- FindStat

Your data matches 55 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St001116

Mapping

Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
St001116: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

0 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
1 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => 2
00 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 3
11 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 2
000 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
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)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 4
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)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => 3
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)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 4
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)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 4
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)
 => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
 => 3
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)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
 => 4
111 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 3
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
 => 3
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],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)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 4
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)
 => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4
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)
 => ([(0,1),(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 4

Description

The game chromatic number of a graph. Two players, Alice and Bob, take turns colouring properly any uncolored vertex of the graph. Alice begins. If it is not possible for either player to colour a vertex, then Bob wins. If the graph is completely colored, Alice wins. The game chromatic number is the smallest number of colours such that Alice has a winning strategy.

Matching statistic: St001646

Mapping

Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St001646: Graphs ⟶ ℤResult quality: 70% ●values known / values provided: 70%●distinct values known / distinct values provided: 100%

Values

0 => 0 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 2 - 2
1 => 1 => ([(0,1)],2)
 => ([(0,1)],2)
 => 0 = 2 - 2
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0 = 2 - 2
01 => 00 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
10 => 11 => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 1 = 3 - 2
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 0 = 2 - 2
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 = 3 - 2
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 = 4 - 2
010 => 000 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 3 - 2
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 = 4 - 2
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 = 4 - 2
101 => 111 => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1 = 3 - 2
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 = 4 - 2
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 = 3 - 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)
 => ([(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 - 2
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 - 2
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)
 => ? = 4 - 2
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)
 => ? = 4 - 2
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)
 => ? = 4 - 2
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)
 => ? = 4 - 2

Description

The number of edges that can be added without increasing the maximal degree of a graph. This statistic is (except for the degenerate case of two vertices) maximized by the star-graph on

$n$ vertices, which has maximal degree

$n-1$ and therefore has statistic

$\binom{n-1}{2}$ .

Matching statistic: St000777
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%

Values

0 => [1] => ([],1)
 => 1 = 2 - 1
1 => [1] => ([],1)
 => 1 = 2 - 1
00 => [2] => ([],2)
 => ? = 2 - 1
01 => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
10 => [1,1] => ([(0,1)],2)
 => 2 = 3 - 1
11 => [2] => ([],2)
 => ? = 2 - 1
000 => [3] => ([],3)
 => ? = 3 - 1
001 => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 4 - 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
011 => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
100 => [1,2] => ([(1,2)],3)
 => ? = 4 - 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2 = 3 - 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => 3 = 4 - 1
111 => [3] => ([],3)
 => ? = 3 - 1
0000 => [4] => ([],4)
 => ? = 3 - 1
1111 => [4] => ([],4)
 => ? = 3 - 1
00000 => [5] => ([],5)
 => ? = 4 - 1
11111 => [5] => ([],5)
 => ? = 4 - 1
000000 => [6] => ([],6)
 => ? = 4 - 1
111111 => [6] => ([],6)
 => ? = 4 - 1

Description

The number of distinct eigenvalues of the distance Laplacian of a connected graph.

Matching statistic: St000259
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%

Values

0 => [1] => ([],1)
 => 0 = 2 - 2
1 => [1] => ([],1)
 => 0 = 2 - 2
00 => [2] => ([],2)
 => ? = 2 - 2
01 => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
10 => [1,1] => ([(0,1)],2)
 => 1 = 3 - 2
11 => [2] => ([],2)
 => ? = 2 - 2
000 => [3] => ([],3)
 => ? = 3 - 2
001 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
011 => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
100 => [1,2] => ([(1,2)],3)
 => ? = 4 - 2
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 3 - 2
110 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 4 - 2
111 => [3] => ([],3)
 => ? = 3 - 2
0000 => [4] => ([],4)
 => ? = 3 - 2
1111 => [4] => ([],4)
 => ? = 3 - 2
00000 => [5] => ([],5)
 => ? = 4 - 2
11111 => [5] => ([],5)
 => ? = 4 - 2
000000 => [6] => ([],6)
 => ? = 4 - 2
111111 => [6] => ([],6)
 => ? = 4 - 2

Description

The diameter of a connected graph. This is the greatest distance between any pair of vertices.

