Your data matches 208 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001217
(load all 84 compositions to match this statistic)
Mapping
St001217: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => 1
[1,0,1,0]
 => 0
[1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => 0
[1,0,1,1,0,0]
 => 0
[1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0]
 => 0
[1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,1,0,0]
 => 0
[1,0,1,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,0]
 => 0
[1,1,0,0,1,0,1,0]
 => 1
[1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,0]
 => 0
[1,1,1,0,0,0,1,0]
 => 1
[1,1,1,0,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => 1
[1,1,0,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,0,1,0,1,1,0,0,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,0,1,0,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => 0
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St000745
(load all 16 compositions to match this statistic)
Mapping
Mp00033: Dyck paths ā€”to two-row standard tableauāŸ¶ Standard tableaux
St000745: Standard tableaux āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1],[2]]
 => 2 = 1 + 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => 2 = 1 + 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => 1 = 0 + 1
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 2 = 1 + 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 1 = 0 + 1
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 1 = 0 + 1
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => 1 = 0 + 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000237
(load all 2 compositions to match this statistic)
Mapping
Mp00033: Dyck paths ā€”to two-row standard tableauāŸ¶ Standard tableaux
Mp00081: Standard tableaux ā€”reading word permutationāŸ¶ Permutations
St000237: Permutations āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1],[2]]
 => [2,1] => 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => [2,4,1,3] => 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => [3,4,1,2] => 0
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => [2,4,6,1,3,5] => 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => [2,5,6,1,3,4] => 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => [3,4,6,1,2,5] => 0
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => [3,5,6,1,2,4] => 0
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => [4,5,6,1,2,3] => 0
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => [2,4,6,8,1,3,5,7] => 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => [2,4,7,8,1,3,5,6] => 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => [2,5,6,8,1,3,4,7] => 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => [2,5,7,8,1,3,4,6] => 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => [2,6,7,8,1,3,4,5] => 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => [3,4,6,8,1,2,5,7] => 0
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => [3,4,7,8,1,2,5,6] => 0
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => [3,5,6,8,1,2,4,7] => 0
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => [3,5,7,8,1,2,4,6] => 0
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => [3,6,7,8,1,2,4,5] => 0
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => [4,5,6,8,1,2,3,7] => 0
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => [4,5,7,8,1,2,3,6] => 0
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => [4,6,7,8,1,2,3,5] => 0
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => [5,6,7,8,1,2,3,4] => 0
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => [2,4,6,8,10,1,3,5,7,9] => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => [2,4,6,9,10,1,3,5,7,8] => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => [2,4,7,8,10,1,3,5,6,9] => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => [2,4,7,9,10,1,3,5,6,8] => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => [2,4,8,9,10,1,3,5,6,7] => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => [2,5,6,8,10,1,3,4,7,9] => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => [2,5,6,9,10,1,3,4,7,8] => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => [2,5,7,8,10,1,3,4,6,9] => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => [2,5,7,9,10,1,3,4,6,8] => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => [2,5,8,9,10,1,3,4,6,7] => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => [2,6,7,8,10,1,3,4,5,9] => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => [2,6,7,9,10,1,3,4,5,8] => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => [2,6,8,9,10,1,3,4,5,7] => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => [2,7,8,9,10,1,3,4,5,6] => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => [3,4,6,8,10,1,2,5,7,9] => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => [3,4,6,9,10,1,2,5,7,8] => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => [3,4,7,8,10,1,2,5,6,9] => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => [3,4,7,9,10,1,2,5,6,8] => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => [3,4,8,9,10,1,2,5,6,7] => 0
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => [3,5,6,8,10,1,2,4,7,9] => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => [3,5,6,9,10,1,2,4,7,8] => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => [3,5,7,8,10,1,2,4,6,9] => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => [3,5,7,9,10,1,2,4,6,8] => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => [3,5,8,9,10,1,2,4,6,7] => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => [3,6,7,8,10,1,2,4,5,9] => 0
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => [3,6,7,9,10,1,2,4,5,8] => 0
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => [3,6,8,9,10,1,2,4,5,7] => 0
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => [3,7,8,9,10,1,2,4,5,6] => 0
Description
The number of small exceedances. This is the number of indices $i$ such that $\pi_i=i+1$.
