Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000761
Mp00268: Binary words zeros to flag zerosBinary words
Mp00200: Binary words twistBinary words
Mp00097: Binary words delta morphismInteger compositions
St000761: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 => [1] => 0
1 => 1 => 0 => [1] => 0
00 => 10 => 00 => [2] => 0
01 => 00 => 10 => [1,1] => 0
10 => 01 => 11 => [2] => 0
11 => 11 => 01 => [1,1] => 0
000 => 010 => 110 => [2,1] => 0
001 => 110 => 010 => [1,1,1] => 0
010 => 100 => 000 => [3] => 0
011 => 000 => 100 => [1,2] => 1
100 => 101 => 001 => [2,1] => 0
101 => 001 => 101 => [1,1,1] => 0
110 => 011 => 111 => [3] => 0
111 => 111 => 011 => [1,2] => 1
0000 => 1010 => 0010 => [2,1,1] => 0
0001 => 0010 => 1010 => [1,1,1,1] => 0
0010 => 0110 => 1110 => [3,1] => 0
0011 => 1110 => 0110 => [1,2,1] => 1
0100 => 0100 => 1100 => [2,2] => 0
0101 => 1100 => 0100 => [1,1,2] => 1
0110 => 1000 => 0000 => [4] => 0
0111 => 0000 => 1000 => [1,3] => 1
1000 => 0101 => 1101 => [2,1,1] => 0
1001 => 1101 => 0101 => [1,1,1,1] => 0
1010 => 1001 => 0001 => [3,1] => 0
1011 => 0001 => 1001 => [1,2,1] => 1
1100 => 1011 => 0011 => [2,2] => 0
1101 => 0011 => 1011 => [1,1,2] => 1
1110 => 0111 => 1111 => [4] => 0
1111 => 1111 => 0111 => [1,3] => 1
Description
The number of ascents in an integer composition. A composition has an ascent, or rise, at position $i$ if $a_i < a_{i+1}$.
Matching statistic: St001037
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00122: Dyck paths Elizalde-Deutsch bijectionDyck paths
St001037: Dyck paths ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,0,1,0]
=> 0
1 => [1,1] => [1,0,1,0]
=> [1,1,0,0]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [1,1,1,0,0,0]
=> 0
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 0
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> 1
Description
The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path.
Matching statistic: St001423
Mp00104: Binary words reverseBinary words
Mp00234: Binary words valleys-to-peaksBinary words
Mp00278: Binary words rowmotionBinary words
St001423: Binary words ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 => 1 => 0
1 => 1 => 1 => 1 => 0
00 => 00 => 01 => 10 => 0
01 => 10 => 11 => 11 => 0
10 => 01 => 10 => 01 => 0
11 => 11 => 11 => 11 => 0
000 => 000 => 001 => 010 => 0
001 => 100 => 101 => 110 => 0
010 => 010 => 101 => 110 => 0
011 => 110 => 111 => 111 => 1
100 => 001 => 010 => 100 => 0
101 => 101 => 110 => 011 => 0
110 => 011 => 101 => 110 => 0
111 => 111 => 111 => 111 => 1
0000 => 0000 => 0001 => 0010 => 0
0001 => 1000 => 1001 => 0110 => 0
0010 => 0100 => 1001 => 0110 => 0
0011 => 1100 => 1101 => 1110 => 1
0100 => 0010 => 0101 => 1010 => 0
0101 => 1010 => 1101 => 1110 => 1
0110 => 0110 => 1011 => 1101 => 0
0111 => 1110 => 1111 => 1111 => 1
1000 => 0001 => 0010 => 0100 => 0
1001 => 1001 => 1010 => 1100 => 0
1010 => 0101 => 1010 => 1100 => 0
1011 => 1101 => 1110 => 0111 => 1
1100 => 0011 => 0101 => 1010 => 0
1101 => 1011 => 1101 => 1110 => 1
1110 => 0111 => 1011 => 1101 => 0
1111 => 1111 => 1111 => 1111 => 1
Description
The number of distinct cubes in a binary word. A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uuu$ is a factor of the word.
