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Your data matches 24 different statistics following compositions of up to 3 maps.
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Matching statistic: St000929
(load all 216 compositions to match this statistic)
(load all 216 compositions to match this statistic)
Mp00079: Set partitions āshapeā¶ Integer partitions
St000929: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
St000929: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> 0
{{1},{2}}
=> [1,1]
=> 1
{{1,2,3}}
=> [3]
=> 0
{{1,2},{3}}
=> [2,1]
=> 0
{{1,3},{2}}
=> [2,1]
=> 0
{{1},{2,3}}
=> [2,1]
=> 0
{{1},{2},{3}}
=> [1,1,1]
=> 1
{{1,2,3,4}}
=> [4]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> 0
{{1,2},{3},{4}}
=> [2,1,1]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> 1
{{1,2,3,4,5}}
=> [5]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> 0
Description
The constant term of the character polynomial of an integer partition.
The definition of the character polynomial can be found in [1]. Indeed, this constant term is $0$ for partitions $\lambda \neq 1^n$ and $1$ for $\lambda = 1^n$.
Matching statistic: St001198
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St001198: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St001198: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2] => [1] => [1,0]
=> ? = 0 + 2
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> ? = 1 + 2
{{1,2,3}}
=> [3] => [1] => [1,0]
=> ? = 0 + 2
{{1,2},{3}}
=> [2,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2}}
=> [2,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3}}
=> [1,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,3,4}}
=> [4] => [1] => [1,0]
=> ? = 0 + 2
{{1,2,3},{4}}
=> [3,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3}}
=> [3,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2},{3,4}}
=> [2,2] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3,4},{2}}
=> [3,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4}}
=> [2,2] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,4},{2,3}}
=> [2,2] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1},{2,3,4}}
=> [1,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1},{2},{3,4}}
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
{{1,2,3,4,5}}
=> [5] => [1] => [1,0]
=> ? = 0 + 2
{{1,2,3,4},{5}}
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,3,5},{4}}
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,3},{4,5}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3,5}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,4,5},{2}}
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3,4},{2,5}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,4,5},{2,3}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,5},{2,3,4}}
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3,4,5}}
=> [1,4] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3,4},{5}}
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1,5},{2,3},{4}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1},{2,3,5},{4}}
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,4,5},{2},{3}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,4},{2,5},{3}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
{{1,2,3,4,5,6}}
=> [6] => [1] => [1,0]
=> ? = 0 + 2
{{1,2,3},{4,5,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2,4},{3,5,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2,5},{3,4,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2,6},{3,4,5}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2,5,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,3,5},{2,4,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,3,6},{2,4,5}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,4,5},{2,3,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,4,6},{2,3,5}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,4},{2,3},{5,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,5,6},{2,3,4}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,5},{2,3},{4,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,6},{2,3},{4,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,4},{2,5},{3,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,4},{2,6},{3,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,5},{2,4},{3,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,6},{2,4},{3,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,5},{2,6},{3,4}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,6},{2,5},{3,4}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,2,3,4,5,6,7}}
=> [7] => [1] => [1,0]
=> ? = 0 + 2
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,2},{3,4},{5,6},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5,6},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3},{5,6},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2,3},{4,6},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,6},{2,3},{4,5},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,7},{2,3},{4,5},{6,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,8},{2,3},{4,5},{6,7}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,8},{2,4},{3,5},{6,7}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,7},{2,4},{3,5},{6,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,6},{2,4},{3,5},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
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 $eA$.
