Your data matches 145 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001960
(load all 17 compositions to match this statistic)
Mapping
St001960: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 1
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 1
[3,2,4,1] => 1
[3,4,1,2] => 0
[3,4,2,1] => 1
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 1
[4,2,3,1] => 1
[4,3,1,2] => 1
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 1
[1,2,5,3,4] => 1
[1,2,5,4,3] => 2
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 1
[1,3,5,4,2] => 2
[1,4,2,3,5] => 1
[1,4,2,5,3] => 2
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 1
[1,4,5,3,2] => 2
Description
The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000053
(load all 2 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000053: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
Description
The number of valleys of the Dyck path.
Matching statistic: St000272
(load all 6 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000272: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => ([],2)
 => 0
[2,1] => [2] => ([],2)
 => 0
[1,2,3] => [3] => ([],3)
 => 0
[1,3,2] => [1,2] => ([(1,2)],3)
 => 1
[2,1,3] => [3] => ([],3)
 => 0
[2,3,1] => [3] => ([],3)
 => 0
[3,1,2] => [3] => ([],3)
 => 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [4] => ([],4)
 => 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
 => 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
 => 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
 => 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [4] => ([],4)
 => 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,3,1,4] => [4] => ([],4)
 => 0
[2,3,4,1] => [4] => ([],4)
 => 0
[2,4,1,3] => [4] => ([],4)
 => 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,4] => [4] => ([],4)
 => 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [4] => ([],4)
 => 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => [4] => ([],4)
 => 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
 => 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
 => 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
 => 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
 => 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
 => 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
 => 1
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The treewidth of a graph. A graph has treewidth zero if and only if it has no edges. A connected graph has treewidth at most one if and only if it is a tree. A connected graph has treewidth at most two if and only if it is a series-parallel graph.
Matching statistic: St000306
(load all 2 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000306: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
Description
The bounce count of a Dyck path. For a Dyck path $D$ of length $2n$, this is the number of points $(i,i)$ for $1 \leq i < n$ that are touching points of the [[Mp00099|bounce path]] of $D$.
Matching statistic: St000362
(load all 6 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000362: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => ([],2)
 => 0
[2,1] => [2] => ([],2)
 => 0
[1,2,3] => [3] => ([],3)
 => 0
[1,3,2] => [1,2] => ([(1,2)],3)
 => 1
[2,1,3] => [3] => ([],3)
 => 0
[2,3,1] => [3] => ([],3)
 => 0
[3,1,2] => [3] => ([],3)
 => 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [4] => ([],4)
 => 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
 => 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
 => 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
 => 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [4] => ([],4)
 => 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,3,1,4] => [4] => ([],4)
 => 0
[2,3,4,1] => [4] => ([],4)
 => 0
[2,4,1,3] => [4] => ([],4)
 => 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,4] => [4] => ([],4)
 => 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [4] => ([],4)
 => 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => [4] => ([],4)
 => 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
 => 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
 => 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
 => 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
 => 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
 => 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
 => 1
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The size of a minimal vertex cover of a graph. A '''vertex cover''' of a graph $G$ is a subset $S$ of the vertices of $G$ such that each edge of $G$ contains at least one vertex of $S$. Finding a minimal vertex cover is an NP-hard optimization problem.
Matching statistic: St000536
(load all 6 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000536: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => ([],2)
 => 0
[2,1] => [2] => ([],2)
 => 0
[1,2,3] => [3] => ([],3)
 => 0
[1,3,2] => [1,2] => ([(1,2)],3)
 => 1
[2,1,3] => [3] => ([],3)
 => 0
[2,3,1] => [3] => ([],3)
 => 0
[3,1,2] => [3] => ([],3)
 => 0
[3,2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 1
[1,2,3,4] => [4] => ([],4)
 => 0
[1,2,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[1,3,2,4] => [1,3] => ([(2,3)],4)
 => 1
[1,3,4,2] => [1,3] => ([(2,3)],4)
 => 1
[1,4,2,3] => [1,3] => ([(2,3)],4)
 => 1
[1,4,3,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[2,1,3,4] => [4] => ([],4)
 => 0
[2,1,4,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[2,3,1,4] => [4] => ([],4)
 => 0
[2,3,4,1] => [4] => ([],4)
 => 0
[2,4,1,3] => [4] => ([],4)
 => 0
[2,4,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[3,1,2,4] => [4] => ([],4)
 => 0
[3,1,4,2] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,2,1,4] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,2,4,1] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[3,4,1,2] => [4] => ([],4)
 => 0
[3,4,2,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,1,2,3] => [4] => ([],4)
 => 0
[4,1,3,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,2,1,3] => [2,2] => ([(1,3),(2,3)],4)
 => 1
[4,2,3,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 1
[4,3,1,2] => [1,3] => ([(2,3)],4)
 => 1
[4,3,2,1] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,2,3,4,5] => [5] => ([],5)
 => 0
[1,2,3,5,4] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 1
[1,2,4,3,5] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,4,5,3] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,3,4] => [2,3] => ([(2,4),(3,4)],5)
 => 1
[1,2,5,4,3] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,2,4,5] => [1,4] => ([(3,4)],5)
 => 1
[1,3,2,5,4] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,3,4,2,5] => [1,4] => ([(3,4)],5)
 => 1
[1,3,4,5,2] => [1,4] => ([(3,4)],5)
 => 1
[1,3,5,2,4] => [1,4] => ([(3,4)],5)
 => 1
[1,3,5,4,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,2,3,5] => [1,4] => ([(3,4)],5)
 => 1
[1,4,2,5,3] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,2,5] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,3,5,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,4,5,2,3] => [1,4] => ([(3,4)],5)
 => 1
[1,4,5,3,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The pathwidth of a graph.
