Your data matches 145 different statistics following compositions of up to 3 maps.
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Matching statistic: St001420
(load all 36 compositions to match this statistic)
Mapping
St001420: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 = 2 - 2
1 => 0 = 2 - 2
00 => 0 = 2 - 2
01 => 1 = 3 - 2
10 => 1 = 3 - 2
11 => 0 = 2 - 2
000 => 0 = 2 - 2
001 => 1 = 3 - 2
010 => 1 = 3 - 2
011 => 1 = 3 - 2
100 => 1 = 3 - 2
101 => 1 = 3 - 2
110 => 1 = 3 - 2
111 => 0 = 2 - 2
0000 => 0 = 2 - 2
0001 => 1 = 3 - 2
0010 => 1 = 3 - 2
0011 => 2 = 4 - 2
0100 => 1 = 3 - 2
0101 => 2 = 4 - 2
0110 => 1 = 3 - 2
0111 => 1 = 3 - 2
1000 => 1 = 3 - 2
1001 => 1 = 3 - 2
1010 => 2 = 4 - 2
1011 => 1 = 3 - 2
1100 => 2 = 4 - 2
1101 => 1 = 3 - 2
1110 => 1 = 3 - 2
1111 => 0 = 2 - 2
Description
Half the length of a longest factor which is its own reverse-complement of a binary word.
Matching statistic: St000291
(load all 3 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
St000291: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 = 2 - 2
1 => 1 => 0 = 2 - 2
00 => 00 => 0 = 2 - 2
01 => 10 => 1 = 3 - 2
10 => 10 => 1 = 3 - 2
11 => 11 => 0 = 2 - 2
000 => 000 => 0 = 2 - 2
001 => 100 => 1 = 3 - 2
010 => 100 => 1 = 3 - 2
011 => 110 => 1 = 3 - 2
100 => 100 => 1 = 3 - 2
101 => 110 => 1 = 3 - 2
110 => 110 => 1 = 3 - 2
111 => 111 => 0 = 2 - 2
0000 => 0000 => 0 = 2 - 2
0001 => 1000 => 1 = 3 - 2
0010 => 1000 => 1 = 3 - 2
0011 => 1010 => 2 = 4 - 2
0100 => 1000 => 1 = 3 - 2
0101 => 1100 => 1 = 3 - 2
0110 => 1010 => 2 = 4 - 2
0111 => 1110 => 1 = 3 - 2
1000 => 1000 => 1 = 3 - 2
1001 => 1010 => 2 = 4 - 2
1010 => 1100 => 1 = 3 - 2
1011 => 1110 => 1 = 3 - 2
1100 => 1010 => 2 = 4 - 2
1101 => 1110 => 1 = 3 - 2
1110 => 1110 => 1 = 3 - 2
1111 => 1111 => 0 = 2 - 2
Description
The number of descents of a binary word.
Matching statistic: St001738
(load all 6 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001738: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => ([],2)
 => 2
1 => [1,1] => ([(0,1)],2)
 => 2
00 => [3] => ([],3)
 => 2
01 => [2,1] => ([(0,2),(1,2)],3)
 => 3
10 => [1,2] => ([(1,2)],3)
 => 3
11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
000 => [4] => ([],4)
 => 2
001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
010 => [2,2] => ([(1,3),(2,3)],4)
 => 3
011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
100 => [1,3] => ([(2,3)],4)
 => 3
101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 3
110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3
111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
0000 => [5] => ([],5)
 => 2
0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
0100 => [2,3] => ([(2,4),(3,4)],5)
 => 3
0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
1000 => [1,4] => ([(3,4)],5)
 => 3
1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 4
1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 3
1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
1111 => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
Description
The minimal order of a graph which is not an induced subgraph of the given graph. For example, the graph with two isolated vertices is not an induced subgraph of the complete graph on three vertices. By contrast, the minimal number of vertices of a graph which is not a subgraph of a graph is one plus the clique number [[St000097]].