Matching statistic: St001960
Mp00178: Binary words to compositionInteger compositions
Mp00231: Integer compositions bounce pathDyck paths
Mp00024: Dyck paths to 321-avoiding permutationPermutations
St001960: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [1,2] => 0
1 => [1,1] => [1,0,1,0]
=> [2,1] => 0
00 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
01 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,1,4,2] => 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,3,4] => 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,4,1,3] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,1,4,3] => 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 0
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => 0
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => 0
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => 0
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => 0
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => 1
Description
The number of descents of a permutation minus one if its first entry is not one. This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St000764
Mp00268: Binary words zeros to flag zerosBinary words
Mp00200: Binary words twistBinary words
Mp00097: Binary words delta morphismInteger compositions
St000764: Integer compositions ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 => [1] => 1 = 0 + 1
1 => 1 => 0 => [1] => 1 = 0 + 1
00 => 10 => 00 => [2] => 1 = 0 + 1
01 => 00 => 10 => [1,1] => 1 = 0 + 1
10 => 01 => 11 => [2] => 1 = 0 + 1
11 => 11 => 01 => [1,1] => 1 = 0 + 1
000 => 010 => 110 => [2,1] => 1 = 0 + 1
001 => 110 => 010 => [1,1,1] => 1 = 0 + 1
010 => 100 => 000 => [3] => 1 = 0 + 1
011 => 000 => 100 => [1,2] => 2 = 1 + 1
100 => 101 => 001 => [2,1] => 1 = 0 + 1
101 => 001 => 101 => [1,1,1] => 1 = 0 + 1
110 => 011 => 111 => [3] => 1 = 0 + 1
111 => 111 => 011 => [1,2] => 2 = 1 + 1
0000 => 1010 => 0010 => [2,1,1] => 1 = 0 + 1
0001 => 0010 => 1010 => [1,1,1,1] => 1 = 0 + 1
0010 => 0110 => 1110 => [3,1] => 1 = 0 + 1
0011 => 1110 => 0110 => [1,2,1] => 2 = 1 + 1
0100 => 0100 => 1100 => [2,2] => 1 = 0 + 1
0101 => 1100 => 0100 => [1,1,2] => 2 = 1 + 1
0110 => 1000 => 0000 => [4] => 1 = 0 + 1
0111 => 0000 => 1000 => [1,3] => 2 = 1 + 1
1000 => 0101 => 1101 => [2,1,1] => 1 = 0 + 1
1001 => 1101 => 0101 => [1,1,1,1] => 1 = 0 + 1
1010 => 1001 => 0001 => [3,1] => 1 = 0 + 1
1011 => 0001 => 1001 => [1,2,1] => 2 = 1 + 1
1100 => 1011 => 0011 => [2,2] => 1 = 0 + 1
1101 => 0011 => 1011 => [1,1,2] => 2 = 1 + 1
1110 => 0111 => 1111 => [4] => 1 = 0 + 1
1111 => 1111 => 0111 => [1,3] => 2 = 1 + 1
Description
The number of strong records in an integer composition. A strong record is an element $a_i$ such that $a_i > a_j$ for all $j < i$. In particular, the first part of a composition is a strong record. Theorem 1.1 of [1] provides the generating function for compositions with parts in a given set according to the sum of the parts, the number of parts and the number of strong records.
Mp00105: Binary words complementBinary words
Mp00178: Binary words to compositionInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000260: Graphs ⟶ ℤResult quality: 50% values known / values provided: 50%distinct values known / distinct values provided: 50%
Values
0 => 1 => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
1 => 0 => [2] => ([],2)
=> ? = 0 + 1
00 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
01 => 10 => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
10 => 01 => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
11 => 00 => [3] => ([],3)
=> ? = 0 + 1
000 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
001 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
010 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
011 => 100 => [1,3] => ([(2,3)],4)
=> ? = 1 + 1
100 => 011 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
101 => 010 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
110 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
111 => 000 => [4] => ([],4)
=> ? = 1 + 1
0000 => 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)
=> 1 = 0 + 1
0001 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
0010 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
0011 => 1100 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
0100 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 1 + 1
0110 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
0111 => 1000 => [1,4] => ([(3,4)],5)
=> ? = 1 + 1
1000 => 0111 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
1001 => 0110 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
1010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
1011 => 0100 => [2,3] => ([(2,4),(3,4)],5)
=> ? = 1 + 1
1100 => 0011 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
1101 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 1 + 1
1110 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
1111 => 0000 => [5] => ([],5)
=> ? = 1 + 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Mp00104: Binary words reverseBinary words
Mp00200: Binary words twistBinary words
Mp00262: Binary words poset of factorsPosets
St001964: Posets ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 50%
Values
0 => 0 => 1 => ([(0,1)],2)
=> 0
1 => 1 => 0 => ([(0,1)],2)
=> 0
00 => 00 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
01 => 10 => 00 => ([(0,2),(2,1)],3)
=> 0
10 => 01 => 11 => ([(0,2),(2,1)],3)
=> 0
11 => 11 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
000 => 000 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
001 => 100 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 0
010 => 010 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
011 => 110 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1
100 => 001 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 0
101 => 101 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0
110 => 011 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 0
111 => 111 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1
0000 => 0000 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
0001 => 1000 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
0010 => 0100 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0
0011 => 1100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
0100 => 0010 => 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
0101 => 1010 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1
0110 => 0110 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
0111 => 1110 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1
1000 => 0001 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0
1001 => 1001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0
1010 => 0101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
1011 => 1101 => 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)
=> ? = 1
1100 => 0011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0
1101 => 1011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1
1110 => 0111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
1111 => 1111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1
Description
The interval resolution global dimension of a poset. This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
Mp00104: Binary words reverseBinary words
Mp00200: Binary words twistBinary words