Matching statistic: St001206
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St001206: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
St001206: Dyck paths ā¶ ā¤Result quality: 50% āvalues known / values provided: 86%ādistinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2] => [1] => [1,0]
=> ? = 0 + 2
{{1},{2}}
=> [1,1] => [2] => [1,1,0,0]
=> ? = 1 + 2
{{1,2,3}}
=> [3] => [1] => [1,0]
=> ? = 0 + 2
{{1,2},{3}}
=> [2,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2}}
=> [2,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3}}
=> [1,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2},{3}}
=> [1,1,1] => [3] => [1,1,1,0,0,0]
=> ? = 1 + 2
{{1,2,3,4}}
=> [4] => [1] => [1,0]
=> ? = 0 + 2
{{1,2,3},{4}}
=> [3,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3}}
=> [3,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2},{3,4}}
=> [2,2] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2},{3},{4}}
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3,4},{2}}
=> [3,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4}}
=> [2,2] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,3},{2},{4}}
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,4},{2,3}}
=> [2,2] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1},{2,3,4}}
=> [1,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3},{4}}
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1,4},{2},{3}}
=> [2,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1},{2,4},{3}}
=> [1,2,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1},{2},{3,4}}
=> [1,1,2] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 1 + 2
{{1,2,3,4,5}}
=> [5] => [1] => [1,0]
=> ? = 0 + 2
{{1,2,3,4},{5}}
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,3,5},{4}}
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,3},{4,5}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,4,5},{3}}
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3,5}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2,5},{3,4}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2},{3,4,5}}
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,3,4,5},{2}}
=> [4,1] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3,4},{2,5}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3,5},{2,4}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4,5}}
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,3] => [1,0,1,1,1,0,0,0]
=> 2 = 0 + 2
{{1,4,5},{2,3}}
=> [3,2] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,4},{2,3,5}}
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,5},{2,3,4}}
=> [2,3] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3,4,5}}
=> [1,4] => [1,1] => [1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3,4},{5}}
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1,5},{2,3},{4}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1},{2,3,5},{4}}
=> [1,3,1] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,1,2] => [1,0,1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,4,5},{2},{3}}
=> [3,1,1] => [1,2] => [1,0,1,1,0,0]
=> 2 = 0 + 2
{{1,4},{2,5},{3}}
=> [2,2,1] => [2,1] => [1,1,0,0,1,0]
=> 2 = 0 + 2
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,1] => [1,0,1,0,1,0]
=> 2 = 0 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [5] => [1,1,1,1,1,0,0,0,0,0]
=> ? = 1 + 2
{{1,2,3,4,5,6}}
=> [6] => [1] => [1,0]
=> ? = 0 + 2
{{1,2,3},{4,5,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2,4},{3,5,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2,5},{3,4,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2,6},{3,4,5}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2,5,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,3,5},{2,4,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,3,6},{2,4,5}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,4,5},{2,3,6}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,4,6},{2,3,5}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,4},{2,3},{5,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,5,6},{2,3,4}}
=> [3,3] => [2] => [1,1,0,0]
=> ? = 0 + 2
{{1,5},{2,3},{4,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,6},{2,3},{4,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,4},{2,5},{3,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,4},{2,6},{3,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,5},{2,4},{3,6}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,6},{2,4},{3,5}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,5},{2,6},{3,4}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1,6},{2,5},{3,4}}
=> [2,2,2] => [3] => [1,1,1,0,0,0]
=> ? = 0 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,2,3,4,5,6,7}}
=> [7] => [1] => [1,0]
=> ? = 0 + 2
{{1},{2},{3},{4},{5},{6},{7}}
=> [1,1,1,1,1,1,1] => [7] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 1 + 2
{{1,2},{3,4},{5,6},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5,6},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3},{5,6},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2,3},{4,6},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,6},{2,3},{4,5},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,7},{2,3},{4,5},{6,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,8},{2,3},{4,5},{6,7}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,8},{2,4},{3,5},{6,7}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,7},{2,4},{3,5},{6,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,6},{2,4},{3,5},{7,8}}
=> [2,2,2,2] => [4] => [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
Description
The 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 $eA$.
Matching statistic: St001139
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00121: Dyck paths āCori-Le Borgne involutionā¶ Dyck paths
St001139: Dyck paths ā¶ ā¤Result quality: 85% āvalues known / values provided: 85%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00121: Dyck paths āCori-Le Borgne involutionā¶ Dyck paths
St001139: Dyck paths ā¶ ā¤Result quality: 85% āvalues known / values provided: 85%ādistinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,0,1,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [1,1,0,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 0
{{1},{2,3,4,5,6,7,8}}
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,3,4,5,6,7,8},{2}}
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,4,5,6,7,8},{3}}
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,5,6,7,8},{4}}
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,6,7,8},{5}}
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,7,8},{6}}
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,6,7},{8}}
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,6,8},{7}}
=> [7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8,9}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6},{7,9},{8}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6,9},{7},{8}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5,9},{6},{7},{8}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4,9},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3,9},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2,9},{3},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6},{7,8,9}}
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6},{7},{8,10},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6,9},{7,8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6},{7,10},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4},{5,9},{6,7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6,10},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3},{4,9},{5,6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4},{5,10},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2},{3,9},{4,5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4,10},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1},{2,9},{3,4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3,10},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1,9},{2,3},{4},{5},{6},{7},{8}}
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2,10},{3},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,6,7,8},{9}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2,3,4,5,6,7,8,9}}
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,6,7,8,9},{10}}
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,6,8},{7},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2,3,4,5,6,7,9},{8}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2,3,4,5,6,7,8,9,10}}
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,6,7,8,9,10},{11}}
=> [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,6,7,9},{8},{10}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3,4,5,8},{6,7},{9}}
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 0
{{1,2,3,4,5,7,8},{6},{9}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2,3,4,5,6,9},{7,8}}
=> [6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0,1,0]
=> ? = 0
{{1},{2,3,4,5,6,8,9},{7}}
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2,3,4,5,6,7,8,10},{9}}
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2,3,4,5,6,7,8,9,10,11}}
=> [10,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
{{1,3},{2},{4},{5},{6},{7},{8},{9}}
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 0
Description
The number of occurrences of hills of size 2 in a Dyck path.
