Your data matches 16 different statistics following compositions of up to 3 maps.
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Matching statistic: St000393
(load all 6 compositions to match this statistic)
Mapping
St000393: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 1
1 => 1
00 => 2
01 => 1
10 => 2
11 => 2
000 => 3
001 => 2
010 => 2
011 => 2
100 => 3
101 => 2
110 => 3
111 => 3
0000 => 4
0001 => 3
0010 => 3
0011 => 3
0100 => 3
0101 => 2
0110 => 3
0111 => 3
1000 => 4
1001 => 3
1010 => 3
1011 => 3
1100 => 4
1101 => 3
1110 => 4
1111 => 4
Description
The number of strictly increasing runs in a binary word.
Matching statistic: St001237
(load all 8 compositions to match this statistic)
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001237: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => [2] => [1,1,0,0]
 => 3 = 1 + 2
1 => [1,1] => [1,0,1,0]
 => 3 = 1 + 2
00 => [3] => [1,1,1,0,0,0]
 => 4 = 2 + 2
01 => [2,1] => [1,1,0,0,1,0]
 => 3 = 1 + 2
10 => [1,2] => [1,0,1,1,0,0]
 => 4 = 2 + 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 2 + 2
000 => [4] => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => 4 = 2 + 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => 4 = 2 + 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 4 = 2 + 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => 5 = 3 + 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 4 = 2 + 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 5 = 3 + 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 3 + 2
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 5 = 3 + 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 5 = 3 + 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 5 = 3 + 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 4 = 2 + 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 5 = 3 + 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 3 + 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 6 = 4 + 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 5 = 3 + 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 5 = 3 + 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 6 = 4 + 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 5 = 3 + 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 6 = 4 + 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 4 + 2
Description
The number of simple modules with injective dimension at most one or dominant dimension at least one.
Matching statistic: St000245
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000245: 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] => 1
1 => [1,1] => [1,0,1,0]
 => [3,1,2] => 1
00 => [3] => [1,1,1,0,0,0]
 => [2,3,4,1] => 2
01 => [2,1] => [1,1,0,0,1,0]
 => [2,4,1,3] => 2
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] => 2
000 => [4] => [1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => 3
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => 3
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => 3
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] => 2
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] => 3
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,6,1] => 4
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [2,3,4,6,1,5] => 4
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [2,3,5,1,6,4] => 3
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [2,3,6,1,4,5] => 4
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => 3
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => 3
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => 3
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => 4
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] => 3
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => 3
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => 3
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => 3
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] => 4
Description
The number of ascents of a permutation.
Matching statistic: St000876
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
Mp00316: Binary words —inverse Foata bijection⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
0 => 0 => 1 => 1 => 1
1 => 1 => 0 => 0 => 1
00 => 00 => 10 => 10 => 1
01 => 01 => 11 => 11 => 2
10 => 01 => 11 => 11 => 2
11 => 11 => 01 => 01 => 2
000 => 000 => 100 => 010 => 2
001 => 001 => 101 => 101 => 2
010 => 001 => 101 => 101 => 2
011 => 011 => 111 => 111 => 3
100 => 001 => 101 => 101 => 2
101 => 011 => 111 => 111 => 3
110 => 011 => 111 => 111 => 3
111 => 111 => 011 => 011 => 3
0000 => 0000 => 1000 => 0010 => 3
0001 => 0001 => 1001 => 0101 => 3
0010 => 0001 => 1001 => 0101 => 3
0011 => 0011 => 1011 => 1011 => 3
0100 => 0001 => 1001 => 0101 => 3
0101 => 0101 => 1101 => 1101 => 2
0110 => 0011 => 1011 => 1011 => 3
0111 => 0111 => 1111 => 1111 => 4
1000 => 0001 => 1001 => 0101 => 3
1001 => 0011 => 1011 => 1011 => 3
1010 => 0011 => 1011 => 1011 => 3
1011 => 0111 => 1111 => 1111 => 4
1100 => 0011 => 1011 => 1011 => 3
1101 => 0111 => 1111 => 1111 => 4
1110 => 0111 => 1111 => 1111 => 4
1111 => 1111 => 0111 => 0111 => 4
Description
The number of factors in the Catalan decomposition of a binary word. Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2]. This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000863