Matching statistic: St001165
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001165: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => 1 = 2 - 1
1 => [1] => [1,0]
 => 1 = 2 - 1
00 => [2] => [1,1,0,0]
 => 1 = 2 - 1
01 => [1,1] => [1,0,1,0]
 => 2 = 3 - 1
10 => [1,1] => [1,0,1,0]
 => 2 = 3 - 1
11 => [2] => [1,1,0,0]
 => 1 = 2 - 1
000 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
001 => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => 2 = 3 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => 2 = 3 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
111 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
Description
Number of simple modules with even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St001471
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001471: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => 1 = 2 - 1
1 => [1] => [1,0]
 => 1 = 2 - 1
00 => [2] => [1,1,0,0]
 => 1 = 2 - 1
01 => [1,1] => [1,0,1,0]
 => 2 = 3 - 1
10 => [1,1] => [1,0,1,0]
 => 2 = 3 - 1
11 => [2] => [1,1,0,0]
 => 1 = 2 - 1
000 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
001 => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
010 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
011 => [1,2] => [1,0,1,1,0,0]
 => 2 = 3 - 1
100 => [1,2] => [1,0,1,1,0,0]
 => 2 = 3 - 1
101 => [1,1,1] => [1,0,1,0,1,0]
 => 2 = 3 - 1
110 => [2,1] => [1,1,0,0,1,0]
 => 2 = 3 - 1
111 => [3] => [1,1,1,0,0,0]
 => 1 = 2 - 1
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 2 = 3 - 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 3 - 1
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
Description
The magnitude of a Dyck path. The magnitude of a finite dimensional algebra with invertible Cartan matrix C is defined as the sum of all entries of the inverse of C. We define the magnitude of a Dyck path as the magnitude of the corresponding LNakayama algebra.
Matching statistic: St001486
(load all 2 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St001486: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => 1 = 2 - 1
1 => [1] => [1] => 1 = 2 - 1
00 => [2] => [1] => 1 = 2 - 1
01 => [1,1] => [2] => 2 = 3 - 1
10 => [1,1] => [2] => 2 = 3 - 1
11 => [2] => [1] => 1 = 2 - 1
000 => [3] => [1] => 1 = 2 - 1
001 => [2,1] => [1,1] => 2 = 3 - 1
010 => [1,1,1] => [3] => 2 = 3 - 1
011 => [1,2] => [1,1] => 2 = 3 - 1
100 => [1,2] => [1,1] => 2 = 3 - 1
101 => [1,1,1] => [3] => 2 = 3 - 1
110 => [2,1] => [1,1] => 2 = 3 - 1
111 => [3] => [1] => 1 = 2 - 1
0000 => [4] => [1] => 1 = 2 - 1
0001 => [3,1] => [1,1] => 2 = 3 - 1
0010 => [2,1,1] => [1,2] => 3 = 4 - 1
0011 => [2,2] => [2] => 2 = 3 - 1
0100 => [1,1,2] => [2,1] => 3 = 4 - 1
0101 => [1,1,1,1] => [4] => 2 = 3 - 1
0110 => [1,2,1] => [1,1,1] => 2 = 3 - 1
0111 => [1,3] => [1,1] => 2 = 3 - 1
1000 => [1,3] => [1,1] => 2 = 3 - 1
1001 => [1,2,1] => [1,1,1] => 2 = 3 - 1
1010 => [1,1,1,1] => [4] => 2 = 3 - 1
1011 => [1,1,2] => [2,1] => 3 = 4 - 1
1100 => [2,2] => [2] => 2 = 3 - 1
1101 => [2,1,1] => [1,2] => 3 = 4 - 1
1110 => [3,1] => [1,1] => 2 = 3 - 1
1111 => [4] => [1] => 1 = 2 - 1
Description
The number of corners of the ribbon associated with an integer composition. We associate a ribbon shape to a composition $c=(c_1,\dots,c_n)$ with $c_i$ cells in the $i$-th row from bottom to top, such that the cells in two rows overlap in precisely one cell. This statistic records the total number of corners of the ribbon shape.
Matching statistic: St001642
(load all 5 compositions to match this statistic)
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St001642: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => ([],1)
 => 1 = 2 - 1
1 => [1] => ([],1)
 => 1 = 2 - 1
00 => [2] => ([],2)
 => 2 = 3 - 1
01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
10 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
11 => [2] => ([],2)
 => 2 = 3 - 1
000 => [3] => ([],3)
 => 2 = 3 - 1
001 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
011 => [1,2] => ([(1,2)],3)
 => 2 = 3 - 1
100 => [1,2] => ([(1,2)],3)
 => 2 = 3 - 1
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
110 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
111 => [3] => ([],3)
 => 2 = 3 - 1
0000 => [4] => ([],4)
 => 2 = 3 - 1
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
0011 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
0111 => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
1000 => [1,3] => ([(2,3)],4)
 => 3 = 4 - 1
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 3 = 4 - 1
1100 => [2,2] => ([(1,3),(2,3)],4)
 => 2 = 3 - 1
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
1111 => [4] => ([],4)
 => 2 = 3 - 1
Description
The Prague dimension of a graph. This is the least number of complete graphs such that the graph is an induced subgraph of their (categorical) product. Put differently, this is the least number $n$ such that the graph can be embedded into $\mathbb N^n$, where two points are connected by an edge if and only if they differ in all coordinates.