A hill of size two is a subpath beginning at height zero, consisting of two up steps followed by two down steps.
Matching statistic: St000264
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St000264: Graphs ā¶ ā¤Result quality: 50% āvalues known / values provided: 68%ādistinct values known / distinct values provided: 50%
Mp00133: Integer compositions ādelta morphismā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St000264: Graphs ā¶ ā¤Result quality: 50% āvalues known / values provided: 68%ādistinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2] => [1] => ([],1)
=> ? = 0 + 3
{{1},{2}}
=> [1,1] => [2] => ([],2)
=> ? = 1 + 3
{{1,2,3}}
=> [3] => [1] => ([],1)
=> ? = 0 + 3
{{1,2},{3}}
=> [2,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,3},{2}}
=> [2,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1},{2,3}}
=> [1,2] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1},{2},{3}}
=> [1,1,1] => [3] => ([],3)
=> ? = 1 + 3
{{1,2,3,4}}
=> [4] => [1] => ([],1)
=> ? = 0 + 3
{{1,2,3},{4}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2,4},{3}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2},{3,4}}
=> [2,2] => [2] => ([],2)
=> ? = 0 + 3
{{1,2},{3},{4}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1,3,4},{2}}
=> [3,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,3},{2,4}}
=> [2,2] => [2] => ([],2)
=> ? = 0 + 3
{{1,3},{2},{4}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1,4},{2,3}}
=> [2,2] => [2] => ([],2)
=> ? = 0 + 3
{{1},{2,3,4}}
=> [1,3] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1},{2,3},{4}}
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,4},{2},{3}}
=> [2,1,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1},{2,4},{3}}
=> [1,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2},{3,4}}
=> [1,1,2] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 3
{{1,2,3,4,5}}
=> [5] => [1] => ([],1)
=> ? = 0 + 3
{{1,2,3,4},{5}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3,5},{4}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3},{4,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1,2,4,5},{3}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2,4},{3,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1,2,5},{3,4}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2},{3,4,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 0 + 3
{{1,3,4,5},{2}}
=> [4,1] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,3,4},{2,5}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1,3,5},{2,4}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,3},{2,4,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,3] => ([(2,3)],4)
=> ? = 0 + 3
{{1,4,5},{2,3}}
=> [3,2] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,4},{2,3,5}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1,5},{2,3,4}}
=> [2,3] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1},{2,3,4,5}}
=> [1,4] => [1,1] => ([(0,1)],2)
=> ? = 0 + 3
{{1},{2,3,4},{5}}
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,5},{2,3},{4}}
=> [2,2,1] => [2,1] => ([(0,2),(1,2)],3)
=> ? = 0 + 3
{{1},{2,3,5},{4}}
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 3
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2,4,5},{3}}
=> [1,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,5},{2},{3,4}}
=> [2,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,2,3},{4,5},{6}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3},{4,6},{5}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,3},{4},{5,6}}
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4},{3,5},{6}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4},{3,6},{5}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,4},{3},{5,6}}
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,5},{3,4},{6}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3,4,5},{6}}
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,6},{3,4},{5}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3,4,6},{5}}
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,5},{3,6},{4}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,5},{3},{4,6}}
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,6},{3,5},{4}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3,5,6},{4}}
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2,6},{3},{4,5}}
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3},{4,5,6}}
=> [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,3,4},{2,5},{6}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,4},{2,6},{5}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,4},{2},{5,6}}
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,5},{2,4},{6}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3},{2,4,5},{6}}
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,6},{2,4},{5}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3},{2,4,6},{5}}
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,5},{2,6},{4}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,5},{2},{4,6}}
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,6},{2,5},{4}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3},{2,5,6},{4}}
=> [2,3,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3,6},{2},{4,5}}
=> [3,1,2] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3},{2},{4,5,6}}
=> [2,1,3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
{{1,3},{2},{4,5},{6}}
=> [2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,3},{2},{4,6},{5}}
=> [2,1,2,1] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,3},{2},{4},{5,6}}
=> [2,1,1,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 0 + 3
{{1,4,5},{2,3},{6}}
=> [3,2,1] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 0 + 3
Description
The girth of a graph, which is not a tree.
This is the length of the shortest cycle in the graph.
