Your data matches 13 different statistics following compositions of up to 3 maps.
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Matching statistic: St000392
(load all 3 compositions to match this statistic)
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00094: Integer compositions —to binary word⟶ Binary words
St000392: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => 1 => 1
[1,0,1,0]
 => [1,1] => [1,1] => 11 => 2
[1,1,0,0]
 => [2] => [2] => 10 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1] => 111 => 3
[1,0,1,1,0,0]
 => [1,2] => [2,1] => 101 => 1
[1,1,0,0,1,0]
 => [2,1] => [1,2] => 110 => 2
[1,1,0,1,0,0]
 => [3] => [3] => 100 => 1
[1,1,1,0,0,0]
 => [3] => [3] => 100 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1] => 1111 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,2,1] => 1101 => 2
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1] => 1011 => 2
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1] => 1001 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1] => 1001 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,2] => 1110 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2] => 1010 => 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [1,3] => 1100 => 2
[1,1,0,1,0,1,0,0]
 => [4] => [4] => 1000 => 1
[1,1,0,1,1,0,0,0]
 => [4] => [4] => 1000 => 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,3] => 1100 => 2
[1,1,1,0,0,1,0,0]
 => [4] => [4] => 1000 => 1
[1,1,1,0,1,0,0,0]
 => [4] => [4] => 1000 => 1
[1,1,1,1,0,0,0,0]
 => [4] => [4] => 1000 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1] => 11111 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,1,2,1] => 11101 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,2,1,1] => 11011 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,3,1] => 11001 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,3,1] => 11001 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1] => 10111 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => 10101 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1] => 10011 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1] => 10001 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1] => 10001 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1] => 10011 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1] => 10001 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1] => 10001 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1] => 10001 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => 11110 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,2,2] => 11010 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1,2] => 10110 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2] => 10010 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2] => 10010 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [1,1,3] => 11100 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [2,3] => 10100 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [1,4] => 11000 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5] => 10000 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5] => 10000 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [1,4] => 11000 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5] => 10000 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5] => 10000 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5] => 10000 => 1
Description
The length of the longest run of ones in a binary word.
Matching statistic: St000723
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000723: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => ([],1)
 => 1
[1,0,1,0]
 => [1,1] => [2] => ([],2)
 => 2
[1,1,0,0]
 => [2] => [1,1] => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => ([],3)
 => 3
[1,0,1,1,0,0]
 => [1,2] => [1,2] => ([(1,2)],3)
 => 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => ([(0,2),(1,2)],3)
 => 2
[1,1,0,1,0,0]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => ([],4)
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => ([(2,3)],4)
 => 2
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => ([],5)
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => ([(3,4)],5)
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 1
Description
The maximal cardinality of a set of vertices with the same neighbourhood in a graph. The set of so called mating graphs, for which this statistic equals $1$, is enumerated by [1].