Matching statistic: St000285
(load all 2 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000285: Integer compositions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%

Values

=> [1] => [1] => ? = 2 - 2
=> [1] => [1] => ? = 2 - 2
=> [2] => [1] => ? = 2 - 2
=> [1,1] => [2] => 1 = 3 - 2
=> [1,1] => [2] => 1 = 3 - 2
=> [2] => [1] => ? = 2 - 2
=> [3] => [1] => ? = 3 - 2
=> [2,1] => [1,1] => 2 = 4 - 2
=> [1,1,1] => [3] => 1 = 3 - 2
=> [1,2] => [1,1] => 2 = 4 - 2
=> [1,2] => [1,1] => 2 = 4 - 2
=> [1,1,1] => [3] => 1 = 3 - 2
=> [2,1] => [1,1] => 2 = 4 - 2
=> [3] => [1] => ? = 3 - 2
=> [4] => [1] => ? = 3 - 2
=> [4] => [1] => ? = 3 - 2
=> [5] => [1] => ? = 4 - 2
=> [5] => [1] => ? = 4 - 2
000000 => [6] => [1] => ? = 4 - 2
111111 => [6] => [1] => ? = 4 - 2

Description

The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions.

Matching statistic: St001118
(load all 2 compositions to match this statistic)

Mapping

Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%

Values

0 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 2
1 => ([(0,1)],2)
 => ([],2)
 => ? = 2 - 2
00 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 3 - 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 1 = 3 - 2
11 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 2 - 2
000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 3 - 2
001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1 = 3 - 2
011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 1 = 3 - 2
110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => 2 = 4 - 2
111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 3 - 2
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 3 - 2
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 3 - 2
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 4 - 2
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
 => ([],6)
 => ? = 4 - 2
000000 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ? = 4 - 2
111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
 => ([],7)
 => ? = 4 - 2

Description

The acyclic chromatic index of a graph. An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest. The smallest number of colours such that such a colouring exists is the acyclic chromatic index.

Matching statistic: St000455
(load all 4 compositions to match this statistic)

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%

Values

0 => [1] => ([],1)
 => ? = 2 - 4
1 => [1] => ([],1)
 => ? = 2 - 4
00 => [2] => ([],2)
 => ? = 2 - 4
01 => [1,1] => ([(0,1)],2)
 => -1 = 3 - 4
10 => [1,1] => ([(0,1)],2)
 => -1 = 3 - 4
11 => [2] => ([],2)
 => ? = 2 - 4
000 => [3] => ([],3)
 => ? = 3 - 4
001 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 4 - 4
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 3 - 4
011 => [1,2] => ([(1,2)],3)
 => 0 = 4 - 4
100 => [1,2] => ([(1,2)],3)
 => 0 = 4 - 4
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => -1 = 3 - 4
110 => [2,1] => ([(0,2),(1,2)],3)
 => 0 = 4 - 4
111 => [3] => ([],3)
 => ? = 3 - 4
0000 => [4] => ([],4)
 => ? = 3 - 4
1111 => [4] => ([],4)
 => ? = 3 - 4
00000 => [5] => ([],5)
 => ? = 4 - 4
11111 => [5] => ([],5)
 => ? = 4 - 4
000000 => [6] => ([],6)
 => ? = 4 - 4
111111 => [6] => ([],6)
 => ? = 4 - 4

Description

The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.

Matching statistic: St000420

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000420: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%

Values

0 => [1] => [1] => [1,0]
 => ? = 2 - 2
1 => [1] => [1] => [1,0]
 => ? = 2 - 2
00 => [2] => [1] => [1,0]
 => ? = 2 - 2
01 => [1,1] => [2] => [1,1,0,0]
 => 1 = 3 - 2
10 => [1,1] => [2] => [1,1,0,0]
 => 1 = 3 - 2
11 => [2] => [1] => [1,0]
 => ? = 2 - 2
000 => [3] => [1] => [1,0]
 => ? = 3 - 2
001 => [2,1] => [1,1] => [1,0,1,0]
 => 2 = 4 - 2
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1 = 3 - 2
011 => [1,2] => [1,1] => [1,0,1,0]
 => 2 = 4 - 2
100 => [1,2] => [1,1] => [1,0,1,0]
 => 2 = 4 - 2
101 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1 = 3 - 2
110 => [2,1] => [1,1] => [1,0,1,0]
 => 2 = 4 - 2
111 => [3] => [1] => [1,0]
 => ? = 3 - 2
0000 => [4] => [1] => [1,0]
 => ? = 3 - 2
1111 => [4] => [1] => [1,0]
 => ? = 3 - 2
00000 => [5] => [1] => [1,0]
 => ? = 4 - 2
11111 => [5] => [1] => [1,0]
 => ? = 4 - 2
000000 => [6] => [1] => [1,0]
 => ? = 4 - 2
111111 => [6] => [1] => [1,0]
 => ? = 4 - 2

Description

The number of Dyck paths that are weakly above a Dyck path.