Matching statistic: St000781
Mp00221: Set partitions āconjugateā¶ Set partitions
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000781: Integer partitions ā¶ ā¤Result quality: 50% āvalues known / values provided: 65%ādistinct values known / distinct values provided: 50%
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000781: Integer partitions ā¶ ā¤Result quality: 50% āvalues known / values provided: 65%ādistinct values known / distinct values provided: 50%
Values
{{1,2}}
=> {{1},{2}}
=> [1,1]
=> [1]
=> 1 = 0 + 1
{{1},{2}}
=> {{1,2}}
=> [2]
=> []
=> ? = 1 + 1
{{1,2,3}}
=> {{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1]
=> [1]
=> 1 = 0 + 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [2,1]
=> [1]
=> 1 = 0 + 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1]
=> [1]
=> 1 = 0 + 1
{{1},{2},{3}}
=> {{1,2,3}}
=> [3]
=> []
=> ? = 1 + 1
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> [3,1]
=> [1]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [2,2]
=> [2]
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> [2,2]
=> [2]
=> 1 = 0 + 1
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> [3,1]
=> [1]
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> [3,1]
=> [1]
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> [2,2]
=> [2]
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> [3,1]
=> [1]
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> [4]
=> []
=> ? = 1 + 1
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> 1 = 0 + 1
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,5},{2,3,4}}
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> [5]
=> []
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6}}
=> {{1,2,3,4,5,6}}
=> [6]
=> []
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> {{1,2,3,4,5,6,7}}
=> [7]
=> []
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1,2,3,4,5,6,7,8}}
=> [8]
=> []
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,6},{7,8}}
=> {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,7},{6},{8}}
=> {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> {{1,2,5,6,7,8},{3},{4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,8},{6,7}}
=> {{1,5,6,7,8},{2,4},{3}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> {{1,5,6,7,8},{2,3},{4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4,5},{6},{7,8}}
=> {{1,3,4,6,7,8},{2},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4,8},{5,6},{7}}
=> {{1,6,7,8},{2,3,5},{4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4},{5},{6},{7,8}}
=> {{1,3,4,5,7,8},{2},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4},{5},{6,8},{7}}
=> {{1,4,5,7,8},{2,3},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4},{5,6}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,5,7,8},{4},{6}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4},{5,6,7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,5,8},{4},{6,7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,5,6,7,8},{4}}
=> {{1,7,8},{2},{3},{4},{5,6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4,5},{6,7,8}}
=> {{1,4,7,8},{2},{3},{5},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,5},{6},{7}}
=> {{1,7,8},{2,3,4,6},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,6,7,8},{4,5}}
=> {{1,7,8},{2},{3},{4,6},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,6},{5},{7}}
=> {{1,7,8},{2,3,6},{4,5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4,6},{5}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4,5,6}}
=> {{1,7,8},{2},{3,6},{4},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,7},{5,6}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4,5,8},{6},{7}}
=> {{1,7,8},{2,3,4},{5},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4,5,8},{6,7}}
=> {{1,7,8},{2,4},{3},{5},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4,5,6,8},{7}}
=> {{1,7,8},{2,3},{4},{5},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,6,7},{5},{8}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,6,7,8},{5}}
=> {{1,6,8},{2},{3},{4,5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,5,6},{7,8}}
=> {{1,3,6,8},{2},{4},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,8},{5,6},{7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,5,8},{6},{7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,8},{5,6,7}}
=> {{1,6,8},{2,5},{3},{4},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,5,8},{6,7}}
=> {{1,6,8},{2,4},{3},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4},{3},{5},{6,7,8}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4},{3},{5,7,8},{6}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4},{3},{5,6,7,8}}
=> {{1,5,8},{2},{3},{4},{6,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,5,8},{3},{4},{6},{7}}
=> {{1,8},{2,3,4},{5,6,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3},{4},{5,6,7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,5,6,8},{3},{4},{7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4,5},{3},{6,8},{7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4,5},{3},{6,7,8}}
=> {{1,4,8},{2},{3},{5},{6,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,6},{3},{4,5},{7,8}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001901
Mp00221: Set partitions āconjugateā¶ Set partitions
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St001901: Integer partitions ā¶ ā¤Result quality: 50% āvalues known / values provided: 65%ādistinct values known / distinct values provided: 50%
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St001901: Integer partitions ā¶ ā¤Result quality: 50% āvalues known / values provided: 65%ādistinct values known / distinct values provided: 50%
Values
{{1,2}}
=> {{1},{2}}
=> [1,1]
=> [1]
=> 1 = 0 + 1
{{1},{2}}
=> {{1,2}}
=> [2]
=> []
=> ? = 1 + 1
{{1,2,3}}
=> {{1},{2},{3}}
=> [1,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3}}
=> {{1,2},{3}}
=> [2,1]
=> [1]
=> 1 = 0 + 1
{{1,3},{2}}
=> {{1},{2,3}}
=> [2,1]
=> [1]
=> 1 = 0 + 1
{{1},{2,3}}
=> {{1,3},{2}}
=> [2,1]
=> [1]
=> 1 = 0 + 1
{{1},{2},{3}}
=> {{1,2,3}}
=> [3]
=> []
=> ? = 1 + 1
{{1,2,3,4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,3},{4}}
=> {{1,2},{3},{4}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2,4},{3}}
=> {{1},{2,3},{4}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3,4}}
=> {{1,3},{2},{4}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> {{1,2,3},{4}}
