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Matching statistic: St000847
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Mp00158: Binary words —alternating inverse⟶ Binary words
St000847: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000847: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1
1 => 1 => 1
00 => 01 => 1
01 => 00 => 1
10 => 11 => 1
11 => 10 => 1
000 => 010 => 2
001 => 011 => 1
010 => 000 => 1
011 => 001 => 1
100 => 110 => 1
101 => 111 => 1
110 => 100 => 1
111 => 101 => 2
0000 => 0101 => 3
1111 => 1010 => 3
00000 => 01010 => 6
11111 => 10101 => 6
000000 => 010101 => 11
111111 => 101010 => 11
Description
The number of standard Young tableaux whose descent set is the binary word.
A descent in a standard Young tableau is an entry $i$ such that $i+1$ appears in a lower row in English notation.
For example, the tableaux $[[1,2,4],[3]]$ and $[[1,2],[3,4]]$ are those with descent set $\{2\}$, corresponding to the binary word $010$.
Matching statistic: St001282
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Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 11
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 11
Description
The number of graphs with the same chromatic polynomial.
Matching statistic: St001740
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Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 6
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 11
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> 11
Description
The number of graphs with the same symmetric edge polytope as the given graph.
The symmetric edge polytope of a graph on $n$ vertices is the polytope in $\mathbb R^n$ defined as the convex hull of $e_i - e_j$ and $e_j - e_i$ for each edge $(i, j)$, where $e_1,\dots, e_n$ denotes the standard basis.
Matching statistic: St000034
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 90%●distinct values known / distinct values provided: 80%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St000034: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 90%●distinct values known / distinct values provided: 80%
Values
0 => [1] => [1,0]
=> [1] => 0 = 1 - 1
1 => [1] => [1,0]
=> [1] => 0 = 1 - 1
00 => [2] => [1,1,0,0]
=> [2,1] => 0 = 1 - 1
01 => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
10 => [1,1] => [1,0,1,0]
=> [1,2] => 0 = 1 - 1
11 => [2] => [1,1,0,0]
=> [2,1] => 0 = 1 - 1
000 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
001 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 0 = 1 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0 = 1 - 1
110 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0 = 1 - 1
111 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1 = 2 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 3 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 2 = 3 - 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5 = 6 - 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 5 = 6 - 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ? = 11 - 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ? = 11 - 1
Description
The maximum defect over any reduced expression for a permutation and any subexpression.
Matching statistic: St001327
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001327: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 70%●distinct values known / distinct values provided: 40%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001327: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 70%●distinct values known / distinct values provided: 40%
Values
0 => 0 => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
1 => 1 => ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
01 => 00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 0 = 1 - 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 0 = 1 - 1
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1 = 2 - 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 0 = 1 - 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 0 = 1 - 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> 1 = 2 - 1
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 3 - 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 3 - 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? = 6 - 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? = 6 - 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(2,11),(3,10),(4,9),(5,8),(6,7)],12)
=> ? = 11 - 1
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(2,11),(3,10),(4,9),(5,8),(6,7)],12)
=> ? = 11 - 1
Description
The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph.
A graph is a split graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ is an edge and $(b,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001329
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001329: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 70%●distinct values known / distinct values provided: 40%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001329: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 70%●distinct values known / distinct values provided: 40%
Values
0 => 0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
1 => 1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
01 => 00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 6 - 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 6 - 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(0,5),(1,2),(1,3),(2,8),(2,9),(3,8),(3,9),(4,10),(4,11),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11)],12)
=> ? = 11 - 1
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(0,5),(1,2),(1,3),(2,8),(2,9),(3,8),(3,9),(4,10),(4,11),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11)],12)
=> ? = 11 - 1
Description
The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph.
A graph is outerplanar if and only if in any linear ordering of its vertices, there are no four vertices $a < b < c < d$ such that $(a,c)$ and $(b,d)$ are edges. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Matching statistic: St001656
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 70%●distinct values known / distinct values provided: 40%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001656: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 70%●distinct values known / distinct values provided: 40%
Values
0 => 0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
1 => 1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 2 = 1 + 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
01 => 00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2 = 1 + 1
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3 = 2 + 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2 = 1 + 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 3 = 2 + 1
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 + 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 + 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 6 + 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 6 + 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(0,5),(1,2),(1,3),(2,8),(2,9),(3,8),(3,9),(4,10),(4,11),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11)],12)
=> ? = 11 + 1
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(0,5),(1,2),(1,3),(2,8),(2,9),(3,8),(3,9),(4,10),(4,11),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11)],12)
=> ? = 11 + 1
Description
The monophonic position number of a graph.
A subset $M$ of the vertex set of a graph is a monophonic position set if no three vertices of $M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.
