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Your data matches 55 different statistics following compositions of up to 3 maps.
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Matching statistic: St001524
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
St001524: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 1
01 => 0
10 => 0
11 => 1
000 => 1
001 => 0
010 => 1
011 => 0
100 => 0
101 => 1
110 => 0
111 => 1
0000 => 2
0001 => 1
0010 => 1
0011 => 0
0100 => 1
0101 => 0
0110 => 2
0111 => 1
1000 => 1
1001 => 2
1010 => 0
1011 => 1
1100 => 0
1101 => 1
1110 => 1
1111 => 2
00000 => 2
00001 => 1
00010 => 1
00011 => 0
00100 => 2
00101 => 1
00110 => 1
00111 => 0
01000 => 1
01001 => 0
01010 => 2
01011 => 1
01100 => 1
01101 => 0
01110 => 2
01111 => 1
10000 => 1
10001 => 2
10010 => 0
10011 => 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≤n/2 such that wi=wn+1−i.
Matching statistic: St001092
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001092: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> []
=> 0
1 => [1,1] => [1,0,1,0]
=> [1]
=> 0
00 => [3] => [1,1,1,0,0,0]
=> []
=> 0
01 => [2,1] => [1,1,0,0,1,0]
=> [2]
=> 1
10 => [1,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
11 => [1,1,1] => [1,0,1,0,1,0]
=> [2,1]
=> 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> []
=> 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 0
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> 0
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 0
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 2
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> 0
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,4]
=> 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3,3]
=> 0
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [5,3,3]
=> 0
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [4,4,3]
=> 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2]
=> 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> 2
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2]
=> 2
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2]
=> 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2]
=> 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2]
=> 2
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 2
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> 0
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> 0
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> 1
Description
The number of distinct even parts of a partition.
See Section 3.3.1 of [1].
Matching statistic: St001115
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
St001115: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> [2,1] => 0
1 => [1,1] => [1,0,1,0]
=> [1,2] => 0
00 => [3] => [1,1,1,0,0,0]
=> [3,2,1] => 1
01 => [2,1] => [1,1,0,0,1,0]
=> [2,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
=> [1,3,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> [1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 0
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 0
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => 2
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => 0
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => 0
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => 0
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => 2
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => 2
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,3,2,1,6,5] => 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,3,2,1,5,6] => 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,2,1,6,5,4] => 2
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [3,2,1,5,4,6] => 2
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [3,2,1,4,6,5] => 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [3,2,1,4,5,6] => 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,5,4,3,6] => 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5] => 0
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,5,6] => 0
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,3,6,5,4] => 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,3,5,4,6] => 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,3,4,6,5] => 0
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [2,1,3,4,5,6] => 0
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => 2
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,5,4,3,2,6] => 2
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,4,3,2,6,5] => 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,4,3,2,5,6] => 1
Description
The number of even descents of a permutation.
Matching statistic: St001153
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001153: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001153: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1,0]
=> {{1}}
=> 0
1 => [1] => [1,0]
=> {{1}}
=> 0
00 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
01 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
10 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1
11 => [2] => [1,1,0,0]
=> {{1,2}}
=> 0
000 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
001 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
010 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
011 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
100 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1
101 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 1
110 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 0
111 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 0
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 1
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 0
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 0
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 1
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 2
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 2
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 2
Description
The number of blocks with even minimum in a set partition.