Matching statistic: St001933
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
St001933: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [[1],[]]
 => [1]
 => 1
[1,0,1,0]
 => [1,1] => [[1,1],[]]
 => [1,1]
 => 2
[1,1,0,0]
 => [2] => [[2],[]]
 => [2]
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [[1,1,1],[]]
 => [1,1,1]
 => 3
[1,0,1,1,0,0]
 => [1,2] => [[2,1],[]]
 => [2,1]
 => 1
[1,1,0,0,1,0]
 => [2,1] => [[2,2],[1]]
 => [2,2]
 => 2
[1,1,0,1,0,0]
 => [3] => [[3],[]]
 => [3]
 => 1
[1,1,1,0,0,0]
 => [3] => [[3],[]]
 => [3]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [[1,1,1,1],[]]
 => [1,1,1,1]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [[2,1,1],[]]
 => [2,1,1]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [[2,2,1],[1]]
 => [2,2,1]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3] => [[3,1],[]]
 => [3,1]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [[3,1],[]]
 => [3,1]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [[2,2,2],[1,1]]
 => [2,2,2]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [[3,2],[1]]
 => [3,2]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 2
[1,1,0,1,0,1,0,0]
 => [4] => [[4],[]]
 => [4]
 => 1
[1,1,0,1,1,0,0,0]
 => [4] => [[4],[]]
 => [4]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [[3,3],[2]]
 => [3,3]
 => 2
[1,1,1,0,0,1,0,0]
 => [4] => [[4],[]]
 => [4]
 => 1
[1,1,1,0,1,0,0,0]
 => [4] => [[4],[]]
 => [4]
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => [[4],[]]
 => [4]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [[1,1,1,1,1],[]]
 => [1,1,1,1,1]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [[2,1,1,1],[]]
 => [2,1,1,1]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [[2,2,1,1],[1]]
 => [2,2,1,1]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [[3,1,1],[]]
 => [3,1,1]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [[2,2,2,1],[1,1]]
 => [2,2,2,1]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [[3,2,1],[1]]
 => [3,2,1]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [[4,1],[]]
 => [4,1]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [[4,1],[]]
 => [4,1]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [[3,3,1],[2]]
 => [3,3,1]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [[4,1],[]]
 => [4,1]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [[4,1],[]]
 => [4,1]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [[4,1],[]]
 => [4,1]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
 => [2,2,2,2]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [[3,2,2],[1,1]]
 => [3,2,2]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [[3,3,2],[2,1]]
 => [3,3,2]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [[4,2],[1]]
 => [4,2]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [[4,2],[1]]
 => [4,2]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [[3,3,3],[2,2]]
 => [3,3,3]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [[4,3],[2]]
 => [4,3]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [[4,4],[3]]
 => [4,4]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [[5],[]]
 => [5]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [[5],[]]
 => [5]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [[4,4],[3]]
 => [4,4]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [[5],[]]
 => [5]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [[5],[]]
 => [5]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [[5],[]]
 => [5]
 => 1
Description
The largest multiplicity of a part in an integer partition.
Matching statistic: St000969
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000969: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1] => [1,1] => [1,0,1,0]
 => 3 = 2 + 1
[1,1,0,0]
 => [2] => [2] => [1,1,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,0]
 => [3] => [3] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3] => [3] => [1,1,1,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 4 = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 3 = 2 + 1
[1,1,1,0,0,1,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 6 = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 4 = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 4 = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3 = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 5 = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 3 = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 4 = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3 = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 2 = 1 + 1
Description
We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. Then we calculate the global dimension of that CNakayama algebra.
Matching statistic: St001330
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St001330: Graphs ⟶ ℤResult quality: 33% ●values known / values provided: 33%●distinct values known / distinct values provided: 100%
Values
[1,0]
 => [1] => ([],1)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0]
 => [1,1] => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 3 = 2 + 1
[1,1,0,0]
 => [2] => ([],2)
 => ([(0,2),(1,2)],3)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 4 = 3 + 1
[1,0,1,1,0,0]
 => [1,2] => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 1 + 1
[1,1,0,0,1,0]
 => [2,1] => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ? = 2 + 1
[1,1,0,1,0,0]
 => [3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3] => ([],3)
 => ([(0,3),(1,3),(2,3)],4)
 => 2 = 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),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5 = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,3] => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,3] => ([(2,3)],4)
 => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,2] => ([(1,3),(2,3)],4)
 => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,1,0,0]
 => [4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4] => ([],4)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 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),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 6 = 5 + 1
[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)
 => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[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)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[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)
 => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => ([(3,4)],5)
 => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[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)
 => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[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)
 => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => ([(2,4),(3,4)],5)
 => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[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)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [5] => ([],5)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => 7 = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,2] => ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,2,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,2,1,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,3,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,4] => ([(3,4),(3,5),(4,5)],6)
 => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [6] => ([],6)
 => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
 => 2 = 1 + 1
Description
The hat guessing number of a graph. Suppose that each vertex of a graph corresponds to a player, wearing a hat whose color is arbitrarily chosen from a set of $q$ possible colors. Each player can see the hat colors of his neighbors, but not his own hat color. All of the players are asked to guess their own hat colors simultaneously, according to a predetermined guessing strategy and the hat colors they see, where no communication between them is allowed. The hat guessing number $HG(G)$ of a graph $G$ is the largest integer $q$ such that there exists a guessing strategy guaranteeing at least one correct guess for any hat assignment of $q$ possible colors. Because it suffices that a single player guesses correctly, the hat guessing number of a graph is the maximum of the hat guessing numbers of its connected components.