Matching statistic: St000711
(load all 5 compositions to match this statistic)
Mapping
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
St000711: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,2] => [2,1] => 0
[2,1] => [2,1] => [1,2] => 0
[1,2,3] => [1,2,3] => [2,3,1] => 0
[1,3,2] => [1,3,2] => [3,2,1] => 1
[2,1,3] => [2,1,3] => [1,3,2] => 0
[2,3,1] => [3,2,1] => [2,1,3] => 0
[3,1,2] => [2,3,1] => [1,2,3] => 0
[3,2,1] => [3,1,2] => [3,1,2] => 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,1] => 0
[1,2,4,3] => [1,2,4,3] => [2,4,3,1] => 1
[1,3,2,4] => [1,3,2,4] => [3,2,4,1] => 1
[1,3,4,2] => [1,4,3,2] => [4,3,2,1] => 1
[1,4,2,3] => [1,3,4,2] => [4,2,3,1] => 1
[1,4,3,2] => [1,4,2,3] => [3,4,2,1] => 2
[2,1,3,4] => [2,1,3,4] => [1,3,4,2] => 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,2] => 1
[2,3,1,4] => [3,2,1,4] => [2,1,4,3] => 0
[2,3,4,1] => [4,2,3,1] => [2,3,1,4] => 0
[2,4,1,3] => [3,2,4,1] => [2,1,3,4] => 0
[2,4,3,1] => [4,2,1,3] => [2,4,1,3] => 1
[3,1,2,4] => [2,3,1,4] => [1,2,4,3] => 0
[3,1,4,2] => [3,4,1,2] => [4,1,2,3] => 1
[3,2,1,4] => [3,1,2,4] => [3,1,4,2] => 1
[3,2,4,1] => [4,3,2,1] => [3,2,1,4] => 1
[3,4,1,2] => [2,4,3,1] => [1,3,2,4] => 0
[3,4,2,1] => [4,1,3,2] => [4,3,1,2] => 1
[4,1,2,3] => [2,3,4,1] => [1,2,3,4] => 0
[4,1,3,2] => [3,4,2,1] => [3,1,2,4] => 1
[4,2,1,3] => [3,1,4,2] => [4,1,3,2] => 1
[4,2,3,1] => [4,3,1,2] => [4,2,1,3] => 1
[4,3,1,2] => [2,4,1,3] => [1,4,2,3] => 1
[4,3,2,1] => [4,1,2,3] => [3,4,1,2] => 2
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,1] => 0
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,1] => 1
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,1] => 1
[1,2,4,5,3] => [1,2,5,4,3] => [2,5,4,3,1] => 1
[1,2,5,3,4] => [1,2,4,5,3] => [2,5,3,4,1] => 1
[1,2,5,4,3] => [1,2,5,3,4] => [2,4,5,3,1] => 2
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,1] => 1
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,1] => 2
[1,3,4,2,5] => [1,4,3,2,5] => [4,3,2,5,1] => 1
[1,3,4,5,2] => [1,5,3,4,2] => [5,3,4,2,1] => 1
[1,3,5,2,4] => [1,4,3,5,2] => [5,3,2,4,1] => 1
[1,3,5,4,2] => [1,5,3,2,4] => [4,3,5,2,1] => 2
[1,4,2,3,5] => [1,3,4,2,5] => [4,2,3,5,1] => 1
[1,4,2,5,3] => [1,4,5,2,3] => [4,5,2,3,1] => 2
[1,4,3,2,5] => [1,4,2,3,5] => [3,4,2,5,1] => 2
[1,4,3,5,2] => [1,5,4,3,2] => [5,4,3,2,1] => 2
[1,4,5,2,3] => [1,3,5,4,2] => [5,2,4,3,1] => 1
[1,4,5,3,2] => [1,5,2,4,3] => [3,5,4,2,1] => 2
Description
The number of big exceedences of a permutation. A big exceedence of a permutation $\pi$ is an index $i$ such that $\pi(i) - i > 1$. This statistic is equidistributed with either of the numbers of big descents, big ascents, and big deficiencies.
Matching statistic: St001142
(load all 2 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001142: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
Description
The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001169
(load all 2 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001169: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
Description
Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra.