Matching statistic: St000297
(load all 22 compositions to match this statistic)
Mapping
Mp00033: Dyck paths ā€”to two-row standard tableauāŸ¶ Standard tableaux
Mp00134: Standard tableaux ā€”descent wordāŸ¶ Binary words
St000297: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [[1],[2]]
 => 1 => 1
[1,0,1,0]
 => [[1,3],[2,4]]
 => 101 => 1
[1,1,0,0]
 => [[1,2],[3,4]]
 => 010 => 0
[1,0,1,0,1,0]
 => [[1,3,5],[2,4,6]]
 => 10101 => 1
[1,0,1,1,0,0]
 => [[1,3,4],[2,5,6]]
 => 10010 => 1
[1,1,0,0,1,0]
 => [[1,2,5],[3,4,6]]
 => 01001 => 0
[1,1,0,1,0,0]
 => [[1,2,4],[3,5,6]]
 => 01010 => 0
[1,1,1,0,0,0]
 => [[1,2,3],[4,5,6]]
 => 00100 => 0
[1,0,1,0,1,0,1,0]
 => [[1,3,5,7],[2,4,6,8]]
 => 1010101 => 1
[1,0,1,0,1,1,0,0]
 => [[1,3,5,6],[2,4,7,8]]
 => 1010010 => 1
[1,0,1,1,0,0,1,0]
 => [[1,3,4,7],[2,5,6,8]]
 => 1001001 => 1
[1,0,1,1,0,1,0,0]
 => [[1,3,4,6],[2,5,7,8]]
 => 1001010 => 1
[1,0,1,1,1,0,0,0]
 => [[1,3,4,5],[2,6,7,8]]
 => 1000100 => 1
[1,1,0,0,1,0,1,0]
 => [[1,2,5,7],[3,4,6,8]]
 => 0100101 => 0
[1,1,0,0,1,1,0,0]
 => [[1,2,5,6],[3,4,7,8]]
 => 0100010 => 0
[1,1,0,1,0,0,1,0]
 => [[1,2,4,7],[3,5,6,8]]
 => 0101001 => 0
[1,1,0,1,0,1,0,0]
 => [[1,2,4,6],[3,5,7,8]]
 => 0101010 => 0
[1,1,0,1,1,0,0,0]
 => [[1,2,4,5],[3,6,7,8]]
 => 0100100 => 0
[1,1,1,0,0,0,1,0]
 => [[1,2,3,7],[4,5,6,8]]
 => 0010001 => 0
[1,1,1,0,0,1,0,0]
 => [[1,2,3,6],[4,5,7,8]]
 => 0010010 => 0
[1,1,1,0,1,0,0,0]
 => [[1,2,3,5],[4,6,7,8]]
 => 0010100 => 0
[1,1,1,1,0,0,0,0]
 => [[1,2,3,4],[5,6,7,8]]
 => 0001000 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [[1,3,5,7,9],[2,4,6,8,10]]
 => 101010101 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [[1,3,5,7,8],[2,4,6,9,10]]
 => 101010010 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [[1,3,5,6,9],[2,4,7,8,10]]
 => 101001001 => 1
[1,0,1,0,1,1,0,1,0,0]
 => [[1,3,5,6,8],[2,4,7,9,10]]
 => 101001010 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [[1,3,5,6,7],[2,4,8,9,10]]
 => 101000100 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [[1,3,4,7,9],[2,5,6,8,10]]
 => 100100101 => 1
[1,0,1,1,0,0,1,1,0,0]
 => [[1,3,4,7,8],[2,5,6,9,10]]
 => 100100010 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [[1,3,4,6,9],[2,5,7,8,10]]
 => 100101001 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [[1,3,4,6,8],[2,5,7,9,10]]
 => 100101010 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [[1,3,4,6,7],[2,5,8,9,10]]
 => 100100100 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [[1,3,4,5,9],[2,6,7,8,10]]
 => 100010001 => 1
[1,0,1,1,1,0,0,1,0,0]
 => [[1,3,4,5,8],[2,6,7,9,10]]
 => 100010010 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [[1,3,4,5,7],[2,6,8,9,10]]
 => 100010100 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [[1,3,4,5,6],[2,7,8,9,10]]
 => 100001000 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [[1,2,5,7,9],[3,4,6,8,10]]
 => 010010101 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [[1,2,5,7,8],[3,4,6,9,10]]
 => 010010010 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [[1,2,5,6,9],[3,4,7,8,10]]
 => 010001001 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [[1,2,5,6,8],[3,4,7,9,10]]
 => 010001010 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [[1,2,5,6,7],[3,4,8,9,10]]
 => 010000100 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [[1,2,4,7,9],[3,5,6,8,10]]
 => 010100101 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [[1,2,4,7,8],[3,5,6,9,10]]
 => 010100010 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [[1,2,4,6,9],[3,5,7,8,10]]
 => 010101001 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [[1,2,4,6,8],[3,5,7,9,10]]
 => 010101010 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [[1,2,4,6,7],[3,5,8,9,10]]
 => 010100100 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [[1,2,4,5,9],[3,6,7,8,10]]
 => 010010001 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [[1,2,4,5,8],[3,6,7,9,10]]
 => 010010010 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [[1,2,4,5,7],[3,6,8,9,10]]
 => 010010100 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [[1,2,4,5,6],[3,7,8,9,10]]
 => 010001000 => 0
Description
The number of leading ones in a binary word.