=> [3,1]
=> [1]
=> 1 = 0 + 1
{{1,3,4},{2}}
=> {{1},{2},{3,4}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,3},{2,4}}
=> {{1,3},{2,4}}
=> [2,2]
=> [2]
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> {{1,2},{3,4}}
=> [2,2]
=> [2]
=> 1 = 0 + 1
{{1,4},{2,3}}
=> {{1},{2,4},{3}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1},{2,3,4}}
=> {{1,4},{2},{3}}
=> [2,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1},{2,3},{4}}
=> {{1,2,4},{3}}
=> [3,1]
=> [1]
=> 1 = 0 + 1
{{1,4},{2},{3}}
=> {{1},{2,3,4}}
=> [3,1]
=> [1]
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> {{1,4},{2,3}}
=> [2,2]
=> [2]
=> 1 = 0 + 1
{{1},{2},{3,4}}
=> {{1,3,4},{2}}
=> [3,1]
=> [1]
=> 1 = 0 + 1
{{1},{2},{3},{4}}
=> {{1,2,3,4}}
=> [4]
=> []
=> ? = 1 + 1
{{1,2,3,4,5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
{{1,2,3,4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> {{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> {{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> {{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> {{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> {{1},{2,3,4},{5}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> {{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> {{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> {{1,2,3,4},{5}}
=> [4,1]
=> [1]
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,3,4},{2,5}}
=> {{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> {{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3,5},{2,4}}
=> {{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> {{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> {{1,2,4},{3,5}}
=> [3,2]
=> [2]
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> {{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2,5},{4}}
=> {{1,4},{2,3,5}}
=> [3,2]
=> [2]
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> {{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> {{1,2,3},{4,5}}
=> [3,2]
=> [2]
=> 1 = 0 + 1
{{1,4,5},{2,3}}
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> {{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> {{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
{{1,5},{2,3,4}}
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1},{2,3,4,5}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1},{2,3,4},{5}}
=> {{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,1]
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> {{1,2,3,4,5}}
=> [5]
=> []
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6}}
=> {{1,2,3,4,5,6}}
=> [6]
=> []
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6},{7}}
=> {{1,2,3,4,5,6,7}}
=> [7]
=> []
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1,2,3,4,5,6,7,8}}
=> [8]
=> []
=> ? = 1 + 1
{{1},{2},{3},{4},{5},{6,7,8}}
=> {{1,4,5,6,7,8},{2},{3}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,6},{7,8}}
=> {{1,3,5,6,7,8},{2},{4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,7},{6},{8}}
=> {{1,2,5,6,7,8},{3,4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,7,8},{6}}
=> {{1,5,6,7,8},{2},{3,4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,6,7},{8}}
=> {{1,2,5,6,7,8},{3},{4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,8},{6,7}}
=> {{1,5,6,7,8},{2,4},{3}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4},{5,6,8},{7}}
=> {{1,5,6,7,8},{2,3},{4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4,5},{6},{7,8}}
=> {{1,3,4,6,7,8},{2},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3},{4,8},{5,6},{7}}
=> {{1,6,7,8},{2,3,5},{4}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4},{5},{6},{7,8}}
=> {{1,3,4,5,7,8},{2},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4},{5},{6,8},{7}}
=> {{1,4,5,7,8},{2,3},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4},{5},{6},{7}}
=> {{1,7,8},{2,3,4,5,6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4},{5,6}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,5,7,8},{4},{6}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4},{5,6,7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,5,8},{4},{6,7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,5,6,7,8},{4}}
=> {{1,7,8},{2},{3},{4},{5,6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4,5},{6,7,8}}
=> {{1,4,7,8},{2},{3},{5},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,5},{6},{7}}
=> {{1,7,8},{2,3,4,6},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,6,7,8},{4,5}}
=> {{1,7,8},{2},{3},{4,6},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,6},{5},{7}}
=> {{1,7,8},{2,3,6},{4,5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4,6},{5}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4,5,6}}
=> {{1,7,8},{2},{3,6},{4},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,7},{5,6}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4,5,8},{6},{7}}
=> {{1,7,8},{2,3,4},{5},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4,5,8},{6,7}}
=> {{1,7,8},{2,4},{3},{5},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,4,5,6,8},{7}}
=> {{1,7,8},{2,3},{4},{5},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,6,7},{5},{8}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,6,7,8},{5}}
=> {{1,6,8},{2},{3},{4,5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,5,6},{7,8}}
=> {{1,3,6,8},{2},{4},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,8},{5,6},{7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,5,8},{6},{7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,8},{5,6,7}}
=> {{1,6,8},{2,5},{3},{4},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3},{4,5,8},{6,7}}
=> {{1,6,8},{2,4},{3},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4},{3},{5},{6,7,8}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4},{3},{5,7,8},{6}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4},{3},{5,6,7,8}}
=> {{1,5,8},{2},{3},{4},{6,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,5,8},{3},{4},{6},{7}}
=> {{1,8},{2,3,4},{5,6,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3},{4},{5,6,7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,5,6,8},{3},{4},{7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4,5},{3},{6,8},{7}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,4,5},{3},{6,7,8}}
=> {{1,4,8},{2},{3},{5},{6,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,6},{3},{4,5},{7,8}}
=> ?
