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Your data matches 31 different statistics following compositions of up to 3 maps.
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Matching statistic: St001323
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Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> 0
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)
=> 0
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 0
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)
=> 0
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)
=> 0
0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1
00000 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1
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)
=> 1
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)
=> 1
Description
The independence gap of a graph.
This is the difference between the independence number [[St000093]] and the minimal size of a maximally independent set of a graph.
In particular, this statistic is $0$ for well covered graphs
Matching statistic: St001524
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Mp00224: Binary words —runsort⟶ Binary words
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001524: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0 => 1 => 0
1 => 1 => 1 => 0 => 0
00 => 00 => 01 => 11 => 1
01 => 01 => 00 => 10 => 0
10 => 01 => 00 => 10 => 0
11 => 11 => 10 => 00 => 1
000 => 000 => 010 => 110 => 0
001 => 001 => 011 => 111 => 1
010 => 001 => 011 => 111 => 1
011 => 011 => 001 => 101 => 1
100 => 001 => 011 => 111 => 1
101 => 011 => 001 => 101 => 1
110 => 011 => 001 => 101 => 1
111 => 111 => 101 => 001 => 0
0000 => 0000 => 0101 => 1101 => 1
1111 => 1111 => 1010 => 0010 => 1
00000 => 00000 => 01010 => 11010 => 1
11111 => 11111 => 10101 => 00101 => 1
000000 => 000000 => 010101 => 110101 => 1
111111 => 111111 => 101010 => 001010 => 1
Description
The degree of symmetry of a binary word.
For a binary word $w$ of length $n$, this is the number of positions $i\leq n/2$ such that $w_i = w_{n+1-i}$.
Matching statistic: St000741
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(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1 = 0 + 1
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
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)
=> ? = 0 + 1
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)
=> ? = 0 + 1
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2 = 1 + 1
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
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)
=> 2 = 1 + 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)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 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)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 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)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 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)
=> ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)
=> 2 = 1 + 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)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 2 = 1 + 1
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
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)
=> 2 = 1 + 1
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)
=> 2 = 1 + 1
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)
=> ? = 1 + 1
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)
=> ? = 1 + 1
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)
=> ? = 1 + 1
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)
=> ? = 1 + 1
Description
The Colin de Verdière graph invariant.
Matching statistic: St001195
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001195: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? = 0
1 => [1] => [1,0]
=> [1,0]
=> ? = 0
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> ? = 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> ? = 1
000 => [3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
111 => [3] => [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 1
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001199
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(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? = 0
1 => [1] => [1,0]
=> [1,0]
=> ? = 0
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 1
000 => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 1
111 => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 1
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001198
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(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
1 => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0 + 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0 + 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
000 => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
111 => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
Description
The 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 $eA$.
Matching statistic: St001206
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(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001206: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 60%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
1 => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0 + 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0 + 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
000 => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
111 => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 2 = 1 + 1
Description
The 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 $eA$.
Matching statistic: St000264
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
0 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 2
1 => ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> ? = 0 + 2
00 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ? = 0 + 2
11 => ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ? = 1 + 2
000 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 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)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> 3 = 1 + 2
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 = 1 + 2
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)
=> 3 = 1 + 2
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)
=> 3 = 1 + 2
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 = 1 + 2
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)
=> 3 = 1 + 2
111 => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 0 + 2
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 = 1 + 2
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 = 1 + 2
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)
=> 3 = 1 + 2
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)
=> 3 = 1 + 2
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)
=> 3 = 1 + 2
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)
=> 3 = 1 + 2
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St001200
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00228: Dyck paths —reflect parallelogram polyomino⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 50%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
1 => [1] => [1,0]
=> [1,0]
=> ? = 0 + 1
00 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
01 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0 + 1
10 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> ? = 0 + 1
11 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 2 = 1 + 1
000 => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
001 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
011 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
100 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 2 = 1 + 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> 2 = 1 + 1
110 => [2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
111 => [3] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ? = 0 + 1
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
11111 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 1
000000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
111111 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 1 + 1
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000260
Mp00200: Binary words —twist⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1] => ([],1)
=> 0
1 => 0 => [1] => ([],1)
=> 0
00 => 10 => [1,1] => ([(0,1)],2)
=> 1
01 => 11 => [2] => ([],2)
=> ? = 0
10 => 00 => [2] => ([],2)
=> ? = 0
11 => 01 => [1,1] => ([(0,1)],2)
=> 1
000 => 100 => [1,2] => ([(1,2)],3)
=> ? = 0
001 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
010 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> 1
011 => 111 => [3] => ([],3)
=> ? = 1
100 => 000 => [3] => ([],3)
=> ? = 1
101 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> 1
110 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
111 => 011 => [1,2] => ([(1,2)],3)
=> ? = 0
0000 => 1000 => [1,3] => ([(2,3)],4)
=> ? = 1
1111 => 0111 => [1,3] => ([(2,3)],4)
=> ? = 1
00000 => 10000 => [1,4] => ([(3,4)],5)
=> ? = 1
11111 => 01111 => [1,4] => ([(3,4)],5)
=> ? = 1
000000 => 100000 => [1,5] => ([(4,5)],6)
=> ? = 1
111111 => 011111 => [1,5] => ([(4,5)],6)
=> ? = 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 21 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000285The size of the preimage of the map 'to inverse des composition' from Parking functions to Integer compositions. St000455The second largest eigenvalue of a graph if it is integral. St000939The number of characters of the symmetric group whose value on the partition is positive. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000438The position of the last up step in a Dyck path. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000456The monochromatic index of a connected graph. St000762The sum of the positions of the weak records of an integer composition. St001118The acyclic chromatic index of a graph. St001498The normalised height of a Nakayama algebra with magnitude 1. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000454The largest eigenvalue of a graph if it is integral. St000464The Schultz index of a connected graph. 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. St001545The second Elser number of a connected graph. St000806The semiperimeter of the associated bargraph. St001060The distinguishing index of a graph.
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