Matching statistic: St001276
(load all 5 compositions to match this statistic)
Mapping
Mp00199: Dyck paths ā€”prime Dyck pathāŸ¶ Dyck paths
Mp00120: Dyck paths ā€”Lalanne-Kreweras involutionāŸ¶ Dyck paths
St001276: Dyck paths āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1,1,0,0]
 => [1,0,1,0]
 => 1
[1,0,1,0]
 => [1,1,0,1,0,0]
 => [1,1,0,1,0,0]
 => 1
[1,1,0,0]
 => [1,1,1,0,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 1
[1,0,1,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => [1,1,0,1,0,0,1,0]
 => 0
[1,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,0,0]
 => 0
[1,1,0,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[1,1,1,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 1
[1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 0
[1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 0
[1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 0
[1,1,0,0,1,1,0,0]
 => [1,1,1,0,0,1,1,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 0
[1,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 1
[1,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[1,1,1,0,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 0
[1,1,1,0,0,1,0,0]
 => [1,1,1,1,0,0,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 0
[1,1,1,1,0,0,0,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 0
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,1,1,0,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,0,1,0,1,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,0,1,0,0]
 => 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,0,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,0,1,0,1,0]
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,0,1,0]
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,0,1,0,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,0,1,0]
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,1,0,1,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,0,1,1,0,1,0,0]
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,1,0,1,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,0,1,0,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,1,0,1,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,0,1,0,1,0,1,0]
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [1,1,1,0,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [1,1,1,0,0,1,0,1,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [1,1,1,0,0,1,1,0,0,1,0,0]
 => [1,0,1,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [1,1,1,0,0,1,1,0,1,0,0,0]
 => [1,0,1,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,1,1,0,0,0,0]
 => [1,0,1,1,0,1,0,0,1,0,1,0]
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,1,0,1,0,1,0,0,0]
 => [1,1,1,1,0,1,0,1,0,0,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [1,1,1,0,1,0,1,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 0
[1,1,0,1,1,0,0,0,1,0]
 => [1,1,1,0,1,1,0,0,0,1,0,0]
 => [1,1,0,1,0,1,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [1,1,1,0,1,1,0,0,1,0,0,0]
 => [1,1,0,1,1,0,1,0,0,1,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [1,1,1,0,1,1,0,1,0,0,0,0]
 => [1,1,1,0,1,0,1,0,0,1,0,0]
 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [1,1,1,0,1,1,1,0,0,0,0,0]
 => [1,1,0,1,0,1,0,0,1,0,1,0]
 => 0
Description
The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. Generalising the notion of k-regular modules from simple to arbitrary indecomposable modules, we call an indecomposable module $M$ over an algebra $A$ k-regular in case it has projective dimension k and $Ext_A^i(M,A)=0$ for $i \neq k$ and $Ext_A^k(M,A)$ is 1-dimensional. The number of Dyck paths where the statistic returns 0 might be given by [[OEIS:A035929]] .