Matching statistic: St000456
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00074: Posets —to graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 32%●distinct values known / distinct values provided: 14%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => ? = 1
[1,0,1,0]
 => [2,1] => ([],2)
 => ([],2)
 => ? = 2
[1,1,0,0]
 => [1,2] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
[1,0,1,0,1,0]
 => [3,2,1] => ([],3)
 => ([],3)
 => ? = 3
[1,0,1,1,0,0]
 => [2,3,1] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1
[1,1,0,0,1,0]
 => [3,1,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2
[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)
 => ? = 4
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 2
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 2
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => ([(1,2),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => ([(2,3)],4)
 => ([(2,3)],4)
 => ? = 3
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => ([(1,3),(2,3)],4)
 => ([(1,3),(2,3)],4)
 => ? = 2
[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)
 => ? = 2
[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)
 => ? = 5
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 3
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 2
[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)
 => ? = 2
[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)
 => ? = 2
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 3
[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)
 => ? = 1
[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)
 => ? = 2
[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)
 => ? = 1
[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)
 => ? = 1
[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)
 => ? = 2
[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)
 => ? = 1
[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)
 => ? = 1
[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)
 => ? = 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => ([(3,4)],5)
 => ([(3,4)],5)
 => ? = 4
[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)
 => ? = 2
[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)
 => ? = 2
[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)
 => ? = 1
[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)
 => ? = 1
[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)
 => ? = 3
[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)
 => ? = 1
[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)
 => ? = 2
[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)
 => ? = 2
[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
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => ([(2,3),(3,4)],5)
 => ([(2,4),(3,4)],5)
 => ? = 3
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
 => ([(0,1),(2,4),(3,4)],5)
 => ? = 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => ([(1,4),(2,4),(4,3)],5)
 => ([(1,4),(2,4),(3,4)],5)
 => ? = 2
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => ([(0,4),(1,3),(2,3),(3,4)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => ([(0,4),(1,4),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => ([(1,4),(3,2),(4,3)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
 => ([(0,4),(1,4),(2,3),(3,4)],5)
 => 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => ([],6)
 => ([],6)
 => ? = 6
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => ([(3,4),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => ([(4,5)],6)
 => ([(4,5)],6)
 => ? = 3
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => ([(2,5),(3,4)],6)
 => ([(2,5),(3,4)],6)
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => ([(3,5),(4,5)],6)
 => ([(3,5),(4,5)],6)
 => ? = 2
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,1,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,1,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,1,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,1,6] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => ([(0,5),(1,4),(2,4),(3,5),(4,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => ([(0,5),(1,4),(2,5),(3,2),(4,3)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1,2,6] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [4,5,3,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,4,1,2,6] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,5,1,2,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6] => ([(0,3),(1,4),(2,5),(3,5),(4,2)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,5,2,1,3,6] => ([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,4,6] => ([(0,5),(1,5),(2,5),(3,4),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => ([(0,5),(1,5),(2,5),(3,5),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,2,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,3,1,4,6] => ([(0,5),(1,4),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,3,1,5,6] => ([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,4,1,5,6] => ([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6] => ([(0,5),(1,4),(2,5),(4,2),(5,3)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => 1
Description
The monochromatic index of a connected graph. This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path. For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001024
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00173: Integer compositions —rotate front to back⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001024: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 1
[1,0,1,0]
 => [1,1] => [1,1] => [1,0,1,0]
 => 2
[1,1,0,0]
 => [2] => [2] => [1,1,0,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [1,1,1] => [1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0]