Matching statistic: St001413
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00094: Integer compositions —to binary word⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [1] => 1 => 1 => 0
1 => [1] => 1 => 1 => 0
00 => [2] => 10 => 10 => 0
01 => [1,1] => 11 => 11 => 1
10 => [1,1] => 11 => 11 => 1
11 => [2] => 10 => 10 => 0
000 => [3] => 100 => 010 => 0
001 => [2,1] => 101 => 101 => 0
010 => [1,1,1] => 111 => 111 => 1
011 => [1,2] => 110 => 110 => 1
100 => [1,2] => 110 => 110 => 1
101 => [1,1,1] => 111 => 111 => 1
110 => [2,1] => 101 => 101 => 0
111 => [3] => 100 => 010 => 0
0000 => [4] => 1000 => 0010 => 1
0001 => [3,1] => 1001 => 0101 => 0
0010 => [2,1,1] => 1011 => 1011 => 0
0011 => [2,2] => 1010 => 1100 => 1
0100 => [1,1,2] => 1110 => 1110 => 1
0101 => [1,1,1,1] => 1111 => 1111 => 2
0110 => [1,2,1] => 1101 => 1101 => 1
0111 => [1,3] => 1100 => 0110 => 2
1000 => [1,3] => 1100 => 0110 => 2
1001 => [1,2,1] => 1101 => 1101 => 1
1010 => [1,1,1,1] => 1111 => 1111 => 2
1011 => [1,1,2] => 1110 => 1110 => 1
1100 => [2,2] => 1010 => 1100 => 1
1101 => [2,1,1] => 1011 => 1011 => 0
1110 => [3,1] => 1001 => 0101 => 0
1111 => [4] => 1000 => 0010 => 1
00000 => [5] => 10000 => 00010 => 1
00001 => [4,1] => 10001 => 00101 => 1
00010 => [3,1,1] => 10011 => 01011 => 0
00011 => [3,2] => 10010 => 10100 => 0
00100 => [2,1,2] => 10110 => 11010 => 1
00101 => [2,1,1,1] => 10111 => 10111 => 0
00110 => [2,2,1] => 10101 => 11001 => 1
00111 => [2,3] => 10100 => 01100 => 2
01000 => [1,1,3] => 11100 => 01110 => 0
01001 => [1,1,2,1] => 11101 => 11101 => 1
01010 => [1,1,1,1,1] => 11111 => 11111 => 2
01011 => [1,1,1,2] => 11110 => 11110 => 2
01100 => [1,2,2] => 11010 => 11100 => 1
01101 => [1,2,1,1] => 11011 => 11011 => 1
01110 => [1,3,1] => 11001 => 01101 => 2
01111 => [1,4] => 11000 => 00110 => 1
10000 => [1,4] => 11000 => 00110 => 1
10001 => [1,3,1] => 11001 => 01101 => 2
10010 => [1,2,1,1] => 11011 => 11011 => 1
10011 => [1,2,2] => 11010 => 11100 => 1
Description
Half the length of the longest even length palindromic prefix of a binary word.
Matching statistic: St001151
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St001151: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
=> {{1,2}}
=> 1 = 0 + 1
1 => [1,1] => [1,0,1,0]
=> {{1},{2}}
=> 1 = 0 + 1
00 => [3] => [1,1,1,0,0,0]
=> {{1,2,3}}
=> 1 = 0 + 1
01 => [2,1] => [1,1,0,0,1,0]
=> {{1,2},{3}}
=> 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
=> {{1},{2,3}}
=> 1 = 0 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
=> {{1},{2},{3}}
=> 2 = 1 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
=> {{1,2,3,4}}
=> 1 = 0 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
=> {{1,2,3},{4}}
=> 1 = 0 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
=> {{1},{2,3,4}}
=> 1 = 0 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> {{1},{2,3},{4}}
=> 1 = 0 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> {{1},{2},{3,4}}
=> 2 = 1 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4}}
=> 2 = 1 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> {{1,2,3,4,5}}
=> 1 = 0 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> {{1,2,3},{4,5}}
=> 1 = 0 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> {{1,2,3},{4},{5}}
=> 2 = 1 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> {{1,2},{3,4,5}}
=> 2 = 1 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> {{1,2},{3,4},{5}}
=> 3 = 2 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> {{1,2},{3},{4,5}}
=> 2 = 1 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5}}
=> 3 = 2 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> {{1},{2,3,4,5}}
=> 1 = 0 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> {{1},{2,3,4},{5}}
=> 2 = 1 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> {{1},{2,3},{4,5}}
=> 1 = 0 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> {{1},{2,3},{4},{5}}
=> 2 = 1 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> {{1},{2},{3,4,5}}
=> 2 = 1 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> {{1},{2},{3,4},{5}}
=> 3 = 2 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> {{1},{2},{3},{4,5}}
=> 2 = 1 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> {{1},{2},{3},{4},{5}}
=> 3 = 2 + 1
00000 => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> {{1,2,3,4,5,6}}
=> 1 = 0 + 1
00001 => [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> {{1,2,3,4,5},{6}}
=> 1 = 0 + 1
00010 => [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> {{1,2,3,4},{5,6}}
=> 2 = 1 + 1
00011 => [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> {{1,2,3,4},{5},{6}}
=> 2 = 1 + 1
00100 => [3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> {{1,2,3},{4,5,6}}
=> 1 = 0 + 1
00101 => [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> {{1,2,3},{4,5},{6}}