Matching statistic: St000326
(load all 10 compositions to match this statistic)
Mapping
Mp00093: Dyck paths ā€”to binary wordāŸ¶ Binary words
Mp00135: Binary words ā€”rotate front-to-backāŸ¶ Binary words
St000326: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => 10 => 01 => 2 = 1 + 1
[1,0,1,0]
 => 1010 => 0101 => 2 = 1 + 1
[1,1,0,0]
 => 1100 => 1001 => 1 = 0 + 1
[1,0,1,0,1,0]
 => 101010 => 010101 => 2 = 1 + 1
[1,0,1,1,0,0]
 => 101100 => 011001 => 2 = 1 + 1
[1,1,0,0,1,0]
 => 110010 => 100101 => 1 = 0 + 1
[1,1,0,1,0,0]
 => 110100 => 101001 => 1 = 0 + 1
[1,1,1,0,0,0]
 => 111000 => 110001 => 1 = 0 + 1
[1,0,1,0,1,0,1,0]
 => 10101010 => 01010101 => 2 = 1 + 1
[1,0,1,0,1,1,0,0]
 => 10101100 => 01011001 => 2 = 1 + 1
[1,0,1,1,0,0,1,0]
 => 10110010 => 01100101 => 2 = 1 + 1
[1,0,1,1,0,1,0,0]
 => 10110100 => 01101001 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => 10111000 => 01110001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => 11001010 => 10010101 => 1 = 0 + 1
[1,1,0,0,1,1,0,0]
 => 11001100 => 10011001 => 1 = 0 + 1
[1,1,0,1,0,0,1,0]
 => 11010010 => 10100101 => 1 = 0 + 1
[1,1,0,1,0,1,0,0]
 => 11010100 => 10101001 => 1 = 0 + 1
[1,1,0,1,1,0,0,0]
 => 11011000 => 10110001 => 1 = 0 + 1
[1,1,1,0,0,0,1,0]
 => 11100010 => 11000101 => 1 = 0 + 1
[1,1,1,0,0,1,0,0]
 => 11100100 => 11001001 => 1 = 0 + 1
[1,1,1,0,1,0,0,0]
 => 11101000 => 11010001 => 1 = 0 + 1
[1,1,1,1,0,0,0,0]
 => 11110000 => 11100001 => 1 = 0 + 1
[1,0,1,0,1,0,1,0,1,0]
 => 1010101010 => 0101010101 => 2 = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
 => 1010101100 => 0101011001 => 2 = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
 => 1010110010 => 0101100101 => 2 = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
 => 1010110100 => 0101101001 => 2 = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
 => 1010111000 => 0101110001 => 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
 => 1011001010 => 0110010101 => 2 = 1 + 1
[1,0,1,1,0,0,1,1,0,0]
 => 1011001100 => 0110011001 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => 1011010010 => 0110100101 => 2 = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
 => 1011010100 => 0110101001 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => 1011011000 => 0110110001 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => 1011100010 => 0111000101 => 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
 => 1011100100 => 0111001001 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => 1011101000 => 0111010001 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => 1011110000 => 0111100001 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => 1100101010 => 1001010101 => 1 = 0 + 1
[1,1,0,0,1,0,1,1,0,0]
 => 1100101100 => 1001011001 => 1 = 0 + 1
[1,1,0,0,1,1,0,0,1,0]
 => 1100110010 => 1001100101 => 1 = 0 + 1
[1,1,0,0,1,1,0,1,0,0]
 => 1100110100 => 1001101001 => 1 = 0 + 1
[1,1,0,0,1,1,1,0,0,0]
 => 1100111000 => 1001110001 => 1 = 0 + 1
[1,1,0,1,0,0,1,0,1,0]
 => 1101001010 => 1010010101 => 1 = 0 + 1
[1,1,0,1,0,0,1,1,0,0]
 => 1101001100 => 1010011001 => 1 = 0 + 1
[1,1,0,1,0,1,0,0,1,0]
 => 1101010010 => 1010100101 => 1 = 0 + 1
[1,1,0,1,0,1,0,1,0,0]
 => 1101010100 => 1010101001 => 1 = 0 + 1
[1,1,0,1,0,1,1,0,0,0]
 => 1101011000 => 1010110001 => 1 = 0 + 1
[1,1,0,1,1,0,0,0,1,0]
 => 1101100010 => 1011000101 => 1 = 0 + 1
[1,1,0,1,1,0,0,1,0,0]
 => 1101100100 => 1011001001 => 1 = 0 + 1
[1,1,0,1,1,0,1,0,0,0]
 => 1101101000 => 1011010001 => 1 = 0 + 1
[1,1,0,1,1,1,0,0,0,0]
 => 1101110000 => 1011100001 => 1 = 0 + 1
Description
The position of the first one in a binary word after appending a 1 at the end. Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000096
Mapping
Mp00025: Dyck paths ā€”to 132-avoiding permutationāŸ¶ Permutations
Mp00065: Permutations ā€”permutation posetāŸ¶ Posets
Mp00074: Posets ā€”to graphāŸ¶ Graphs
St000096: Graphs āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => [2,1] => ([],2)
 => ([],2)
 => 0
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => ([],3)
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
Description
The number of spanning trees of a graph. A subgraph $H \subseteq G$ is a spanning tree if $V(H)=V(G)$ and $H$ is a tree (i.e. $H$ is connected and contains no cycles).