Matching statistic: St000678

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000678: Dyck paths ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%

Values

0 => [1] => [1] => [1,0]
 => ? = 2 - 2
1 => [1] => [1] => [1,0]
 => ? = 2 - 2
00 => [2] => [1] => [1,0]
 => ? = 2 - 2
01 => [1,1] => [2] => [1,1,0,0]
 => 1 = 3 - 2
10 => [1,1] => [2] => [1,1,0,0]
 => 1 = 3 - 2
11 => [2] => [1] => [1,0]
 => ? = 2 - 2
000 => [3] => [1] => [1,0]
 => ? = 3 - 2
001 => [2,1] => [1,1] => [1,0,1,0]
 => 2 = 4 - 2
010 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1 = 3 - 2
011 => [1,2] => [1,1] => [1,0,1,0]
 => 2 = 4 - 2
100 => [1,2] => [1,1] => [1,0,1,0]
 => 2 = 4 - 2
101 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 1 = 3 - 2
110 => [2,1] => [1,1] => [1,0,1,0]
 => 2 = 4 - 2
111 => [3] => [1] => [1,0]
 => ? = 3 - 2
0000 => [4] => [1] => [1,0]
 => ? = 3 - 2
1111 => [4] => [1] => [1,0]
 => ? = 3 - 2
00000 => [5] => [1] => [1,0]
 => ? = 4 - 2
11111 => [5] => [1] => [1,0]
 => ? = 4 - 2
000000 => [6] => [1] => [1,0]
 => ? = 4 - 2
111111 => [6] => [1] => [1,0]
 => ? = 4 - 2

Description

The number of up steps after the last double rise of a Dyck path.

Matching statistic: St000706

Mapping

Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00040: Integer compositions —to partition⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%

Values

0 => [1] => [1] => [1]
 => ? = 2 - 2
1 => [1] => [1] => [1]
 => ? = 2 - 2
00 => [2] => [1] => [1]
 => ? = 2 - 2
01 => [1,1] => [2] => [2]
 => 1 = 3 - 2
10 => [1,1] => [2] => [2]
 => 1 = 3 - 2
11 => [2] => [1] => [1]
 => ? = 2 - 2
000 => [3] => [1] => [1]
 => ? = 3 - 2
001 => [2,1] => [1,1] => [1,1]
 => 2 = 4 - 2
010 => [1,1,1] => [3] => [3]
 => 1 = 3 - 2
011 => [1,2] => [1,1] => [1,1]
 => 2 = 4 - 2
100 => [1,2] => [1,1] => [1,1]
 => 2 = 4 - 2
101 => [1,1,1] => [3] => [3]
 => 1 = 3 - 2
110 => [2,1] => [1,1] => [1,1]
 => 2 = 4 - 2
111 => [3] => [1] => [1]
 => ? = 3 - 2
0000 => [4] => [1] => [1]
 => ? = 3 - 2
1111 => [4] => [1] => [1]
 => ? = 3 - 2
00000 => [5] => [1] => [1]
 => ? = 4 - 2
11111 => [5] => [1] => [1]
 => ? = 4 - 2
000000 => [6] => [1] => [1]
 => ? = 4 - 2
111111 => [6] => [1] => [1]
 => ? = 4 - 2

Description

The product of the factorials of the multiplicities of an integer partition.

The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.

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. St000993The multiplicity of the largest part of an integer partition. St001500The global dimension of magnitude 1 Nakayama algebras. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001568The smallest positive integer that does not appear twice in the partition. St001808The box weight or horizontal decoration of a Dyck path. St001914The size of the orbit of an integer partition in Bulgarian solitaire. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000567The sum of the products of all pairs of parts. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. 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. 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. St001281The normalized isoperimetric number of a graph. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001592The maximal number of simple paths between any two different vertices of a graph. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000460The hook length of the last cell along the main diagonal of an integer partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001360The number of covering relations in Young's lattice below a partition. St001378The product of the cohook lengths of the integer partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. 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. St000454The largest eigenvalue of a graph if it is integral. St000806The semiperimeter of the associated bargraph. St001060The distinguishing index of a graph. St000464The Schultz index of a connected graph. 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

$eA$ . 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

$eA$ . St001545The second Elser number of a connected graph. St000456The monochromatic index of a connected graph. St000699The toughness times the least common multiple of 1,. St000762The sum of the positions of the weak records of an integer composition. St001199The dominant dimension of

$eAe$ for the corresponding Nakayama algebra

$A$ with minimal faithful projective-injective module

$eA$ . St001498The normalised height of a Nakayama algebra with magnitude 1.