=> ?
=> ?
=> ? = 0 + 1
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St001204
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00296: Dyck paths āKnuth-Krattenthalerā¶ Dyck paths
St001204: Dyck paths ā¶ ā¤Result quality: 60% āvalues known / values provided: 60%ādistinct values known / distinct values provided: 100%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00296: Dyck paths āKnuth-Krattenthalerā¶ Dyck paths
St001204: Dyck paths ā¶ ā¤Result quality: 60% āvalues known / values provided: 60%ādistinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
{{1},{2}}
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
{{1,2,3}}
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
{{1,2},{3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
{{1,3},{2}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
{{1},{2,3}}
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 0
{{1},{2},{3}}
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 1
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3},{4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,2,4},{3}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,2},{3,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
{{1,2},{3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1,3,4},{2}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1,3},{2,4}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
{{1,3},{2},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1,4},{2,3}}
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> 0
{{1},{2,3,4}}
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 0
{{1},{2,3},{4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1,4},{2},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1},{2,4},{3}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1},{2},{3,4}}
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 0
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
{{1,2,3},{4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,2,3},{4},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
{{1,2,4},{3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,2,4},{3},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
{{1,2,5},{3,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,2},{3,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
{{1,2,5},{3},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 0
{{1,3,4},{2,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,3,4},{2},{5}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
{{1,3,5},{2,4}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,3},{2,4,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
{{1,3,5},{2},{4}}
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 0
{{1,4,5},{2,3}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,4},{2,3,5}}
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 0
{{1,2,3,4,5,6,7}}
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,5,6},{7}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,5,7},{6}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,5},{6},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,6,7},{5}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,6},{5},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4,7},{5},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,5,6,7},{4}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,5,6},{4},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,5,7},{4},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,6,7},{4},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,4,5,6,7},{3}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,4,5,6},{3},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,4,5,7},{3},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,4,6,7},{3},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,4,7},{3},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,4},{3},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,5,6,7},{3},{4}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,5,6},{3},{4},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,5,7},{3},{4},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,5},{3},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,6,7},{3},{4},{5}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,6},{3},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2,7},{3},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,4,5,6,7},{2}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,4,5,6},{2},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,4,5,7},{2},{6}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,4,5},{2},{6},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,4,6,7},{2},{5}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,4,6},{2},{5},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,4,7},{2},{5},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,4},{2},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,5,6,7},{2},{4}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,5,6},{2},{4},{7}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,5,7},{2},{4},{6}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,5},{2},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,6,7},{2},{4},{5}}
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,6},{2},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3,7},{2},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1,3},{2},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 0
{{1},{2,3,4,5,6,7}}
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 0
{{1},{2,3,4,5,6},{7}}
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> ? = 0
Description
Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra.
Associate to this special CNakayama algebra a Dyck path as follows:
In the list L delete the first entry $c_0$ and substract from all other entries $n$ā1 and then append the last element 1. The result is a Kupisch series of an LNakayama algebra.
The statistic gives the $(t-1)/2$ when $t$ is the projective dimension of the simple module $S_{n-2}$.
