Your data matches 8 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000516
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000516: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => [2,3,1] => 0
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 0
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 0
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 0
10 => [1,2] => [1,0,1,1,0,0]
 => [3,1,4,2] => 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [4,1,2,3] => 0
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => 0
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => 0
=> [1] => [1,0]
 => [2,1] => 0
Description
The number of stretching pairs of a permutation. This is the number of pairs $(i,j)$ with $\pi(i) < i < j < \pi(j)$.
Matching statistic: St001438
(load all 2 compositions to match this statistic)
Mapping
Mp00104: Binary words —reverse⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => [2] => [[2],[]]
 => 0
1 => 1 => [1,1] => [[1,1],[]]
 => 0
00 => 00 => [3] => [[3],[]]
 => 0
01 => 10 => [1,2] => [[2,1],[]]
 => 0
10 => 01 => [2,1] => [[2,2],[1]]
 => 1
11 => 11 => [1,1,1] => [[1,1,1],[]]
 => 0
000 => 000 => [4] => [[4],[]]
 => 0
001 => 100 => [1,3] => [[3,1],[]]
 => 0
010 => 010 => [2,2] => [[3,2],[1]]
 => 1
011 => 110 => [1,1,2] => [[2,1,1],[]]
 => 0
100 => 001 => [3,1] => [[3,3],[2]]
 => 2
101 => 101 => [1,2,1] => [[2,2,1],[1]]
 => 1
110 => 011 => [2,1,1] => [[2,2,2],[1,1]]
 => 2
111 => 111 => [1,1,1,1] => [[1,1,1,1],[]]
 => 0
0000 => 0000 => [5] => [[5],[]]
 => 0
0001 => 1000 => [1,4] => [[4,1],[]]
 => 0
0010 => 0100 => [2,3] => [[4,2],[1]]
 => 1
0011 => 1100 => [1,1,3] => [[3,1,1],[]]
 => 0
0100 => 0010 => [3,2] => [[4,3],[2]]
 => 2
0101 => 1010 => [1,2,2] => [[3,2,1],[1]]
 => 1
0110 => 0110 => [2,1,2] => [[3,2,2],[1,1]]
 => 2
0111 => 1110 => [1,1,1,2] => [[2,1,1,1],[]]
 => 0
1000 => 0001 => [4,1] => [[4,4],[3]]
 => 3
1001 => 1001 => [1,3,1] => [[3,3,1],[2]]
 => 2
1010 => 0101 => [2,2,1] => [[3,3,2],[2,1]]
 => 3
1011 => 1101 => [1,1,2,1] => [[2,2,1,1],[1]]
 => 1
1100 => 0011 => [3,1,1] => [[3,3,3],[2,2]]
 => 4
1101 => 1011 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => 2
1110 => 0111 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => 3
1111 => 1111 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => 0
=> => [1] => [[1],[]]
 => 0
Description
The number of missing boxes of a skew partition.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
Mapping
St000293: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 0
1 => 0
00 => 0
01 => 0
10 => 1
11 => 0
000 => 0
001 => 0
010 => 1
011 => 0
100 => 2
101 => 1
110 => 2
111 => 0
0000 => 0
0001 => 0
0010 => 1
0011 => 0
0100 => 2
0101 => 1
0110 => 2
0111 => 0
1000 => 3
1001 => 2
1010 => 3
1011 => 1
1100 => 4
1101 => 2
1110 => 3
1111 => 0
=> ? = 0
Description
The number of inversions of a binary word.
Matching statistic: St000290
(load all 4 compositions to match this statistic)
Mapping
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000290: Binary words ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 0
1 => 1 => 0
00 => 00 => 0
01 => 01 => 0
10 => 10 => 1
11 => 11 => 0
000 => 000 => 0
001 => 001 => 0
010 => 100 => 1
011 => 011 => 0
100 => 010 => 2
101 => 101 => 1
110 => 110 => 2
111 => 111 => 0
0000 => 0000 => 0
0001 => 0001 => 0
0010 => 1000 => 1
0011 => 0011 => 0
0100 => 0100 => 2
0101 => 1001 => 1
0110 => 1100 => 2
0111 => 0111 => 0
1000 => 0010 => 3
1001 => 0101 => 2
1010 => 0110 => 3
1011 => 1011 => 1
1100 => 1010 => 4
1101 => 1101 => 2
1110 => 1110 => 3
1111 => 1111 => 0
=> ? => ? = 0
Description
The major index of a binary word. This is the sum of the positions of descents, i.e., a one followed by a zero. For words of length $n$ with $a$ zeros, the generating function for the major index is the $q$-binomial coefficient $\binom{n}{a}_q$.
Matching statistic: St000589
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000589: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 0
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 0
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 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]
 => {{1,2,3,4}}
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
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]
 => {{1,2,3,4,5}}
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
=> [1] => [1,0]
 => {{1}}
 => ? = 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is maximal, (2,3) are consecutive in a block.