 => [1,2] => [2,1] => [1,1,0,0,1,0]
 => 1
[1,1,0,0,1,0]
 => [2,1] => [1,2] => [1,0,1,1,0,0]
 => 2
[1,1,0,1,0,0]
 => [3] => [3] => [1,1,1,0,0,0]
 => 1
[1,1,1,0,0,0]
 => [3] => [3] => [1,1,1,0,0,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [3,1] => [1,1,1,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [2,2] => [1,1,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => [4] => [1,1,1,1,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => [1,1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => [1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,2,1] => [1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,3] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,3] => [1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,2,1,1] => [1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,2,2] => [1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,3,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,4] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,4] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,3,1] => [1,1,3,1,1] => [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,4] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,4] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,4] => [1,1,4,1] => [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1] => [1,2,1,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,2,1,2] => [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,2,2,1] => [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,2,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,2,3] => [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,3,1,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,3,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,3,1,1] => [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,3,2] => [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,4,1] => [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,5] => [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,2,1,1,1,1] => [2,1,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,1,1,2] => [2,1,1,2,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0]
 => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,2,1,2,1] => [2,1,2,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0]
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,2,1,3] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,2,1,3] => [2,1,3,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0]
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,2,2,1,1] => [2,2,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0]
 => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,2,2,2] => [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
 => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,2,3,1] => [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
 => ? = 2
Description
Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001210
Mapping
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001210: Dyck paths ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 86%
Values
[1,0]
 => [1] => [1] => [1,0]
 => 1
[1,0,1,0]
 => [1,1] => [2] => [1,1,0,0]
 => 2
[1,1,0,0]
 => [2] => [1,1] => [1,0,1,0]
 => 1
[1,0,1,0,1,0]
 => [1,1,1] => [3] => [1,1,1,0,0,0]
 => 3
[1,0,1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0,1,0]
 => 2
[1,1,0,1,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,1,1,0,0,0]
 => [3] => [1,1,1] => [1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0]
 => [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
 => 4
[1,0,1,0,1,1,0,0]
 => [1,1,2] => [1,3] => [1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0]
 => [1,2,1] => [2,2] => [1,1,0,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0]
 => [1,3] => [1,1,2] => [1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0,1,0]
 => [2,1,1] => [3,1] => [1,1,1,0,0,0,1,0]
 => 3
[1,1,0,0,1,1,0,0]
 => [2,2] => [1,2,1] => [1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,0,1,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,1,0,0,0,1,0]
 => [3,1] => [2,1,1] => [1,1,0,0,1,0,1,0]
 => 2
[1,1,1,0,0,1,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,1,0,1,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,1,1,1,0,0,0,0]
 => [4] => [1,1,1,1] => [1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
 => 5
[1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,2] => [1,4] => [1,0,1,1,1,1,0,0,0,0]
 => 3
[1,0,1,0,1,1,0,0,1,0]
 => [1,1,2,1] => [2,3] => [1,1,0,0,1,1,1,0,0,0]
 => 2
[1,0,1,0,1,1,0,1,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,0,1,1,1,0,0,0]
 => [1,1,3] => [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
 => 2
[1,0,1,1,0,0,1,0,1,0]
 => [1,2,1,1] => [3,2] => [1,1,1,0,0,0,1,1,0,0]
 => 3
[1,0,1,1,0,0,1,1,0,0]
 => [1,2,2] => [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,0,1,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,3,1] => [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
 => 2
[1,0,1,1,1,0,0,1,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,4] => [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
 => 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,1,1] => [4,1] => [1,1,1,1,0,0,0,0,1,0]
 => 4
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,2] => [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,0,1,0]
 => [2,2,1] => [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
 => 2
[1,1,0,0,1,1,0,1,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,3] => [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
 => 1
[1,1,0,1,0,0,1,0,1,0]
 => [3,1,1] => [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
 => 3
[1,1,0,1,0,0,1,1,0,0]
 => [3,2] => [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
 => 1