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St000863: 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] => 2 = 1 + 1
1 => [1,1] => [1,0,1,0]
 => [2,1] => 2 = 1 + 1
00 => [3] => [1,1,1,0,0,0]
 => [1,2,3] => 3 = 2 + 1
01 => [2,1] => [1,1,0,0,1,0]
 => [1,3,2] => 2 = 1 + 1
10 => [1,2] => [1,0,1,1,0,0]
 => [2,1,3] => 3 = 2 + 1
11 => [1,1,1] => [1,0,1,0,1,0]
 => [2,3,1] => 3 = 2 + 1
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,2,3,4] => 4 = 3 + 1
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,2,4,3] => 3 = 2 + 1
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,3,2,4] => 3 = 2 + 1
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,3,4,2] => 3 = 2 + 1
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [2,1,3,4] => 4 = 3 + 1
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [2,1,4,3] => 3 = 2 + 1
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [2,3,1,4] => 4 = 3 + 1
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [2,3,4,1] => 4 = 3 + 1
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => 5 = 4 + 1
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,2,3,5,4] => 4 = 3 + 1
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,2,4,3,5] => 4 = 3 + 1
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,2,4,5,3] => 4 = 3 + 1
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,3,2,4,5] => 4 = 3 + 1
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,3,2,5,4] => 3 = 2 + 1
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,3,4,2,5] => 4 = 3 + 1
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,3,4,5,2] => 4 = 3 + 1
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [2,1,3,4,5] => 5 = 4 + 1
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [2,1,3,5,4] => 4 = 3 + 1
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [2,1,4,3,5] => 4 = 3 + 1
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [2,1,4,5,3] => 4 = 3 + 1
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [2,3,1,4,5] => 5 = 4 + 1
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [2,3,1,5,4] => 4 = 3 + 1
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [2,3,4,1,5] => 5 = 4 + 1
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [2,3,4,5,1] => 5 = 4 + 1
Description
The length of the first row of the shifted shape of a permutation. The diagram of a strict partition $\lambda_1 < \lambda_2 < \dots < \lambda_\ell$ of $n$ is a tableau with $\ell$ rows, the $i$-th row being indented by $i$ cells. A shifted standard Young tableau is a filling of such a diagram, where entries in rows and columns are strictly increasing. The shifted Robinson-Schensted algorithm [1] associates to a permutation a pair $(P, Q)$ of standard shifted Young tableaux of the same shape, where off-diagonal entries in $Q$ may be circled. This statistic records the length of the first row of $P$ and $Q$.
Matching statistic: St001179
Mapping
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001179: 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]
 => 3 = 1 + 2
1 => [1,1] => [1,0,1,0]
 => [1,1,0,0]
 => 3 = 1 + 2
00 => [3] => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 3 = 1 + 2
01 => [2,1] => [1,1,0,0,1,0]
 => [1,1,0,0,1,0]
 => 4 = 2 + 2
10 => [1,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 4 = 2 + 2
11 => [1,1,1] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 4 = 2 + 2
000 => [4] => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 4 = 2 + 2
001 => [3,1] => [1,1,1,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0]
 => 4 = 2 + 2
010 => [2,2] => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 4 = 2 + 2
011 => [2,1,1] => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0]
 => 5 = 3 + 2
100 => [1,3] => [1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 4 = 2 + 2
101 => [1,2,1] => [1,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 5 = 3 + 2
110 => [1,1,2] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 5 = 3 + 2
111 => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 5 = 3 + 2
0000 => [5] => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 5 = 3 + 2
0001 => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => [1,1,0,0,1,0,1,0,1,0]
 => 5 = 3 + 2
0010 => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 5 = 3 + 2
0011 => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 5 = 3 + 2
0100 => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 5 = 3 + 2
0101 => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0,1,0]
 => 5 = 3 + 2
0110 => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 5 = 3 + 2
0111 => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0,1,0]
 => 6 = 4 + 2
1000 => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 2 + 2
1001 => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => 5 = 3 + 2
1010 => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 5 = 3 + 2
1011 => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 6 = 4 + 2
1100 => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 5 = 3 + 2
1101 => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => 6 = 4 + 2
1110 => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 6 = 4 + 2
1111 => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 6 = 4 + 2
Description
Number of indecomposable injective modules with projective dimension at most 2 in the corresponding Nakayama algebra.
