative entries in a signed permutation. St001433The flag major index of a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001470The cyclic holeyness of a permutation. St001485The modular major index of a binary word. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001535The number of cyclic alignments of a permutation. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001556The number of inversions of the third entry of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001584The area statistic between a Dyck path and its bounce path. St001596The number of two-by-two squares inside a skew partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001689The number of celebrities in a graph. St001697The shifted natural comajor index of a standard Young tableau. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001727The number of invisible inversions of a permutation. St001728The number of invisible descents of a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001745The number of occurrences of the arrow pattern 13 with an arrow from 1 to 2 in a permutation. St001764The number of non-convex subsets of vertices in a graph. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001793The difference between the clique number and the chromatic number of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001810The number of fixed points of a permutation smaller than its largest moved point. St001819The flag Denert index of a signed permutation. St001821The sorting index of a signed permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001836The number of occurrences of a 213 pattern in the restricted growth word of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001863The number of weak excedances of a signed permutation. St001905The number of preferred parking spots in a parking function less than the index of the car. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001911A descent variant minus the number of inversions. St001935The number of ascents in a parking function. St001946The number of descents in a parking function. St001947The number of ties in a parking function. St001948The number of augmented double ascents of a permutation. St000014The number of parking functions supported by a Dyck path. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000026The position of the first return of a Dyck path. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000047The number of standard immaculate tableaux of a given shape. St000056The decomposition (or block) number of a permutation. St000092The number of outer peaks of a permutation. St000110The number of permutations less than or equal to a permutation in left weak order. St000124The cardinality of the preimage of the Simion-Schmidt map. St000154The sum of the descent bottoms of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000163The size of the orbit of the set partition under rotation. 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. St000305The inverse major index of a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000383The last part of an integer composition. St000392The length of the longest run of ones in a binary word. St000461The rix statistic of a permutation. St000493The los statistic of a set partition. St000545The number of parabolic double cosets with minimal element being the given permutation. St000570The Edelman-Greene number of a permutation. St000573The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton and 2 a maximal element. St000624The normalized sum of the minimal distances to a greater element. St000655The length of the minimal rise of a Dyck path. St000657The smallest part of an integer composition. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000694The number of affine bounded permutations that project to a given permutation. St000735The last entry on the main diagonal of a standard tableau. St000740The last entry of a permutation. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000762The sum of the positions of the weak records of an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000765The number of weak records in an integer composition. St000775The multiplicity of the largest eigenvalue in a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000788The number of nesting-similar perfect matchings of a perfect matching. St000789The number of crossing-similar perfect matchings of a perfect matching. St000796The stat' of a permutation. St000797The stat`` of a permutation. St000798The makl of a permutation. St000805The number of peaks of the associated bargraph. St000833The comajor index of a permutation. St000836The number of descents of distance 2 of a permutation. St000868The aid statistic in the sense of Shareshian-Wachs. St000876The number of factors in the Catalan decomposition of a binary word. 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. St000899The maximal number of repetitions of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000904The maximal number of repetitions of an integer composition. St000905The number of different multiplicities of parts of an integer composition. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000942The number of critical left to right maxima of the parking functions. St000958The number of Bruhat factorizations of a permutation. 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 $c_{n−1}$. St000982The length of the longest constant subword. St000988The orbit size of a permutation under Foata's bijection. St000990The first ascent of a permutation. St000991The number of right-to-left minima of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001010Number of indecomposable injective modules with projective dimension g-1 when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001041The depth of the label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001044The number of pairs whose larger element is at most one more than half the size of the perfect matching. St001075The minimal size of a block of a set partition. St001080The minimal length of a factorization of a permutation using the transposition (12) and the cycle (1,. St001081The number of minimal length factorizations of a permutation into star transpositions. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001162The minimum jump of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001170Number of indecomposable injective modules whose socle has projective dimension at most g-1 when g denotes the global dimension in the corresponding Nakayama algebra. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001220The width of a permutation. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001260The permanent of an alternating sign matrix. St001267The length of the Lyndon factorization of the binary word. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001272The number of graphs with the same degree sequence. St001282The number of graphs with the same chromatic polynomial. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001316The domatic number of a graph. St001344The neighbouring number of a permutation. St001372The length of a longest cyclic run of ones of a binary word. St001388The number of non-attacking neighbors of a permutation. St001427The number of descents of a signed permutation. St001437The flex of a binary word. St001461The number of topologically connected components of the chord diagram of a permutation. St001469The holeyness of a permutation. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001523The degree of symmetry of a Dyck path. St001590The crossing number of a perfect matching. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001597The Frobenius rank of a skew partition. St001645The pebbling number of a connected graph. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001761The maximal multiplicity of a letter in a reduced word of a permutation. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001775The degree of the minimal polynomial of the largest eigenvalue of a graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001778The largest greatest common divisor of an element and its image in a permutation. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001853The size of the two-sided Kazhdan-Lusztig cell, St001859The number of factors of the Stanley symmetric function associated with a permutation. St001865The number of alignments of a signed permutation. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001874Lusztig's a-function for the symmetric group. St001885The number of binary words with the same proper border set. St001904The length of the initial strictly increasing segment of a parking function. St001928The number of non-overlapping descents in a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000058The order of a permutation. St000071The number of maximal chains in a poset. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000099The number of valleys of a permutation, including the boundary. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000417The size of the automorphism group of the ordered tree. St000527The width of the poset. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000758The length of the longest staircase fitting into an integer composition. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000893The number of distinct diagonal sums of an alternating sign matrix. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001042The size of the automorphism group of the leaf labelled binary unordered tree associated with the perfect matching. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001375The pancake length of a permutation. St001545The second Elser number of a connected graph. St001637The number of (upper) dissectors of a poset. St001741The largest integer such that all patterns of this size are contained in the permutation. St001760The number of prefix or suffix reversals needed to sort a permutation. St001817The number of flag weak exceedances of a signed permutation. St001486The number of corners of the ribbon associated with an integer composition. St000401The size of the symmetry class of a permutation. St001722The number of minimal chains with small intervals between a binary word and the top element. St000039The number of crossings of a permutation. St000043The number of crossings plus two-nestings of a perfect matching. St000089The absolute variation of a composition. St000226The convexity of a permutation. St000234The number of global ascents of a permutation. St000247The number of singleton blocks of a set partition. St000304The load of a permutation. St000338The number of pixed points of a permutation. St000341The non-inversion sum of a permutation. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000423The number of occurrences of the pattern 123 or of the pattern 132 in a permutation. St000427The number of occurrences of the pattern 123 or of the pattern 231 in a permutation. St000428The number of occurrences of the pattern 123 or of the pattern 213 in a permutation. St000430The number of occurrences of the pattern 123 or of the pattern 312 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000471The sum of the ascent tops of a permutation. St000472The sum of the ascent bottoms of a permutation. St000491The number of inversions of a set partition. St000498The lcs statistic of a set partition. St000500Eigenvalues of the random-to-random operator acting on the regular representation. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000565The major index of a set partition. St000575The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element and 2 a singleton. St000576The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal and 2 a minimal element. St000578The number of occurrences of the pattern {{1},{2}} such that 1 is a singleton. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000589The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000606The number of occurrences of the pattern {{1},{2,3}} such that 1,3 are maximal, (2,3) are consecutive in a block. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000611The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000837The number of ascents of distance 2 of a permutation. St000872The number of very big descents of a permutation. St000961The shifted major index of a permutation. St000962The 3-shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001130The number of two successive successions in a permutation. St001298The number of repeated entries in the Lehmer code of a permutation. St001332The number of steps on the non-negative side of the walk associated with the permutation. St001402The number of separators in a permutation. St001403The number of vertical separators in a permutation. St001511The minimal number of transpositions needed to sort a permutation in either direction. St001519The pinnacle sum of a permutation. St001536The number of cyclic misalignments of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001639The number of alternating subsets such that applying the permutation does not yield an alternating subset. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001781The interlacing number of a set partition. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St001857The number of edges in the reduced word graph of a signed permutation. St000075The orbit size of a standard tableau under promotion. St000078The number of alternating sign matrices whose left key is the permutation. St000210Minimum over maximum difference of elements in cycles. St000253The crossing number of a set partition. St000280The size of the preimage of the map 'to labelling permutation' from Parking functions to Permutations. St000308The height of the tree associated to a permutation. St000490The intertwining number of a set partition. St000502The number of successions of a set partitions. St000530The number of permutations with the same descent word as the given permutation. St000574The number of occurrences of the pattern {{1},{2}} such that 1 is a minimal and 2 a maximal element. St000654The first descent of a permutation. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000727The largest label of a leaf in the binary search tree associated with the permutation. St000729The minimal arc length of a set partition. St000756The sum of the positions of the left to right maxima of a permutation. St000887The maximal number of nonzero entries on a diagonal of a permutation matrix. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001246The maximal difference between two consecutive entries of a permutation. St001346The number of parking functions that give the same permutation. St001641The number of ascent tops in the flattened set partition such that all smaller elements appear before. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001806The upper middle entry of a permutation. St001958The degree of the polynomial interpolating the values of a permutation. St000084The number of subtrees. St000241The number of cyclical small excedances. St000248The number of anti-singletons of a set partition. St000249The number of singletons (St000247) plus the number of antisingletons (St000248) of a set partition. St000328The maximum number of child nodes in a tree. St000485The length of the longest cycle of a permutation. St000487The length of the shortest cycle of a permutation. St000504The cardinality of the first block of a set partition. St000823The number of unsplittable factors of the set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St001062The maximal size of a block of a set partition. St000029The depth of a permutation. St000030The sum of the descent differences of a permutations. St000051The size of the left subtree of a binary tree. St000055The inversion sum of a permutation. St000073The number of boxed entries. St000077The number of boxed and circled entries. St000080The rank of the poset. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000111The sum of the descent tops (or Genocchi descents) of a permutation. St000117The number of centered tunnels of a Dyck path. St000118The number of occurrences of the contiguous pattern [.,[.,[.,.]]] in a binary tree. St000121The number of occurrences of the contiguous pattern [.,[.,[.,[.,.]]]] in a binary tree. St000122The number of occurrences of the contiguous pattern [.,[.,[[.,.],.]]] in a binary tree. St000125The number of occurrences of the contiguous pattern [.,[[[.,.],.],. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000129The number of occurrences of the contiguous pattern [.,[.,[[[.,.],.],.]]] in a binary tree. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000156The Denert index of a permutation. St000173The segment statistic of a semistandard tableau. St000174The flush statistic of a semistandard tableau. St000196The number of occurrences of the contiguous pattern [[.,.],[.,. St000204The number of internal nodes of a binary tree. St000209Maximum difference of elements in cycles. St000224The sorting index of a permutation. St000252The number of nodes of degree 3 of a binary tree. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000268The number of strongly connected orientations of a graph. St000274The number of perfect matchings of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000309The number of vertices with even degree. St000310The minimal degree of a vertex of a graph. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000334The maz index, the major index of a permutation after replacing fixed points by zeros. St000339The maf index of a permutation. St000344The number of strongly connected outdegree sequences of a graph. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000494The number of inversions of distance at most 3 of a permutation. St000538The number of even inversions of a permutation. St000595The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal. St000599The number of occurrences of the pattern {{1},{2,3}} such that (2,3) are consecutive in a block. St000605The number of occurrences of the pattern {{1},{2,3}} such that 3 is maximal, (2,3) are consecutive in a block. St000612The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block. St000614The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, 3 is maximal, (2,3) are consecutive in a block. St000616The inversion index of a permutation. St000637The length of the longest cycle in a graph. St000653The last descent of a permutation. St000674The number of hills of a Dyck path. St000677The standardized bi-alternating inversion number of a permutation. St000680The Grundy value for Hackendot on posets. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000699The toughness times the least common multiple of 1,. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000794The mak of a permutation. St000795The mad of a permutation. St000809The reduced reflection length of the permutation. St000830The total displacement of a permutation. St000932The number of occurrences of the pattern UDU in a Dyck path. St000948The chromatic discriminant of a graph. St000951The dimension of $Ext^{1}(D(A),A)$ of the corresponding LNakayama algebra. St000957The number of Bruhat lower covers of a permutation. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001073The number of nowhere zero 3-flows of a graph. St001076The minimal length of a factorization of a permutation into transpositions that are cyclic shifts of (12). St001077The prefix exchange distance of a permutation. St001079The minimal length of a factorization of a permutation using the permutations (12)(34). St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001114The number of odd descents of a permutation. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001153The number of blocks with even minimum in a set partition. St001160The number of proper blocks (or intervals) of a permutations. St001161The major index north count of a Dyck path. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. 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. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St001309The number of four-cliques in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001319The minimal number of occurrences of the star-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001347The number of pairs of vertices of a graph having the same neighbourhood. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001379The number of inversions plus the major index of a permutation. St001412Number of minimal entries in the Bruhat order matrix of a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001521Half the total irregularity of a graph. St001522The total irregularity of a graph. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St001565The number of arithmetic progressions of length 2 in a permutation. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001574The minimal number of edges to add or remove to make a graph regular. St001575The minimal number of edges to add or remove to make a graph edge transitive. St001576The minimal number of edges to add or remove to make a graph vertex transitive. St001577The minimal number of edges to add or remove to make a graph a cograph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001579The number of cyclically simple transpositions decreasing the number of cyclic descents needed to sort a permutation. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001631The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset. St001646The number of edges that can be added without increasing the maximal degree of a graph. St001669The number of single rises in a Dyck path. St001671Haglund's hag of a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001691The number of kings in a graph. St001692The number of vertices with higher degree than the average degree in a graph. St001708The number of pairs of vertices of different degree in a graph. St001718The number of non-empty open intervals in a poset. St001731The factorization defect of a permutation. St001736The total number of cycles in a graph. St001742The difference of the maximal and the minimal degree in a graph. St001759The Rajchgot index of a permutation. St001783The number of odd automorphisms of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001797The number of overfull subgraphs of a graph. St001799The number of proper separations of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001846The number of elements which do not have a complement in the lattice. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001871The number of triconnected components of a graph. St001903The number of fixed points of a parking function. St001927Sparre Andersen's number of positives of a signed permutation. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000020The rank of the permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000037The sign of a permutation. St000040The number of regions of the inversion arrangement of a permutation. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000061The number of nodes on the left branch of a binary tree. St000074The number of special entries. St000082The number of elements smaller than a binary tree in Tamari order. St000109The number of elements less than or equal to the given element in Bruhat order. St000201The number of leaf nodes in a binary tree. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000211The rank of the set partition. St000230Sum of the minimal elements of the blocks of a set partition. St000251The number of nonsingleton blocks of a set partition. St000254The nesting number of a set partition. St000260The radius of a connected graph. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000273The domination number of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000335The difference of lower and upper interactions. St000349The number of different adjacency matrices of a graph. St000362The size of a minimal vertex cover of a graph. St000363The number of minimal vertex covers of a graph. St000385The number of vertices with out-degree 1 in a binary tree. St000387The matching number of a graph. St000388The number of orbits of vertices of a graph under automorphisms. St000396The register function (or Horton-Strahler number) of a binary tree. St000409The number of pitchforks in a binary tree. St000420The number of Dyck paths that are weakly above a Dyck path. St000460The hook length of the last cell along the main diagonal of an integer partition. St000474Dyson's crank of a partition. St000492The rob statistic of a set partition. St000501The size of the first part in the decomposition of a permutation. St000503The maximal difference between two elements in a common block. St000505The biggest entry in the block containing the 1. St000529The number of permutations whose descent word is the given binary word. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000558The number of occurrences of the pattern {{1,2}} in a set partition. St000568The hook number of a binary tree. St000577The number of occurrences of the pattern {{1},{2}} such that 1 is a maximal element. St000617The number of global maxima of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St000690The size of the conjugacy class of a permutation. St000701The protection number of a binary tree. St000717The number of ordinal summands of a poset. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000730The maximal arc length of a set partition. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000832The number of permutations obtained by reversing blocks of three consecutive numbers. St000844The size of the largest block in the direct sum decomposition of a permutation. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000906The length of the shortest maximal chain in a poset. St000916The packing number of a graph. St000919The number of maximal left branches of a binary tree. St000946The sum of the skew hook positions in a Dyck path. St000952Gives the number of irreducible factors of the Coxeter polynomial of the Dyck path over the rational numbers. St000959The number of strong Bruhat factorizations of a permutation. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000971The smallest closer of a set partition. St000985The number of positive eigenvalues of the adjacency matrix of the graph. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001050The number of terminal closers of a set partition. St001056The Grundy value for the game of deleting vertices of a graph until it has no edges. St001057The Grundy value of the game of creating an independent set in a graph. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001151The number of blocks with odd minimum. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001191Number of simple modules $S$ with $Ext_A^i(S,A)=0$ for all $i=0,1,...,g-1$ in the corresponding Nakayama algebra $A$ with global dimension $g$. St001248Sum of the even parts of a partition. St001262The dimension of the maximal parabolic seaweed algebra corresponding to the partition. St001277The degeneracy of a graph. St001285The number of primes in the column sums of the two line notation of a permutation. St001287The number of primes obtained by multiplying preimage and image of a permutation and subtracting one. St001322The size of a minimal independent dominating set in a graph. St001333The cardinality of a minimal edge-isolating set of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001339The irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001349The number of different graphs obtained from the given graph by removing an edge. St001352The number of internal nodes in the modular decomposition of a graph. St001354The number of series nodes in the modular decomposition of a graph. St001358The largest degree of a regular subgraph of a graph. St001360The number of covering relations in Young's lattice below a partition. St001363The Euler characteristic of a graph according to Knill. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001389The number of partitions of the same length below the given integer partition. St001393The induced matching number of a graph. St001405The number of bonds in a permutation. St001463The number of distinct columns in the nullspace of a graph. St001464The number of bases of the positroid corresponding to the permutation, with all fixed points counterclockwise. St001468The smallest fixpoint of a permutation. St001471The magnitude of a Dyck path. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001481The minimal height of a peak of a Dyck path. St001497The position of the largest weak excedence of a permutation. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001512The minimum rank of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St001527The cyclic permutation representation number of an integer partition. St001528The number of permutations such that the product with the permutation has the same number of fixed points. St001546The number of monomials in the Tutte polynomial of a graph. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001607The number of coloured graphs such that the multiplicities of colours are given by a partition. St001611The number of multiset partitions such that the multiplicities of elements are given by a partition. St001642The Prague dimension of a graph. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St001694The number of maximal dissociation sets in a graph. St001717The largest size of an interval in a poset. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001732The number of peaks visible from the left. St001733The number of weak left to right maxima of a Dyck path. St001734The lettericity of a graph. St001739The number of graphs with the same edge polytope as the given graph. St001743The discrepancy of a graph. St001765The number of connected components of the friends and strangers graph. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001792The arboricity of a graph. St001800The number of 3-Catalan paths having this Dyck path as first and last coordinate projections. St001807The lower middle entry of a permutation. St001808The box weight or horizontal decoration of a Dyck path. St001812The biclique partition number of a graph. St001813The product of the sizes of the principal order filters in a poset. St001829The common independence number of a graph. St001854The size of the left Kazhdan-Lusztig cell, St001914The size of the orbit of an integer partition in Bulgarian solitaire. St001917The order of toric promotion on the set of labellings of a graph. St001949The rigidity index of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St001955The number of natural descents for set-valued two row standard Young tableaux. St000064The number of one-box pattern of a permutation. St000145The Dyson rank of a partition. St000172The Grundy number of a graph. St000244The cardinality of the automorphism group of a graph. St000250The number of blocks (St000105) plus the number of antisingletons (St000248) of a set partition. St000258The burning number of a graph. St000299The number of nonisomorphic vertex-induced subtrees. St000364The exponent of the automorphism group of a graph. St000469The distinguishing number of a graph. St000489The number of cycles of a permutation of length at most 3. St000636The hull number of a graph. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000643The size of the largest orbit of antichains under Panyushev complementation. St000676The number of odd rises of a Dyck path. St000724The label of the leaf of the path following the smaller label in the increasing binary tree associated to a permutation. St000822The Hadwiger number of the graph. St000839The largest opener of a set partition. St000907The number of maximal antichains of minimal length in a poset. St000917The open packing number of a graph. St000918The 2-limited packing number of a graph. St000969We 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 $c_{n-1}$. St000981The length of the longest zigzag subpath. St001029The size of the core of a graph. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001093The detour number of a graph. St001109The number of proper colourings of a graph with as few colours as possible. St001111The weak 2-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001126Number of simple module that are 1-regular 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. St001261The Castelnuovo-Mumford regularity of a graph. St001280The number of parts of an integer partition that are at least two. St001288The number of primes obtained by multiplying preimage and image of a permutation and adding one. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001315The dissociation number of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001415The length of the longest palindromic prefix of a binary word. St001439The number of even weak deficiencies and of odd weak exceedences. St001494The Alon-Tarsi number of a graph. St001541The Gini index of an integer partition. St001566The length of the longest arithmetic progression in a permutation. St001580The acyclic chromatic number of a graph. St001654The monophonic hull number of a graph. St001670The connected partition number of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001707The length of a longest path in a graph such that the remaining vertices can be partitioned into two sets of the same size without edges between them. St001716The 1-improper chromatic number of a graph. St001720The minimal length of a chain of small intervals in a lattice. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001782The order of rowmotion on the set of order ideals of a poset. St001826The maximal number of leaves on a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001943The sum of the squares of the hook lengths of an integer partition. St000176The total number of tiles in the Gelfand-Tsetlin pattern. St001240The number of indecomposable modules e_i J^2 that have injective dimension at most one in the corresponding Nakayama algebra St001247The number of parts of a partition that are not congruent 2 modulo 3. St001249Sum of the odd parts of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. 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.
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