=> 1 = 0 + 1
00110 => [3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> {{1,2,3},{4},{5,6}}
=> 2 = 1 + 1
00111 => [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> {{1,2,3},{4},{5},{6}}
=> 2 = 1 + 1
01000 => [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> {{1,2},{3,4,5,6}}
=> 2 = 1 + 1
01001 => [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> {{1,2},{3,4,5},{6}}
=> 2 = 1 + 1
01010 => [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> {{1,2},{3,4},{5,6}}
=> 3 = 2 + 1
01011 => [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> {{1,2},{3,4},{5},{6}}
=> 3 = 2 + 1
01100 => [2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> {{1,2},{3},{4,5,6}}
=> 2 = 1 + 1
01101 => [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> {{1,2},{3},{4,5},{6}}
=> 2 = 1 + 1
01110 => [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> {{1,2},{3},{4},{5,6}}
=> 3 = 2 + 1
01111 => [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> {{1,2},{3},{4},{5},{6}}
=> 3 = 2 + 1
10000 => [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> {{1},{2,3,4,5,6}}
=> 1 = 0 + 1
10001 => [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> {{1},{2,3,4,5},{6}}
=> 1 = 0 + 1
10010 => [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> {{1},{2,3,4},{5,6}}
=> 2 = 1 + 1
10011 => [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> {{1},{2,3,4},{5},{6}}
=> 2 = 1 + 1
Description
The number of blocks with odd minimum.
See [[St000746]] for the analogous statistic on perfect matchings.
Matching statistic: St001114
(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
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001114: Permutations ⟶ ℤResult quality: 97% ●values known / values provided: 97%●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]
=> [1,2] => 0
01 => [1,1] => [1,0,1,0]
=> [2,1] => 1
10 => [1,1] => [1,0,1,0]
=> [2,1] => 1
11 => [2] => [1,1,0,0]
=> [1,2] => 0
000 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
001 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
010 => [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 1
011 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
100 => [1,2] => [1,0,1,1,0,0]
=> [2,3,1] => 0
101 => [1,1,1] => [1,0,1,0,1,0]
=> [3,2,1] => 1
110 => [2,1] => [1,1,0,0,1,0]
=> [3,1,2] => 1
111 => [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
0000 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
0001 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
0010 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
0011 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
0100 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
0101 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
0110 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
0111 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
1000 => [1,3] => [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 1
1001 => [1,2,1] => [1,0,1,1,0,0,1,0]
=> [4,2,3,1] => 2
1010 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
1011 => [1,1,2] => [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
1100 => [2,2] => [1,1,0,0,1,1,0,0]
=> [3,4,1,2] => 0
1101 => [2,1,1] => [1,1,0,0,1,0,1,0]
=> [4,3,1,2] => 1
1110 => [3,1] => [1,1,1,0,0,0,1,0]
=> [4,1,2,3] => 1
1111 => [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
00000 => [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
00001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => 1
00010 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => 1
00011 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => 0
00100 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => 1
00101 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => 2
00110 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => 2
00111 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 1
01000 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => 1
01001 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => 2
01010 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 2
01011 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
01100 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0
01101 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1
01110 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
01111 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
10000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 0
10001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => 1
10010 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => 1
10011 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => 0
10100 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => 1
10101 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => 2
Description
The number of odd descents of a permutation.