Matching statistic: St000806
Mapping
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
Mp00133: Integer compositions —delta morphism⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 47% ●values known / values provided: 47%●distinct values known / distinct values provided: 50%
Values
0 => [1] => [1] => [1] => ? ∊ {1,1}
1 => [1] => [1] => [1] => ? ∊ {1,1}
00 => [2] => [1] => [1] => ? ∊ {1,2,2,2}
01 => [1,1] => [2] => [1] => ? ∊ {1,2,2,2}
10 => [1,1] => [2] => [1] => ? ∊ {1,2,2,2}
11 => [2] => [1] => [1] => ? ∊ {1,2,2,2}
000 => [3] => [1] => [1] => ? ∊ {2,2,2,2}
001 => [2,1] => [1,1] => [2] => 3
010 => [1,1,1] => [3] => [1] => ? ∊ {2,2,2,2}
011 => [1,2] => [1,1] => [2] => 3
100 => [1,2] => [1,1] => [2] => 3
101 => [1,1,1] => [3] => [1] => ? ∊ {2,2,2,2}
110 => [2,1] => [1,1] => [2] => 3
111 => [3] => [1] => [1] => ? ∊ {2,2,2,2}
0000 => [4] => [1] => [1] => ? ∊ {2,3,3,4,4,4}
0001 => [3,1] => [1,1] => [2] => 3
0010 => [2,1,1] => [1,2] => [1,1] => 3
0011 => [2,2] => [2] => [1] => ? ∊ {2,3,3,4,4,4}
0100 => [1,1,2] => [2,1] => [1,1] => 3
0101 => [1,1,1,1] => [4] => [1] => ? ∊ {2,3,3,4,4,4}
0110 => [1,2,1] => [1,1,1] => [3] => 4
0111 => [1,3] => [1,1] => [2] => 3
1000 => [1,3] => [1,1] => [2] => 3
1001 => [1,2,1] => [1,1,1] => [3] => 4
1010 => [1,1,1,1] => [4] => [1] => ? ∊ {2,3,3,4,4,4}
1011 => [1,1,2] => [2,1] => [1,1] => 3
1100 => [2,2] => [2] => [1] => ? ∊ {2,3,3,4,4,4}
1101 => [2,1,1] => [1,2] => [1,1] => 3
1110 => [3,1] => [1,1] => [2] => 3
1111 => [4] => [1] => [1] => ? ∊ {2,3,3,4,4,4}
Description
The semiperimeter of the associated bargraph. Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
Matching statistic: St000454
(load all 4 compositions to match this statistic)
Mapping
Mp00234: Binary words —valleys-to-peaks⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000454: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 100%
Values
0 => 1 => [1,1] => ([(0,1)],2)
 => 1
1 => 1 => [1,1] => ([(0,1)],2)
 => 1
00 => 01 => [2,1] => ([(0,2),(1,2)],3)
 => ? = 2
01 => 10 => [1,2] => ([(1,2)],3)
 => 1
10 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
11 => 11 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
000 => 001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3}
001 => 010 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3}
010 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3}
011 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3}
100 => 101 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? ∊ {2,2,2,3,3}
101 => 110 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 2
110 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
111 => 111 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
0000 => 0001 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2
0001 => 0010 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
0010 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
0011 => 0101 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
0100 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
0101 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
0110 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
0111 => 1011 => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
1000 => 1001 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
1001 => 1010 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
1010 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
1011 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
1100 => 1101 => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? ∊ {3,3,3,3,3,3,3,3,3,4,4,4}
1101 => 1110 => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
1110 => 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)
 => 4
1111 => 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)
 => 4
Description
The largest eigenvalue of a graph if it is integral. If a graph is $d$-regular, then its largest eigenvalue equals $d$. One can show that the largest eigenvalue always lies between the average degree and the maximal degree. This statistic is undefined if the largest eigenvalue of the graph is not integral.
Matching statistic: St000777
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 75%
Values
0 => 0 => [1] => ([],1)
 => 1
1 => 1 => [1] => ([],1)
 => 1
00 => 00 => [2] => ([],2)
 => ? ∊ {1,2}
01 => 01 => [1,1] => ([(0,1)],2)
 => 2
10 => 01 => [1,1] => ([(0,1)],2)
 => 2
11 => 11 => [2] => ([],2)
 => ? ∊ {1,2}
000 => 000 => [3] => ([],3)
 => ? ∊ {2,2,2,2,3}
001 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 3
010 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 3
011 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3}
100 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 3
101 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3}
110 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3}
111 => 111 => [3] => ([],3)
 => ? ∊ {2,2,2,2,3}
0000 => 0000 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
0001 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
0010 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
0011 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
0100 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
0101 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
0110 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
0111 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
1000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
1001 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
1010 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
1011 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
1100 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
1101 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
1110 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
1111 => 1111 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4}
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000259
Mapping
Mp00224: Binary words —runsort⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 75%
Values
0 => 0 => [1] => ([],1)
 => 0 = 1 - 1
1 => 1 => [1] => ([],1)
 => 0 = 1 - 1
00 => 00 => [2] => ([],2)
 => ? ∊ {1,2} - 1
01 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
10 => 01 => [1,1] => ([(0,1)],2)
 => 1 = 2 - 1
11 => 11 => [2] => ([],2)
 => ? ∊ {1,2} - 1
000 => 000 => [3] => ([],3)
 => ? ∊ {2,2,2,2,3} - 1
001 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
010 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
011 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3} - 1
100 => 001 => [2,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
101 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3} - 1
110 => 011 => [1,2] => ([(1,2)],3)
 => ? ∊ {2,2,2,2,3} - 1
111 => 111 => [3] => ([],3)
 => ? ∊ {2,2,2,2,3} - 1
0000 => 0000 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
0001 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
0010 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
0011 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
0100 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
0101 => 0101 => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1 = 2 - 1
0110 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
0111 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
1000 => 0001 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
1001 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
1010 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
1011 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
1100 => 0011 => [2,2] => ([(1,3),(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
1101 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
1110 => 0111 => [1,3] => ([(2,3)],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
1111 => 1111 => [4] => ([],4)
 => ? ∊ {3,3,3,3,3,3,4,4,4,4,4} - 1
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
The following 6 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000264The girth of a graph, which is not a tree. St001060The distinguishing index of a graph. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001330The hat guessing number of a graph. St000075The orbit size of a standard tableau under promotion.