Matching statistic: St000315
(load all 3 compositions to match this statistic)
Mapping
Mp00100: Dyck paths ā€”touch compositionāŸ¶ Integer compositions
Mp00184: Integer compositions ā€”to threshold graphāŸ¶ Graphs
Mp00154: Graphs ā€”coreāŸ¶ Graphs
St000315: Graphs āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 0
[1,1,0,0]
 => [2] => ([],2)
 => ([],1)
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ([(0,1)],2)
 => 0
[1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 0
[1,1,0,1,0,0]
 => [3] => ([],3)
 => ([],1)
 => 1
[1,1,1,0,0,0]
 => [3] => ([],3)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0]
 => [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)
 => 0
[1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0
[1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ([(0,1)],2)
 => 0
[1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
[1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
[1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => ([],1)
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 0
[1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ([],1)
 => 1
[1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ([],1)
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ([],1)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [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)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => ([(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)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => ([(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)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => ([(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)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [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)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,1),(0,2),(1,2)],3)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [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)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ([],1)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => ([],1)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => ([],1)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => ([],1)
 => 1
Description
The number of isolated vertices of a graph.
Matching statistic: St000390
(load all 5 compositions to match this statistic)
Mapping
Mp00201: Dyck paths ā€”RingelāŸ¶ Permutations
Mp00088: Permutations ā€”Kreweras complementāŸ¶ Permutations
Mp00114: Permutations ā€”connectivity setāŸ¶ Binary words
St000390: Binary words āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [2,1] => [1,2] => 1 => 1
[1,0,1,0]
 => [3,1,2] => [3,1,2] => 00 => 0
[1,1,0,0]
 => [2,3,1] => [1,2,3] => 11 => 1
[1,0,1,0,1,0]
 => [4,1,2,3] => [3,4,1,2] => 000 => 0
[1,0,1,1,0,0]
 => [3,1,4,2] => [3,1,2,4] => 001 => 1
[1,1,0,0,1,0]
 => [2,4,1,3] => [4,2,1,3] => 000 => 0
[1,1,0,1,0,0]
 => [4,3,1,2] => [4,1,3,2] => 000 => 0
[1,1,1,0,0,0]
 => [2,3,4,1] => [1,2,3,4] => 111 => 1
[1,0,1,0,1,0,1,0]
 => [5,1,2,3,4] => [3,4,5,1,2] => 0000 => 0
[1,0,1,0,1,1,0,0]
 => [4,1,2,5,3] => [3,4,1,2,5] => 0001 => 1
[1,0,1,1,0,0,1,0]
 => [3,1,5,2,4] => [3,5,2,1,4] => 0000 => 0
[1,0,1,1,0,1,0,0]