[1,1,0,1,0,1,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,1,0,1,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,0,1,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,1,0,0,0,1,0]
 => [4,1] => [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
 => 2
[1,1,0,1,1,0,0,1,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,1,0,1,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,1,0,1,1,1,0,0,0,0]
 => [5] => [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
 => 1
[1,0,1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 7
[1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,1,2] => [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 5
[1,0,1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,1,2,1] => [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,1,3] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,1,1,1,3] => [1,1,5] => [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => ? = 4
[1,0,1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,1,1,2,1,1] => [3,4] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,1,1,2,2] => [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,1,1,3,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,4] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,0,1,1,0,0,0]
 => [1,1,1,4] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,0,1,0]
 => [1,1,1,3,1] => [2,1,4] => [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,0,0,1,0,0]
 => [1,1,1,4] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,0,1,0,0,0]
 => [1,1,1,4] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => [1,1,1,4] => [1,1,1,4] => [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,0,1,0,1,0,1,0]
 => [1,1,2,1,1,1] => [4,3] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
 => ? = 4
[1,0,1,0,1,1,0,0,1,0,1,1,0,0]
 => [1,1,2,1,2] => [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0,1,0]
 => [1,1,2,2,1] => [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,0,1,0,0]
 => [1,1,2,3] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => [1,1,2,3] => [1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0,1,0]
 => [1,1,3,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,0,1,0,0,1,1,0,0]
 => [1,1,3,2] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,0,1,0]
 => [1,1,4,1] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,0,1,1,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,0,1,0]
 => [1,1,4,1] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,0,1,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,0,1,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,0,1,1,1,0,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,0,1,0,1,0]
 => [1,1,3,1,1] => [3,1,3] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
 => ? = 3
[1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => [1,1,3,2] => [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,0,1,0]
 => [1,1,4,1] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,0,1,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,0,1,1,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,0,1,0]
 => [1,1,4,1] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,0,1,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,0,1,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,0,1,1,0,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,0,1,0]
 => [1,1,4,1] => [2,1,1,3] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,0,1,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,0,1,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,1,0,1,0,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,0,1,1,1,1,1,0,0,0,0,0]
 => [1,1,5] => [1,1,1,1,3] => [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
 => ? = 2
[1,0,1,1,0,0,1,0,1,0,1,0,1,0]
 => [1,2,1,1,1,1] => [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
 => ? = 5
[1,0,1,1,0,0,1,0,1,0,1,1,0,0]
 => [1,2,1,1,2] => [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
 => ? = 3
[1,0,1,1,0,0,1,0,1,1,0,0,1,0]
 => [1,2,1,2,1] => [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,0,1,0,0]
 => [1,2,1,3] => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,0,1,0,1,1,1,0,0,0]
 => [1,2,1,3] => [1,1,3,2] => [1,0,1,0,1,1,1,0,0,0,1,1,0,0]
 => ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0,1,0]
 => [1,2,2,1,1] => [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
 => ? = 3
[1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => [1,2,2,2] => [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
 => ? = 1
[1,0,1,1,0,0,1,1,0,1,0,0,1,0]
 => [1,2,3,1] => [2,1,2,2] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0]
 => ? = 2
Description
Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001545
Mapping
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
Mp00247: Graphs —de-duplicate⟶ Graphs
St001545: Graphs ⟶ ℤResult quality: 14% ●values known / values provided: 21%●distinct values known / distinct values provided: 14%
Values
[1,0]
 => [1] => ([],1)
 => ([],1)
 => ? = 1 + 1
[1,0,1,0]
 => [1,2] => ([],2)
 => ([],1)
 => ? = 2 + 1
[1,1,0,0]
 => [2,1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [1,2,3] => ([],3)
 => ([],1)
 => ? = 3 + 1
[1,0,1,1,0,0]
 => [1,3,2] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,1,0,0,1,0]
 => [2,1,3] => ([(1,2)],3)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,1,0,1,0,0]
 => [2,3,1] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [3,1,2] => ([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [1,2,3,4] => ([],4)
 => ([],1)
 => ? = 4 + 1
[1,0,1,0,1,1,0,0]
 => [1,2,4,3] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0]
 => [1,3,2,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,0,1,0,0]
 => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0]
 => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0]
 => [2,1,3,4] => ([(2,3)],4)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,1,0,0,1,1,0,0]
 => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0]
 => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5] => ([],5)
 => ([],1)
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,5,4] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [1,2,4,3,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [1,2,4,5,3] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [1,3,2,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [1,4,5,2,3] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [2,1,3,4,5] => ([(3,4)],5)
 => ([(1,2)],3)
 => ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ([(1,4),(2,3)],5)
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [2,3,1,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ([(0,3),(1,2)],4)
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
 => ([(1,4),(2,3),(3,4)],5)
 => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [3,4,1,2,5] => ([(1,3),(1,4),(2,3),(2,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,4,5,1,2] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [1,2,3,4,5,6] => ([],6)
 => ([],1)
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [1,2,3,4,6,5] => ([(4,5)],6)
 => ([(1,2)],3)
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [1,2,3,5,4,6] => ([(4,5)],6)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [1,2,3,6,4,5] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [1,2,4,3,5,6] => ([(4,5)],6)
 => ([(1,2)],3)
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [1,2,4,3,6,5] => ([(2,5),(3,4)],6)
 => ([(1,4),(2,3)],5)
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [1,2,4,5,3,6] => ([(3,5),(4,5)],6)
 => ([(1,2)],3)
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [2,3,4,6,1,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [2,3,5,1,6,4] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [2,3,5,6,1,4] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [2,4,1,5,6,3] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [3,1,4,5,6,2] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [3,1,5,6,2,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,1,6,2,4,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [3,4,1,6,2,5] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [3,4,5,6,1,2] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,1)],2)
 => 2 = 1 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,6,1,2,5] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,5,6,1,2,4] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [3,6,1,2,4,5] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,0,1,0,0]
 => [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,0,1,1,0,0,0]
 => [4,1,2,6,3,5] => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,1,0,0,0]
 => [4,1,5,6,2,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,0,1,1,0,0,0,0]
 => [4,1,6,2,3,5] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,1,0,0]
 => [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,1,0,0,0]
 => [4,5,1,6,2,3] => ([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,3),(1,2),(2,3)],4)
 => 2 = 1 + 1
Description
The second Elser number of a connected graph. For a connected graph $G$ the $k$-th Elser number is $$ els_k(G) = (-1)^{|V(G)|+1} \sum_N (-1)^{|E(N)|} |V(N)|^k $$ where the sum is over all nuclei of $G$, that is, the connected subgraphs of $G$ whose vertex set is a vertex cover of $G$. It is clear that this number is even. It was shown in [1] that it is non-negative.
Matching statistic: St001198
Mapping
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001198: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 14%
Values
[1,0]
 => [1] => [1] => [1,0]
 => ? = 1 + 1
[1,0,1,0]
 => [2,1] => [2,1] => [1,1,0,0]
 => ? = 2 + 1
[1,1,0,0]
 => [1,2] => [1,2] => [1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0]
 => [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0]
 => [2,3,1] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,0]
 => [3,1,2] => [3,2,1] => [1,1,1,0,0,0]
 => ? = 2 + 1
[1,1,0,1,0,0]
 => [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0]
 => [1,2,3] => [1,2,3] => [1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0]
 => [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,1,0,0]
 => [3,4,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,0]
 => [4,2,3,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,0]
 => [3,2,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,0]
 => [2,3,4,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,0,1,0]
 => [4,3,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 3 + 1
[1,1,0,0,1,1,0,0]
 => [3,4,1,2] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,0,0,1,0]
 => [4,2,1,3] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,1,0,1,0,1,0,0]
 => [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0]
 => [2,3,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0]
 => [4,1,2,3] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
 => ? = 2 + 1
[1,1,1,0,0,1,0,0]
 => [3,1,2,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0]
 => [2,1,3,4] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0]
 => [1,2,3,4] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
 => [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 5 + 1
[1,0,1,0,1,0,1,1,0,0]
 => [4,5,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0]
 => [5,3,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0]
 => [4,3,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,1,0,0,0]
 => [3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,0,1,0,1,0]
 => [5,4,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,1,0,0,1,1,0,0]
 => [4,5,2,3,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,1,0,0,1,0]