Matching statistic: St000292
(load all 3 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
St000292: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 2 - 2
1 => 1 => 1 => 0 = 2 - 2
00 => 00 => 00 => 0 = 2 - 2
01 => 10 => 01 => 1 = 3 - 2
10 => 10 => 01 => 1 = 3 - 2
11 => 11 => 11 => 0 = 2 - 2
000 => 000 => 000 => 0 = 2 - 2
001 => 100 => 001 => 1 = 3 - 2
010 => 100 => 001 => 1 = 3 - 2
011 => 110 => 101 => 1 = 3 - 2
100 => 100 => 001 => 1 = 3 - 2
101 => 110 => 101 => 1 = 3 - 2
110 => 110 => 101 => 1 = 3 - 2
111 => 111 => 111 => 0 = 2 - 2
0000 => 0000 => 0000 => 0 = 2 - 2
0001 => 1000 => 0001 => 1 = 3 - 2
0010 => 1000 => 0001 => 1 = 3 - 2
0011 => 1010 => 0101 => 2 = 4 - 2
0100 => 1000 => 0001 => 1 = 3 - 2
0101 => 1100 => 1001 => 1 = 3 - 2
0110 => 1010 => 0101 => 2 = 4 - 2
0111 => 1110 => 1101 => 1 = 3 - 2
1000 => 1000 => 0001 => 1 = 3 - 2
1001 => 1010 => 0101 => 2 = 4 - 2
1010 => 1100 => 1001 => 1 = 3 - 2
1011 => 1110 => 1101 => 1 = 3 - 2
1100 => 1010 => 0101 => 2 = 4 - 2
1101 => 1110 => 1101 => 1 = 3 - 2
1110 => 1110 => 1101 => 1 = 3 - 2
1111 => 1111 => 1111 => 0 = 2 - 2
Description
The number of ascents of a binary word.
Matching statistic: St000329
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
 => 0 = 2 - 2
1 => [1] => [1,0]
 => 0 = 2 - 2
00 => [2] => [1,1,0,0]
 => 1 = 3 - 2
01 => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
10 => [1,1] => [1,0,1,0]
 => 0 = 2 - 2
11 => [2] => [1,1,0,0]
 => 1 = 3 - 2
000 => [3] => [1,1,1,0,0,0]
 => 1 = 3 - 2
001 => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
010 => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
011 => [1,2] => [1,0,1,1,0,0]
 => 1 = 3 - 2
100 => [1,2] => [1,0,1,1,0,0]
 => 1 = 3 - 2
101 => [1,1,1] => [1,0,1,0,1,0]
 => 0 = 2 - 2
110 => [2,1] => [1,1,0,0,1,0]
 => 1 = 3 - 2
111 => [3] => [1,1,1,0,0,0]
 => 1 = 3 - 2
0000 => [4] => [1,1,1,1,0,0,0,0]
 => 2 = 4 - 2
0001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
0011 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 4 - 2
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
1000 => [1,3] => [1,0,1,1,1,0,0,0]
 => 1 = 3 - 2
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1 = 3 - 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 0 = 2 - 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1 = 3 - 2
1100 => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 4 - 2
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 1 = 3 - 2
1110 => [3,1] => [1,1,1,0,0,0,1,0]
 => 1 = 3 - 2
1111 => [4] => [1,1,1,1,0,0,0,0]
 => 2 = 4 - 2
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000628
(load all 15 compositions to match this statistic)
Mapping
Mp00261: Binary words —Burrows-Wheeler⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 0 = 2 - 2
1 => 1 => 1 => 0 = 2 - 2
00 => 00 => 00 => 0 = 2 - 2
01 => 10 => 10 => 1 = 3 - 2
10 => 10 => 10 => 1 = 3 - 2
11 => 11 => 11 => 0 = 2 - 2
000 => 000 => 000 => 0 = 2 - 2
001 => 100 => 010 => 1 = 3 - 2
010 => 100 => 010 => 1 = 3 - 2
011 => 110 => 110 => 1 = 3 - 2
100 => 100 => 010 => 1 = 3 - 2
101 => 110 => 110 => 1 = 3 - 2
110 => 110 => 110 => 1 = 3 - 2
111 => 111 => 111 => 0 = 2 - 2
0000 => 0000 => 0000 => 0 = 2 - 2
0001 => 1000 => 0010 => 1 = 3 - 2
0010 => 1000 => 0010 => 1 = 3 - 2
0011 => 1010 => 1100 => 2 = 4 - 2
0100 => 1000 => 0010 => 1 = 3 - 2
0101 => 1100 => 0110 => 1 = 3 - 2
0110 => 1010 => 1100 => 2 = 4 - 2
0111 => 1110 => 1110 => 1 = 3 - 2
1000 => 1000 => 0010 => 1 = 3 - 2
1001 => 1010 => 1100 => 2 = 4 - 2
1010 => 1100 => 0110 => 1 = 3 - 2
1011 => 1110 => 1110 => 1 = 3 - 2
1100 => 1010 => 1100 => 2 = 4 - 2
1101 => 1110 => 1110 => 1 = 3 - 2
1110 => 1110 => 1110 => 1 = 3 - 2
1111 => 1111 => 1111 => 0 = 2 - 2
Description
The balance of a binary word. The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1]. A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