Matching statistic: St000546
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St000546: Permutations ā¶ ā¤Result quality: 31% āvalues known / values provided: 31%ādistinct values known / distinct values provided: 100%
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
Mp00129: Dyck paths āto 321-avoiding permutation (Billey-Jockusch-Stanley)ā¶ Permutations
St000546: Permutations ā¶ ā¤Result quality: 31% āvalues known / values provided: 31%ādistinct values known / distinct values provided: 100%
Values
{{1,2}}
=> [2] => [1,1,0,0]
=> [1,2] => 0
{{1},{2}}
=> [1,1] => [1,0,1,0]
=> [2,1] => 1
{{1,2,3}}
=> [3] => [1,1,1,0,0,0]
=> [1,2,3] => 0
{{1,2},{3}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 0
{{1,3},{2}}
=> [2,1] => [1,1,0,0,1,0]
=> [1,3,2] => 0
{{1},{2,3}}
=> [1,2] => [1,0,1,1,0,0]
=> [2,1,3] => 0
{{1},{2},{3}}
=> [1,1,1] => [1,0,1,0,1,0]
=> [2,3,1] => 1
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
{{1,2,3},{4}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
{{1,2,4},{3}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
{{1,2},{3,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
{{1,2},{3},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
{{1,3,4},{2}}
=> [3,1] => [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
{{1,3},{2,4}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
{{1,3},{2},{4}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
{{1,4},{2,3}}
=> [2,2] => [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 0
{{1},{2,3,4}}
=> [1,3] => [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
{{1},{2,3},{4}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
{{1,4},{2},{3}}
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 0
{{1},{2,4},{3}}
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
{{1},{2},{3,4}}
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 0
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => 1
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
{{1,2,3},{4,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
{{1,2,3},{4},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
{{1,2,4},{3,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
{{1,2,4},{3},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
{{1,2,5},{3,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
{{1,2},{3,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
{{1,2,5},{3},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
{{1,3,4},{2,5}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
{{1,3,4},{2},{5}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
{{1,3,5},{2,4}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
{{1,3},{2,4,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
{{1,3,5},{2},{4}}
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,3,4,5,2] => 0
{{1,4,5},{2,3}}
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => 0
{{1,4},{2,3,5}}
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => 0
{{1},{2,3,4,5,6},{7}}
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 0
{{1},{2,3,4,5,7},{6}}
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 0
{{1},{2,3,4,5},{6,7}}
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
{{1},{2,3,4,5},{6},{7}}
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
{{1},{2,3,4,6,7},{5}}
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 0
{{1},{2,3,4,6},{5,7}}
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
{{1},{2,3,4,6},{5},{7}}
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
{{1},{2,3,4,7},{5,6}}
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
{{1},{2,3,4},{5,6,7}}
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
{{1},{2,3,4},{5,6},{7}}
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
{{1},{2,3,4,7},{5},{6}}
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
{{1},{2,3,4},{5,7},{6}}
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
{{1},{2,3,4},{5},{6,7}}
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 0
{{1},{2,3,4},{5},{6},{7}}
=> [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => ? = 0
{{1},{2,3,5,6,7},{4}}
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [2,1,3,4,5,7,6] => ? = 0
{{1},{2,3,5,6},{4,7}}
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
{{1},{2,3,5,6},{4},{7}}
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
{{1},{2,3,5,7},{4,6}}
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
{{1},{2,3,5},{4,6,7}}
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
{{1},{2,3,5},{4,6},{7}}
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
{{1},{2,3,5,7},{4},{6}}
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
{{1},{2,3,5},{4,7},{6}}
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
{{1},{2,3,5},{4},{6,7}}
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 0
{{1},{2,3,5},{4},{6},{7}}
=> [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => ? = 0
{{1},{2,3,6,7},{4,5}}
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [2,1,3,4,6,5,7] => ? = 0
{{1},{2,3,6},{4,5,7}}
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
{{1},{2,3,6},{4,5},{7}}
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
{{1},{2,3,7},{4,5,6}}
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [2,1,3,5,4,6,7] => ? = 0
{{1},{2,3},{4,5,6,7}}
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,4,3,5,6,7] => ? = 0
{{1},{2,3},{4,5,6},{7}}
=> [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 0
{{1},{2,3,7},{4,5},{6}}
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
{{1},{2,3},{4,5,7},{6}}
=> [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 0
{{1},{2,3},{4,5},{6,7}}
=> [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 0
{{1},{2,3},{4,5},{6},{7}}
=> [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ? = 0
{{1},{2,3,6,7},{4},{5}}
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [2,1,3,4,6,7,5] => ? = 0
{{1},{2,3,6},{4,7},{5}}
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
{{1},{2,3,6},{4},{5,7}}
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 0
{{1},{2,3,6},{4},{5},{7}}
=> [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => ? = 0
{{1},{2,3,7},{4,6},{5}}
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [2,1,3,5,4,7,6] => ? = 0
{{1},{2,3},{4,6,7},{5}}
=> [1,2,3,1] => [1,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [2,1,4,3,5,7,6] => ? = 0
{{1},{2,3},{4,6},{5,7}}
=> [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 0
{{1},{2,3},{4,6},{5},{7}}
=> [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ? = 0
{{1},{2,3,7},{4},{5,6}}
=> [1,3,1,2] => [1,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [2,1,3,5,6,4,7] => ? = 0
{{1},{2,3},{4,7},{5,6}}
=> [1,2,2,2] => [1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,6,5,7] => ? = 0
{{1},{2,3},{4},{5,6,7}}
=> [1,2,1,3] => [1,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [2,1,4,5,3,6,7] => ? = 0
{{1},{2,3},{4},{5,6},{7}}
=> [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => ? = 0
{{1},{2,3,7},{4},{5},{6}}
=> [1,3,1,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [2,1,3,5,6,7,4] => ? = 0
{{1},{2,3},{4,7},{5},{6}}
=> [1,2,2,1,1] => [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [2,1,4,3,6,7,5] => ? = 0
{{1},{2,3},{4},{5,7},{6}}