Matching statistic: St000609
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000609: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 0
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 0
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 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]
 => {{1,2,3,4}}
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
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]
 => {{1,2,3,4,5}}
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
=> [1] => [1,0]
 => {{1}}
 => ? = 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1,2 are minimal.
Matching statistic: St000612
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00138: Dyck paths —to noncrossing partition⟶ Set partitions
St000612: Set partitions ⟶ ℤResult quality: 97% ●values known / values provided: 97%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => {{1,2}}
 => 0
1 => [1,1] => [1,0,1,0]
 => {{1},{2}}
 => 0
00 => [3] => [1,1,1,0,0,0]
 => {{1,2,3}}
 => 0
01 => [2,1] => [1,1,0,0,1,0]
 => {{1,2},{3}}
 => 0
10 => [1,2] => [1,0,1,1,0,0]
 => {{1},{2,3}}
 => 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]
 => {{1,2,3,4}}
 => 0
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => {{1,2,3},{4}}
 => 0
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => {{1,2},{3,4}}
 => 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => {{1,2},{3},{4}}
 => 0
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => {{1},{2,3,4}}
 => 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => {{1},{2,3},{4}}
 => 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => {{1},{2},{3,4}}
 => 2
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]
 => {{1,2,3,4,5}}
 => 0
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => {{1,2,3,4},{5}}
 => 0
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => {{1,2,3},{4,5}}
 => 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => {{1,2,3},{4},{5}}
 => 0
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => {{1,2},{3,4,5}}
 => 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => {{1,2},{3,4},{5}}
 => 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => {{1,2},{3},{4,5}}
 => 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => {{1,2},{3},{4},{5}}
 => 0
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => {{1},{2,3,4,5}}
 => 3
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => {{1},{2,3,4},{5}}
 => 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => {{1},{2,3},{4,5}}
 => 3
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => {{1},{2,3},{4},{5}}
 => 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => {{1},{2},{3,4,5}}
 => 4
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => {{1},{2},{3,4},{5}}
 => 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => {{1},{2},{3},{4,5}}
 => 3
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => {{1},{2},{3},{4},{5}}
 => 0
=> [1] => [1,0]
 => {{1}}
 => ? = 0
Description
The number of occurrences of the pattern {{1},{2,3}} such that 1 is minimal, (2,3) are consecutive in a block.
Matching statistic: St000455
Mapping
Mp00158: Binary words —alternating inverse⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
Mp00198: Posets —incomparability graph⟶ Graphs
St000455: Graphs ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 20%
Values
0 => 0 => ([(0,1)],2)
 => ([],2)
 => ? = 0
1 => 1 => ([(0,1)],2)
 => ([],2)
 => ? = 0
00 => 01 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
01 => 00 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 0
10 => 11 => ([(0,2),(2,1)],3)
 => ([],3)
 => ? = 1
11 => 10 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(2,3)],4)
 => 0
000 => 010 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 0
001 => 011 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 0
010 => 000 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
011 => 001 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 0
100 => 110 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 2
101 => 111 => ([(0,3),(2,1),(3,2)],4)
 => ([],4)
 => ? = 1
110 => 100 => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
 => ([(2,5),(3,4),(4,5)],6)
 => ? = 2
111 => 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
 => ([(2,5),(3,4)],6)
 => 0
0000 => 0101 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ([(2,7),(3,6),(4,5)],8)
 => ? = 0
0001 => 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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ? = 0
0010 => 0111 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 1
0011 => 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)
 => ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 0
0100 => 0001 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 2
0101 => 0000 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 1
0110 => 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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? = 2
0111 => 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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ? = 0
1000 => 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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ? = 3
1001 => 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)
 => ([(2,5),(2,8),(3,4),(3,8),(4,7),(5,7),(6,7),(6,8),(7,8)],9)
 => ? = 2
1010 => 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([],5)
 => ? = 3
1011 => 1110 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 1
1100 => 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)
 => ([(2,5),(3,4),(3,7),(4,6),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 4
1101 => 1000 => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
 => ([(2,7),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
 => ? = 2
1110 => 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)
 => ([(2,6),(3,4),(3,8),(4,7),(5,7),(5,8),(6,8),(7,8)],9)
 => ? = 3
1111 => 1010 => ([(0,1),(0,2),(1,6),(1,7),(2,6),(2,7),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5)],8)
 => ([(2,7),(3,6),(4,5)],8)
 => ? = 0
=> => ?
 => ?
 => ? = 0
Description
The second largest eigenvalue of a graph if it is integral. This statistic is undefined if the second largest eigenvalue of the graph is not integral. Chapter 4 of [1] provides lots of context.