Matching statistic: St000259
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => ([],1)
=> 0
1 => [1] => [1] => ([],1)
=> 0
00 => [2] => [1] => ([],1)
=> 0
01 => [1,1] => [2] => ([],2)
=> ? ∊ {1,1}
10 => [1,1] => [2] => ([],2)
=> ? ∊ {1,1}
11 => [2] => [1] => ([],1)
=> 0
000 => [3] => [1] => ([],1)
=> 0
001 => [2,1] => [1,1] => ([(0,1)],2)
=> 1
010 => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0}
011 => [1,2] => [1,1] => ([(0,1)],2)
=> 1
100 => [1,2] => [1,1] => ([(0,1)],2)
=> 1
101 => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0}
110 => [2,1] => [1,1] => ([(0,1)],2)
=> 1
111 => [3] => [1] => ([],1)
=> 0
0000 => [4] => [1] => ([],1)
=> 0
0001 => [3,1] => [1,1] => ([(0,1)],2)
=> 1
0010 => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,2,2}
0011 => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,1,1,2,2}
0100 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
0101 => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,1,1,2,2}
0110 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
0111 => [1,3] => [1,1] => ([(0,1)],2)
=> 1
1000 => [1,3] => [1,1] => ([(0,1)],2)
=> 1
1001 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
1010 => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,1,1,2,2}
1011 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 2
1100 => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,1,1,2,2}
1101 => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,2,2}
1110 => [3,1] => [1,1] => ([(0,1)],2)
=> 1
1111 => [4] => [1] => ([],1)
=> 0
00000 => [5] => [1] => ([],1)
=> 0
00001 => [4,1] => [1,1] => ([(0,1)],2)
=> 1
00010 => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
00011 => [3,2] => [1,1] => ([(0,1)],2)
=> 1
00100 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
00101 => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
00110 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
00111 => [2,3] => [1,1] => ([(0,1)],2)
=> 1
01000 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
01001 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
01010 => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
01011 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
01100 => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
01101 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
01110 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
01111 => [1,4] => [1,1] => ([(0,1)],2)
=> 1
10000 => [1,4] => [1,1] => ([(0,1)],2)
=> 1
10001 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
10010 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
10011 => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
10100 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 2
10101 => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
10110 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
10111 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 2
11000 => [2,3] => [1,1] => ([(0,1)],2)
=> 1
11001 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 2
11010 => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
11011 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
11100 => [3,2] => [1,1] => ([(0,1)],2)
=> 1
11101 => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1}
11110 => [4,1] => [1,1] => ([(0,1)],2)
=> 1
11111 => [5] => [1] => ([],1)
=> 0
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St000777
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 68% ●values known / values provided: 68%●distinct values known / distinct values provided: 100%
Values
0 => [1] => [1] => ([],1)
=> 1 = 0 + 1
1 => [1] => [1] => ([],1)
=> 1 = 0 + 1
00 => [2] => [1] => ([],1)
=> 1 = 0 + 1
01 => [1,1] => [2] => ([],2)
=> ? ∊ {1,1} + 1
10 => [1,1] => [2] => ([],2)
=> ? ∊ {1,1} + 1
11 => [2] => [1] => ([],1)
=> 1 = 0 + 1
000 => [3] => [1] => ([],1)
=> 1 = 0 + 1
001 => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
010 => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0} + 1
011 => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
100 => [1,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
101 => [1,1,1] => [3] => ([],3)
=> ? ∊ {0,0} + 1
110 => [2,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
111 => [3] => [1] => ([],1)
=> 1 = 0 + 1
0000 => [4] => [1] => ([],1)
=> 1 = 0 + 1
0001 => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
0010 => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,2,2} + 1
0011 => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,1,1,2,2} + 1
0100 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
0101 => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,1,1,2,2} + 1
0110 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
0111 => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
1000 => [1,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
1001 => [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
1010 => [1,1,1,1] => [4] => ([],4)
=> ? ∊ {0,0,1,1,2,2} + 1
1011 => [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
1100 => [2,2] => [2] => ([],2)
=> ? ∊ {0,0,1,1,2,2} + 1
1101 => [2,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,2,2} + 1
1110 => [3,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
1111 => [4] => [1] => ([],1)
=> 1 = 0 + 1
00000 => [5] => [1] => ([],1)
=> 1 = 0 + 1
00001 => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
00010 => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
00011 => [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
00100 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
00101 => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
00110 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
00111 => [2,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
01000 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
01001 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
01010 => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
01011 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
01100 => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
01101 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