Mp00262: Binary words poset of factorsPosets
St000181: Posets ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 50%
Values
0 => 0 => 1 => ([(0,1)],2)
=> 1 = 0 + 1
1 => 1 => 0 => ([(0,1)],2)
=> 1 = 0 + 1
00 => 00 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
01 => 10 => 00 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
10 => 01 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
11 => 11 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
000 => 000 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
001 => 100 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
010 => 010 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
011 => 110 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
100 => 001 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 0 + 1
101 => 101 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
110 => 011 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
111 => 111 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
0000 => 0000 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
0001 => 1000 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
0010 => 0100 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
0011 => 1100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
0100 => 0010 => 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 + 1
0101 => 1010 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
0110 => 0110 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
0111 => 1110 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
1000 => 0001 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
1001 => 1001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
1010 => 0101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
1011 => 1101 => 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)
=> ? = 1 + 1
1100 => 0011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
1101 => 1011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
1110 => 0111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
1111 => 1111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Mp00104: Binary words reverseBinary words
Mp00200: Binary words twistBinary words
Mp00262: Binary words poset of factorsPosets
St001890: Posets ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 50%
Values
0 => 0 => 1 => ([(0,1)],2)
=> 1 = 0 + 1
1 => 1 => 0 => ([(0,1)],2)
=> 1 = 0 + 1
00 => 00 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
01 => 10 => 00 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
10 => 01 => 11 => ([(0,2),(2,1)],3)
=> 1 = 0 + 1
11 => 11 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
000 => 000 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
001 => 100 => 000 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
010 => 010 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
011 => 110 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 1 + 1
100 => 001 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ? = 0 + 1
101 => 101 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 0 + 1
110 => 011 => 111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
111 => 111 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ? = 1 + 1
0000 => 0000 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
0001 => 1000 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
0010 => 0100 => 1100 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 0 + 1
0011 => 1100 => 0100 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
0100 => 0010 => 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 + 1
0101 => 1010 => 0010 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 1 + 1
0110 => 0110 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
0111 => 1110 => 0110 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 1 + 1
1000 => 0001 => 1001 => ([(0,2),(0,3),(1,5),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,6),(8,5)],9)
=> ? = 0 + 1
1001 => 1001 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 0 + 1
1010 => 0101 => 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
1011 => 1101 => 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)
=> ? = 1 + 1
1100 => 0011 => 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 0 + 1
1101 => 1011 => 0011 => ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 1 + 1
1110 => 0111 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
1111 => 1111 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
Description
The maximum magnitude of the Möbius function of a poset. The '''Möbius function''' of a poset is the multiplicative inverse of the zeta function in the incidence algebra. The Möbius value $\mu(x, y)$ is equal to the signed sum of chains from $x$ to $y$, where odd-length chains are counted with a minus sign, so this statistic is bounded above by the total number of chains in the poset.
Matching statistic: St000264
Mp00268: Binary words zeros to flag zerosBinary words
Mp00097: Binary words delta morphismInteger compositions
Mp00184: Integer compositions to threshold graphGraphs
St000264: Graphs ⟶ ℤResult quality: 33% values known / values provided: 33%distinct values known / distinct values provided: 50%
Values
0 => 0 => [1] => ([],1)
=> ? = 0 + 3
1 => 1 => [1] => ([],1)
=> ? = 0 + 3
00 => 10 => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
01 => 00 => [2] => ([],2)
=> ? = 0 + 3
10 => 01 => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
11 => 11 => [2] => ([],2)
=> ? = 0 + 3
000 => 010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
001 => 110 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
010 => 100 => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
011 => 000 => [3] => ([],3)
=> ? = 1 + 3
100 => 101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
101 => 001 => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
110 => 011 => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
111 => 111 => [3] => ([],3)
=> ? = 1 + 3
0000 => 1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
0001 => 0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
0010 => 0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
0011 => 1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
0100 => 0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
0101 => 1100 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
0110 => 1000 => [1,3] => ([(2,3)],4)
=> ? = 0 + 3
0111 => 0000 => [4] => ([],4)
=> ? = 1 + 3
1000 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
1001 => 1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
1010 => 1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
1011 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ? = 1 + 3
1100 => 1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
1101 => 0011 => [2,2] => ([(1,3),(2,3)],4)
=> ? = 1 + 3
1110 => 0111 => [1,3] => ([(2,3)],4)
=> ? = 0 + 3
1111 => 1111 => [4] => ([],4)
=> ? = 1 + 3
Description
The girth of a graph, which is not a tree. This is the length of the shortest cycle in the graph.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001722The number of minimal chains with small intervals between a binary word and the top element. St000455The second largest eigenvalue of a graph if it is integral. St000782The indicator function of whether a given perfect matching is an L & P matching.