Matching statistic: St001871
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001871: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 70%●distinct values known / distinct values provided: 40%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001871: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 70%●distinct values known / distinct values provided: 40%
Values
0 => 0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
1 => 1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 0 = 1 - 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
01 => 00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 0 = 1 - 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0 = 1 - 1
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 0 = 1 - 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> 0 = 1 - 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 2 - 1
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 3 - 1
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 6 - 1
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(0,4),(0,5),(1,2),(1,3),(2,6),(2,7),(3,6),(3,7),(4,8),(4,9),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9)],10)
=> ? = 6 - 1
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(0,5),(1,2),(1,3),(2,8),(2,9),(3,8),(3,9),(4,10),(4,11),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11)],12)
=> ? = 11 - 1
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(0,4),(0,5),(1,2),(1,3),(2,8),(2,9),(3,8),(3,9),(4,10),(4,11),(5,10),(5,11),(6,8),(6,9),(6,10),(6,11),(7,8),(7,9),(7,10),(7,11)],12)
=> ? = 11 - 1
Description
The number of triconnected components of a graph.
A connected graph is '''triconnected''' or '''3-vertex connected''' if it cannot be disconnected by removing two or fewer vertices. An arbitrary connected graph can be decomposed as a union of biconnected (2-vertex connected) graphs, known as '''blocks''', and each biconnected graph can be decomposed as a union of components with are either a cycle (type "S"), a cocyle (type "P"), or triconnected (type "R"). The decomposition of a biconnected graph into these components is known as the '''SPQR-tree''' of the graph.
Matching statistic: St000741
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 50%●distinct values known / distinct values provided: 20%
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000741: Graphs ⟶ ℤResult quality: 20% ●values known / values provided: 50%●distinct values known / distinct values provided: 20%
Values
0 => 0 => ([(0,1)],2)
=> ([],2)
=> 1
1 => 1 => ([(0,1)],2)
=> ([],2)
=> 1
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 1
01 => 00 => ([(0,2),(2,1)],3)
=> ([],3)
=> 1
10 => 11 => ([(0,2),(2,1)],3)
=> ([],3)
=> 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ? = 1
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ? = 2
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
010 => 000 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
101 => 111 => ([(0,3),(2,1),(3,2)],4)
=> ([],4)
=> 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ? = 2
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 3
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
=> ([(2,7),(3,6),(4,5)],8)
=> ? = 3
00000 => 01010 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? = 6
11111 => 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ([(2,9),(3,8),(4,7),(5,6)],10)
=> ? = 6
000000 => 010101 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(2,11),(3,10),(4,9),(5,8),(6,7)],12)
=> ? = 11
111111 => 101010 => ([(0,1),(0,2),(1,10),(1,11),(2,10),(2,11),(4,3),(5,3),(6,8),(6,9),(7,8),(7,9),(8,4),(8,5),(9,4),(9,5),(10,6),(10,7),(11,6),(11,7)],12)
=> ([(2,11),(3,10),(4,9),(5,8),(6,7)],12)
=> ? = 11
Description
The Colin de Verdière graph invariant.
Matching statistic: St000181
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00272: Binary words —Gray next⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 40%●distinct values known / distinct values provided: 20%
Mp00262: Binary words —poset of factors⟶ Posets
St000181: Posets ⟶ ℤResult quality: 20% ●values known / values provided: 40%●distinct values known / distinct values provided: 20%
Values
0 => 1 => ([(0,1)],2)
=> 1
1 => 0 => ([(0,1)],2)
=> 1
00 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
01 => 00 => ([(0,2),(2,1)],3)
=> 1
10 => 11 => ([(0,2),(2,1)],3)
=> 1
11 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1
000 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 2
001 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 1
010 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
011 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
101 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
110 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 2
0000 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
1111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 3
00000 => 10000 => ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 6
11111 => 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 6
000000 => 100000 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 11
111111 => 011111 => ([(0,2),(0,6),(1,8),(2,7),(3,5),(3,9),(4,3),(4,11),(5,1),(5,10),(6,4),(6,7),(7,11),(9,10),(10,8),(11,9)],12)
=> ? = 11
Description
The number of connected components of the Hasse diagram for the poset.
The following 97 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001890The maximum magnitude of the Möbius function of a poset. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001964The interval resolution global dimension of a poset. St000264The girth of a graph, which is not a 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. St000284The Plancherel distribution on integer partitions. St000618The number of self-evacuating tableaux of given shape. St000667The greatest common divisor of the parts of the partition. 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. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. 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. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001621The number of atoms of a lattice. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. 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. St001913The number of preimages of an integer partition in Bulgarian solitaire. 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. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000369The dinv deficit of a Dyck path. St000376The bounce deficit of a Dyck path. St000379The number of Hamiltonian cycles in a graph. 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. 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. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. 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 $eA$. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001625The Möbius invariant of a lattice. St001657The number of twos in an integer partition. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000455The second largest eigenvalue of a graph if it is integral. St000806The semiperimeter of the associated bargraph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001722The number of minimal chains with small intervals between a binary word and the top element. St000456The monochromatic index of a connected graph. St001545The second Elser number of a connected graph. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000762The sum of the positions of the weak records of an integer composition. St000782The indicator function of whether a given perfect matching is an L & P matching. St001118The acyclic chromatic index of a graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000464The Schultz index of a connected graph. St000699The toughness times the least common multiple of 1,. St000817The sum of the entries in the column specified by the composition of the change of basis matrix from dual immaculate quasisymmetric functions to monomial quasisymmetric functions. St000818The sum of the entries in the column specified by the composition of the change of basis matrix from quasisymmetric Schur functions to monomial quasisymmetric functions. St001060The distinguishing index of a graph.
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