 => [5,1,4,2,3] => [3,5,1,4,2] => 0000 => 0
[1,0,1,1,1,0,0,0]
 => [3,1,4,5,2] => [3,1,2,4,5] => 0011 => 1
[1,1,0,0,1,0,1,0]
 => [2,5,1,3,4] => [4,2,5,1,3] => 0000 => 0
[1,1,0,0,1,1,0,0]
 => [2,4,1,5,3] => [4,2,1,3,5] => 0001 => 1
[1,1,0,1,0,0,1,0]
 => [5,3,1,2,4] => [4,5,3,1,2] => 0000 => 0
[1,1,0,1,0,1,0,0]
 => [5,4,1,2,3] => [4,5,1,3,2] => 0000 => 0
[1,1,0,1,1,0,0,0]
 => [4,3,1,5,2] => [4,1,3,2,5] => 0001 => 1
[1,1,1,0,0,0,1,0]
 => [2,3,5,1,4] => [5,2,3,1,4] => 0000 => 0
[1,1,1,0,0,1,0,0]
 => [2,5,4,1,3] => [5,2,1,4,3] => 0000 => 0
[1,1,1,0,1,0,0,0]
 => [5,3,4,1,2] => [5,1,3,4,2] => 0000 => 0
[1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [1,2,3,4,5] => 1111 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [6,1,2,3,4,5] => [3,4,5,6,1,2] => 00000 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [5,1,2,3,6,4] => [3,4,5,1,2,6] => 00001 => 1
[1,0,1,0,1,1,0,0,1,0]
 => [4,1,2,6,3,5] => [3,4,6,2,1,5] => 00000 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [6,1,2,5,3,4] => [3,4,6,1,5,2] => 00000 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [4,1,2,5,6,3] => [3,4,1,2,5,6] => 00011 => 1
[1,0,1,1,0,0,1,0,1,0]
 => [3,1,6,2,4,5] => [3,5,2,6,1,4] => 00000 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [3,1,5,2,6,4] => [3,5,2,1,4,6] => 00001 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [6,1,4,2,3,5] => [3,5,6,4,1,2] => 00000 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [6,1,5,2,3,4] => [3,5,6,1,4,2] => 00000 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [5,1,4,2,6,3] => [3,5,1,4,2,6] => 00001 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [3,1,4,6,2,5] => [3,6,2,4,1,5] => 00000 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [3,1,6,5,2,4] => [3,6,2,1,5,4] => 00000 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [6,1,4,5,2,3] => [3,6,1,4,5,2] => 00000 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [3,1,4,5,6,2] => [3,1,2,4,5,6] => 00111 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,6,1,3,4,5] => [4,2,5,6,1,3] => 00000 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [2,5,1,3,6,4] => [4,2,5,1,3,6] => 00001 => 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,4,1,6,3,5] => [4,2,6,3,1,5] => 00000 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [2,6,1,5,3,4] => [4,2,6,1,5,3] => 00000 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [2,4,1,5,6,3] => [4,2,1,3,5,6] => 00011 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [6,3,1,2,4,5] => [4,5,3,6,1,2] => 00000 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [5,3,1,2,6,4] => [4,5,3,1,2,6] => 00001 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [6,4,1,2,3,5] => [4,5,6,3,1,2] => 00000 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [5,6,1,2,3,4] => [4,5,6,1,2,3] => 00000 => 0
[1,1,0,1,0,1,1,0,0,0]
 => [5,4,1,2,6,3] => [4,5,1,3,2,6] => 00001 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,3,1,6,2,5] => [4,6,3,2,1,5] => 00000 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [6,3,1,5,2,4] => [4,6,3,1,5,2] => 00000 => 0
[1,1,0,1,1,0,1,0,0,0]
 => [6,4,1,5,2,3] => [4,6,1,3,5,2] => 00000 => 0
[1,1,0,1,1,1,0,0,0,0]
 => [4,3,1,5,6,2] => [4,1,3,2,5,6] => 00011 => 1
Description
The number of runs of ones in a binary word.