Matching statistic: St001197
(load all 2 compositions to match this statistic)
Mapping
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001197: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => [2] => [1,1,0,0]
 => 0
[2,1] => [2] => [1,1,0,0]
 => 0
[1,2,3] => [3] => [1,1,1,0,0,0]
 => 0
[1,3,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[2,1,3] => [3] => [1,1,1,0,0,0]
 => 0
[2,3,1] => [3] => [1,1,1,0,0,0]
 => 0
[3,1,2] => [3] => [1,1,1,0,0,0]
 => 0
[3,2,1] => [2,1] => [1,1,0,0,1,0]
 => 1
[1,2,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[1,2,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,3,2,4] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,3,4,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,2,3] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[1,4,3,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[2,1,3,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,1,4,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[2,3,1,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,3,4,1] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,1,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[2,4,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[3,1,2,4] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,1,4,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,1,4] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,2,4,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[3,4,1,2] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[3,4,2,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,1,2,3] => [4] => [1,1,1,1,0,0,0,0]
 => 0
[4,1,3,2] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,2,1,3] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[4,2,3,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[4,3,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 1
[4,3,2,1] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,2,3,4,5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 0
[1,2,3,5,4] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,2,4,3,5] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,4,5,3] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,3,4] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,2,5,4,3] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,3,2,4,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,2,5,4] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,3,4,2,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,4,5,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,2,4] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,3,5,4,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,4,2,3,5] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,2,5,3] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,2,5] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,3,5,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,4,5,2,3] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 1
[1,4,5,3,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
Description
The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 135 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $eA$. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001277The degeneracy of a graph. St001278The number of indecomposable modules that are fixed by $\tau \Omega^1$ composed with its inverse in the corresponding Nakayama algebra. St001358The largest degree of a regular subgraph of a graph. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001812The biclique partition number of a graph. St000010The length of the partition. St000011The number of touch points (or returns) of a Dyck path. St000015The number of peaks of a Dyck path. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000288The number of ones in a binary word. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001116The game chromatic number of a graph. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. 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: St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001494The Alon-Tarsi number of a graph. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001963The tree-depth of a graph. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001258Gives the maximum of injective plus projective dimension of an indecomposable module over the corresponding Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000024The number of double up and double down steps of a Dyck path. St000157The number of descents of a standard tableau. St000171The degree of the graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000394The sum of the heights of the peaks of a Dyck path minus the number of peaks. St000454The largest eigenvalue of a graph if it is integral. St000691The number of changes of a binary word. St000710The number of big deficiencies of a permutation. St000741The Colin de Verdière graph invariant. St000778The metric dimension of a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001119The length of a shortest maximal path in a graph. St001120The length of a longest path in a graph. St001189The number of simple modules with dominant and codominant dimension equal to zero in the Nakayama algebra corresponding to the Dyck path. St001270The bandwidth of a graph. St001357The maximal degree of a regular spanning subgraph of a graph. St001382The number of boxes in the diagram of a partition that do not lie in its Durfee square. St001391The disjunction number of a graph. St001418Half of the global dimension of the stable Auslander algebra 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. St001480The number of simple summands of the module J^2/J^3. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001644The dimension of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001949The rigidity index of a graph. St001962The proper pathwidth of a graph. St000087The number of induced subgraphs. St000093The cardinality of a maximal independent set of vertices of a graph. St000147The largest part of an integer partition. St000228The size of a partition. St000286The number of connected components of the complement of a graph. St000363The number of minimal vertex covers of a graph. St000443The number of long tunnels of a Dyck path. St000459The hook length of the base cell of a partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000469The distinguishing number of a graph. St000619The number of cyclic descents of a permutation. St000636The hull number of a graph. St000722The number of different neighbourhoods in a graph. St000733The row containing the largest entry of a standard tableau. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St000870The product of the hook lengths of the diagonal cells in an integer partition. St000926The clique-coclique number of a graph. St000935The number of ordered refinements of an integer partition. St001007Number of simple modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001088Number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001187The number of simple modules with grade at least one in the corresponding Nakayama algebra. St001224Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. 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. St001316The domatic number of a graph. St001330The hat guessing number of a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001342The number of vertices in the center of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001368The number of vertices of maximal degree in a graph. St001427The number of descents of a signed permutation. St001645The pebbling number of a connected graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a 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. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St000236The number of cyclical small weak excedances. St000300The number of independent sets of vertices of a graph. St000301The number of facets of the stable set polytope of a graph. St000312The number of leaves in a graph. St000445The number of rises of length 1 of a Dyck path. 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. 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. St001964The interval resolution global dimension of a poset. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000806The semiperimeter of the associated bargraph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001176The size of a partition minus its first part. 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. St001896The number of right descents of a signed permutations. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000307The number of rowmotion orbits of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001875The number of simple modules with projective dimension at most 1. 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. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000264The girth of a graph, which is not a tree.