=> [1,2,1,2,1] => [1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3,7,6] => ? = 0
{{1},{2,3},{4},{5},{6,7}}
=> [1,2,1,1,2] => [1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [2,1,4,5,6,3,7] => ? = 0
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St000260
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00041: Integer compositions āconjugateā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St000260: Graphs ā¶ ā¤Result quality: 21% āvalues known / values provided: 21%ādistinct values known / distinct values provided: 50%
Mp00041: Integer compositions āconjugateā¶ Integer compositions
Mp00184: Integer compositions āto threshold graphā¶ Graphs
St000260: Graphs ā¶ ā¤Result quality: 21% āvalues known / values provided: 21%ādistinct values known / distinct values provided: 50%
Values
{{1,2}}
=> [2] => [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
{{1},{2}}
=> [1,1] => [2] => ([],2)
=> ? = 1 + 1
{{1,2,3}}
=> [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 1 = 0 + 1
{{1,2},{3}}
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
{{1,3},{2}}
=> [2,1] => [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
{{1},{2,3}}
=> [1,2] => [1,2] => ([(1,2)],3)
=> ? = 0 + 1
{{1},{2},{3}}
=> [1,1,1] => [3] => ([],3)
=> ? = 1 + 1
{{1,2,3,4}}
=> [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,3},{4}}
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2,4},{3}}
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2},{3,4}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,2},{3},{4}}
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3,4},{2}}
=> [3,1] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2,4}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,3},{2},{4}}
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1,4},{2,3}}
=> [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,3,4}}
=> [1,3] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? = 0 + 1
{{1},{2,3},{4}}
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
{{1,4},{2},{3}}
=> [2,1,1] => [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
{{1},{2,4},{3}}
=> [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
=> ? = 0 + 1
{{1},{2},{3,4}}
=> [1,1,2] => [1,3] => ([(2,3)],4)
=> ? = 0 + 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [4] => ([],4)
=> ? = 1 + 1
{{1,2,3,4,5}}
=> [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 = 0 + 1
{{1,2,3,4},{5}}
=> [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)
=> 1 = 0 + 1
{{1,2,3,5},{4}}
=> [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)
=> 1 = 0 + 1
{{1,2,3},{4,5}}
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,4,5},{3}}
=> [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)
=> 1 = 0 + 1
{{1,2,4},{3,5}}
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,5},{3,4}}
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3,4,5}}
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3,4,5},{2}}
=> [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)
=> 1 = 0 + 1
{{1,3,4},{2,5}}
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3,4},{2},{5}}
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3,5},{2,4}}
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2,4,5}}
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2,4},{5}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3,5},{2},{4}}
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2,5},{4}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4,5},{2,3}}
=> [3,2] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2,3,5}}
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2,3},{5}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,5},{2,3,4}}
=> [2,3] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2,3,4,5}}
=> [1,4] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2,3,4},{5}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1,5},{2,3},{4}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2,3,5},{4}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1,4,5},{2},{3}}
=> [3,1,1] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2,5},{3}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1,5},{2,4},{3}}
=> [2,2,1] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2,4,5},{3}}
=> [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2,4},{3,5}}
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1,5},{2},{3,4}}
=> [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
{{1},{2,5},{3,4}}
=> [1,2,2] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2},{3,4,5}}
=> [1,1,3] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [2,3] => ([(2,4),(3,4)],5)
=> ? = 0 + 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,4] => ([(3,4)],5)
=> ? = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [5] => ([],5)
=> ? = 1 + 1
{{1},{2,3,4,5,6}}
=> [1,5] => [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 + 1
{{1},{2,3,4,5},{6}}
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3,4,6},{5}}
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3,4},{5,6}}
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3,5,6},{4}}
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3,5},{4,6}}
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3,5},{4},{6}}
=> [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3,6},{4,5}}
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3},{4,5,6}}
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3,6},{4},{5}}
=> [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3},{4,6},{5}}
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3},{4},{5,6}}
=> [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,3},{4},{5},{6}}
=> [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4,5,6},{3}}
=> [1,4,1] => [2,1,1,2] => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4,5},{3,6}}
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4,5},{3},{6}}
=> [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4,6},{3,5}}
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4},{3,5,6}}
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4},{3,5},{6}}
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4,6},{3},{5}}
=> [1,3,1,1] => [3,1,2] => ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4},{3,6},{5}}
=> [1,2,2,1] => [2,2,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4},{3},{5,6}}
=> [1,2,1,2] => [1,3,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,4},{3},{5},{6}}
=> [1,2,1,1,1] => [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,5,6},{3,4}}
=> [1,3,2] => [1,2,1,2] => ([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
{{1},{2,5},{3,4,6}}
=> [1,2,3] => [1,1,2,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 0 + 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000234The number of global ascents of a permutation. St000259The diameter of a connected graph. St000455The second largest eigenvalue of a graph if it is integral. St000315The number of isolated vertices of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. 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. St001691The number of kings in a graph. St000056The decomposition (or block) number of a permutation. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching.
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