Matching statistic: St000948
Mapping
Mp00025: Dyck paths ā€”to 132-avoiding permutationāŸ¶ Permutations
Mp00065: Permutations ā€”permutation posetāŸ¶ Posets
Mp00074: Posets ā€”to graphāŸ¶ Graphs
St000948: Graphs āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => 1
[1,0,1,0]
 => [2,1] => ([],2)
 => ([],2)
 => 0
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => ([],3)
 => 0
[1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => 0
[1,1,0,1,0,0]
 => [2,1,3] => ([(0,2),(1,2)],3)
 => ([(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [1,2,3] => ([(0,2),(2,1)],3)
 => ([(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => ([],4)
 => ([],4)
 => 0
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => 0
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => 0
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => 0
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
 => ([(0,3),(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => ([],5)
 => ([],5)
 => 0
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(3,4)],5)
 => 0
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => 0
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => ([(2,4),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => ([(0,4),(1,4),(2,3)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => 0
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => 0
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
Description
The chromatic discriminant of a graph. The chromatic discriminant $\alpha(G)$ is the coefficient of the linear term of the chromatic polynomial $\chi(G,q)$. According to [1], it equals the cardinality of any of the following sets: (1) Acyclic orientations of G with unique sink at $q$, (2) Maximum $G$-parking functions relative to $q$, (3) Minimal $q$-critical states, (4) Spanning trees of G without broken circuits, (5) Conjugacy classes of Coxeter elements in the Coxeter group associated to $G$, (6) Multilinear Lyndon heaps on $G$. In addition, $\alpha(G)$ is also equal to the the dimension of the root space corresponding to the sum of all simple roots in the Kac-Moody Lie algebra associated to the graph.
The following 198 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001271The competition number of a graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001367The smallest number which does not occur as degree of a vertex in a graph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001691The number of kings in a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St000382The first part of an integer composition. St000383The last part of an integer composition. St000678The number of up steps after the last double rise of a Dyck path. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001316The domatic number of a graph. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nāˆ’1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St000990The first ascent of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000989The number of final rises of a permutation. St000654The first descent of a permutation. St000877The depth of the binary word interpreted as a path. St000929The constant term of the character polynomial of an integer partition. St000932The number of occurrences of the pattern UDU in a Dyck path. St001061The number of indices that are both descents and recoils of a permutation. St000061The number of nodes on the left branch of a binary tree. St000504The cardinality of the first block of a set partition. St000917The open packing number of a graph. St001948The number of augmented double ascents of a permutation. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000352The Elizalde-Pak rank of a permutation. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St000234The number of global ascents of a permutation. St000661The number of rises of length 3 of a Dyck path. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001587Half of the largest even part of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000864The number of circled entries of the shifted recording tableau of a permutation. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. St000542The number of left-to-right-minima of a permutation. St000153The number of adjacent cycles of a permutation. St000512The number of invariant subsets of size 3 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. St000928The sum of the coefficients of the character polynomial of an integer partition. St000117The number of centered tunnels of a Dyck path. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000480The number of lower covers of a partition in dominance order. St000481The number of upper covers of a partition in dominance order. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001092The number of distinct even parts of a partition. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001141The number of occurrences of hills of size 3 in a Dyck path. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001423The number of distinct cubes in a binary word. St001524The degree of symmetry of a binary word. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001730The number of times the path corresponding to a binary word crosses the base line. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000944The 3-degree of an integer partition. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001657The number of twos in an integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001939The number of parts that are equal to their multiplicity in the integer partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000546The number of global descents of a permutation. St000658The number of rises of length 2 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001095The number of non-isomorphic posets with precisely one further covering relation. St001139The number of occurrences of hills of size 2 in a Dyck path. St001651The Frankl number of a lattice. St000754The Grundy value for the game of removing nestings in a perfect matching. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001403The number of vertical separators in a permutation. St001570The minimal number of edges to add to make a graph Hamiltonian. St001049The smallest label in the subtree not containing 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000260The radius of a connected graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000456The monochromatic index of a connected graph. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000056The decomposition (or block) number of a permutation. St000286The number of connected components of the complement of a graph. St000314The number of left-to-right-maxima of a permutation. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001593This is the number of standard Young tableaux of the given shifted shape. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001383The BG-rank of an integer partition. St001525The number of symmetric hooks on the diagonal of a partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000137The Grundy value of an integer partition. St000284The Plancherel distribution on integer partitions. St000567The sum of the products of all pairs of parts. St000618The number of self-evacuating tableaux of given shape. 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. St000706The product of the factorials of the multiplicities of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000941The number of characters of the symmetric group whose value on the partition is even. St000993The multiplicity of the largest part of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001128The exponens consonantiae of a partition. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001432The order dimension of the partition. St001561The value of the elementary symmetric function evaluated at 1. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001568The smallest positive integer that does not appear twice in the partition. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001933The largest multiplicity of a part in an integer partition. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001961The sum of the greatest common divisors of all pairs of parts. St000392The length of the longest run of ones in a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000534The number of 2-rises of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000733The row containing the largest entry of a standard tableau. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. 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. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.