The following 135 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001873For a Nakayama algebra corresponding to a Dyck path, we define the matrix C with entries the Hom-spaces between $e_i J$ and $e_j J$ (the radical of the indecomposable projective modules). St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000015The number of peaks of a Dyck path. St000092The number of outer peaks of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000527The width of the poset. St000676The number of odd rises of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000829The Ulam distance of a permutation to the identity permutation. St000983The length of the longest alternating subword. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows: St001488The number of corners of a skew partition. St001512The minimum rank of a graph. St001530The depth of a Dyck path. St001792The arboricity of a graph. St001801Half the number of preimage-image pairs of different parity in a permutation. St001874Lusztig's a-function for the symmetric group. St001955The number of natural descents for set-valued two row standard Young tableaux. St000021The number of descents of a permutation. St000035The number of left outer peaks of a permutation. St000053The number of valleys of the Dyck path. St000120The number of left tunnels of a Dyck path. St000155The number of exceedances (also excedences) of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000353The number of inner valleys of a permutation. St000387The matching number of a graph. St000390The number of runs of ones in a binary word. St000632The jump number of the poset. St000662The staircase size of the code of a permutation. St000691The number of changes of a binary word. St000703The number of deficiencies of a permutation. St000769The major index of a composition regarded as a word. St000884The number of isolated descents of a permutation. St000897The number of different multiplicities of parts of an integer partition. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001205The number of non-simple indecomposable projective-injective modules of the algebra $eAe$ in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001394The genus of a permutation. St001413Half the length of the longest even length palindromic prefix of a binary word. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000216The absolute length of a permutation. St000354The number of recoils of a permutation. St000539The number of odd inversions of a permutation. St000659The number of rises of length at least 2 of a Dyck path. St000741The Colin de Verdière graph invariant. St000809The reduced reflection length of the permutation. St001569The maximal modular displacement of a permutation. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000259The diameter of a connected graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000260The radius of a connected graph. St001335The cardinality of a minimal cycle-isolating set of a graph. St000098The chromatic number of a graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000097The order of the largest clique of the graph. St000298The order dimension or Dushnik-Miller dimension of a poset. St000640The rank of the largest boolean interval in a poset. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element 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. St001746The coalition number of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000307The number of rowmotion orbits of a poset. St000310The minimal degree of a vertex of a graph. St000544The cop number of a graph. St000785The number of distinct colouring schemes of a graph. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001277The degeneracy of a graph. St001330The hat guessing number of a graph. St001358The largest degree of a regular subgraph of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001883The mutual visibility number of a graph. St001951The number of factors in the disjoint direct product decomposition of the automorphism group of a graph. St000272The treewidth of a graph. St000535The rank-width of a graph. St000536The pathwidth of a graph. St000537The cutwidth of a graph. St001270The bandwidth of a graph. St001323The independence gap of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001644The dimension of a graph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph. St001962The proper pathwidth of a graph. St000258The burning number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000778The metric dimension of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001057The Grundy value of the game of creating an independent set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001638The book thickness of a graph. St000806The semiperimeter of the associated bargraph. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St001624The breadth of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001060The distinguishing index of a graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.