01110 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
01111 => [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
10000 => [1,4] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
10001 => [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
10010 => [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
10011 => [1,2,2] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
10100 => [1,1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
10101 => [1,1,1,1,1] => [5] => ([],5)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
10110 => [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
10111 => [1,1,3] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
11000 => [2,3] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
11001 => [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
11010 => [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
11011 => [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
11100 => [3,2] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
11101 => [3,1,1] => [1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,1} + 1
11110 => [4,1] => [1,1] => ([(0,1)],2)
=> 2 = 1 + 1
11111 => [5] => [1] => ([],1)
=> 1 = 0 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000260
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 65% ●values known / values provided: 65%●distinct values known / distinct values provided: 67%
Values
0 => [1] => ([],1)
=> ([],1)
=> 0
1 => [1] => ([],1)
=> ([],1)
=> 0
00 => [2] => ([],2)
=> ([],1)
=> 0
01 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
10 => [1,1] => ([(0,1)],2)
=> ([(0,1)],2)
=> 1
11 => [2] => ([],2)
=> ([],1)
=> 0
000 => [3] => ([],3)
=> ([],1)
=> 0
001 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
010 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
011 => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0}
100 => [1,2] => ([(1,2)],3)
=> ([(1,2)],3)
=> ? ∊ {0,0}
101 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
110 => [2,1] => ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
111 => [3] => ([],3)
=> ([],1)
=> 0
0000 => [4] => ([],4)
=> ([],1)
=> 0
0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
0010 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
0011 => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,2,2,2,2}
0100 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,2}
0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
0110 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
0111 => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,2,2,2,2}
1000 => [1,3] => ([(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,2,2,2,2}
1001 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
1010 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
1011 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,2,2,2,2}
1100 => [2,2] => ([(1,3),(2,3)],4)
=> ([(1,2)],3)
=> ? ∊ {0,0,2,2,2,2}
1101 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
1110 => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
1111 => [4] => ([],4)
=> ([],1)
=> 0
00000 => [5] => ([],5)
=> ([],1)
=> 0
00001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
00010 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
00011 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
00100 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
00101 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
00110 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
00111 => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
01000 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
01001 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
01010 => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
01011 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
01100 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
01101 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
01110 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
01111 => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
10000 => [1,4] => ([(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
10001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
10010 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
10011 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
10100 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
10101 => [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)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
10110 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
10111 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
11000 => [2,3] => ([(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
11001 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1
11010 => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
11011 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
11100 => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,2,2,2,2,2}
11101 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
11110 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
11111 => [5] => ([],5)
=> ([],1)
=> 0
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 45 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000137The Grundy value of an integer partition. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000455The second largest eigenvalue of a graph if it is integral. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000934The 2-degree of an integer partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001399The distinguishing number of a poset. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000544The cop number of a graph. St001774The degree of the minimal polynomial of the smallest eigenvalue of a graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001624The breadth of a lattice. St001644The dimension of a graph. St001330The hat guessing number of a graph. St000741The Colin de Verdière graph invariant. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by 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]/(xn).
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