 => [5,3,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,0,1,0,1,0,0]
 => [4,3,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
 => [3,4,2,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
 => [5,2,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0]
 => [4,2,3,5,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
 => [3,2,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
 => [2,3,4,5,1] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
 => [5,4,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 4 + 1
[1,1,0,0,1,0,1,1,0,0]
 => [4,5,3,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,0,1,1,0,0,1,0]
 => [5,3,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,0,1,1,0,1,0,0]
 => [4,3,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
 => [3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
 => [5,4,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,1,0,1,0,0,1,1,0,0]
 => [4,5,2,1,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
 => [5,3,2,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,0]
 => [4,3,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
 => [3,4,2,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
 => [5,2,3,1,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,1,1,0,0,1,0,0]
 => [4,2,3,1,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
 => [3,2,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
 => [2,3,4,1,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
 => [5,4,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 3 + 1
[1,1,1,0,0,0,1,1,0,0]
 => [4,5,1,2,3] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
 => [5,3,1,2,4] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,1,1,0,0,1,0,1,0,0]
 => [4,3,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
 => [3,4,1,2,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
 => [5,2,1,3,4] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,1,1,0,1,0,0,1,0,0]
 => [4,2,1,3,5] => [4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
 => [3,2,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
 => [2,3,1,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,0,0,1,0]
 => [5,1,2,3,4] => [5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
 => ? = 2 + 1
[1,1,1,1,0,0,0,1,0,0]
 => [4,1,2,3,5] => [4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,0,1,0,0,0]
 => [3,1,2,4,5] => [3,2,1,4,5] => [1,1,1,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,0,1,0,0,0,0]
 => [2,1,3,4,5] => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,1,1,0,0,0,0,0]
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
 => [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 6 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
 => [5,6,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 4 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
 => [6,4,5,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
 => [5,4,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
 => [4,5,6,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
 => [6,5,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 3 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
 => [5,6,3,4,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
 => [6,4,3,5,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 2 + 1
[1,1,0,1,0,1,0,1,0,1,0,0]
 => [5,4,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,0,1,1,0,0,0]
 => [4,5,3,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,1,0,0]
 => [5,3,4,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,0,1,0,0,0]
 => [4,3,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
 => [3,4,5,2,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,1,0,0]
 => [5,4,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,0,1,1,0,0,0]
 => [4,5,2,3,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,1,0,0]
 => [5,3,2,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
 => [4,3,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
 => [3,4,2,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
 => [5,2,3,4,1,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
 => [4,2,3,5,1,6] => [5,3,2,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
 => [3,2,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
 => [2,3,4,5,1,6] => [5,2,3,4,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,1,0,0]
 => [5,4,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,0,1,1,0,0,0]
 => [4,5,3,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,1,0,0]
 => [5,3,4,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,0,1,0,0,0]
 => [4,3,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,0,1,1,1,0,0,0,0]
 => [3,4,5,1,2,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,1,0,0]
 => [5,4,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,0,1,1,0,0,0]
 => [4,5,2,1,3,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,1,0,0]
 => [5,3,2,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,0,1,0,0,0]
 => [4,3,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,0,1,1,0,0,0,0]
 => [3,4,2,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,1,0,0]
 => [5,2,3,1,4,6] => [5,4,3,2,1,6] => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,0,1,0,0,0]
 => [4,2,3,1,5,6] => [4,3,2,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,0,1,0,0,0,0]
 => [3,2,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 1 + 1
[1,1,1,0,1,1,1,0,0,0,0,0]
 => [2,3,4,1,5,6] => [4,2,3,1,5,6] => [1,1,1,1,0,0,0,0,1,0,1,0]
 => 2 = 1 + 1
Description
The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
The following 3 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001235The global dimension of the corresponding Comp-Nakayama algebra. St000649The number of 3-excedences of a permutation.