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Your data matches 42 different statistics following compositions of up to 3 maps.
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Matching statistic: St000621
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000621: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000621: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> [2]
=> 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St000183
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00115: Set partitions āKasraoui-Zengā¶ Set partitions
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000183: Integer partitions ā¶ ā¤Result quality: 29% āvalues known / values provided: 73%ādistinct values known / distinct values provided: 29%
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000183: Integer partitions ā¶ ā¤Result quality: 29% āvalues known / values provided: 73%ādistinct values known / distinct values provided: 29%
Values
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [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
{{1,3},{2},{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}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [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
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1},{2,5},{3},{4}}
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [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},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> {{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> {{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2},{3,4},{5,6}}
=> {{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1,2},{3,4},{5},{6}}
=> {{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> {{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2},{3,5},{4,6}}
=> {{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1,2},{3,5},{4},{6}}
=> {{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,2},{3,6},{4,5}}
=> {{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1,2},{3},{4,5},{6}}
=> {{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> {{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,2},{3,6},{4},{5}}
=> {{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,2},{3},{4,6},{5}}
=> {{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,2},{3},{4},{5,6}}
=> {{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> {{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
{{1,3,4},{2},{5},{6}}
=> {{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,3},{2,4},{5,6}}
=> {{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1,3},{2,4},{5},{6}}
=> {{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,3,5},{2},{4},{6}}
=> {{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,3},{2,5},{4,6}}
=> {{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1,3},{2,5},{4},{6}}
=> {{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,3},{2,6},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1,3},{2},{4,5},{6}}
=> {{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,3,6},{2},{4},{5}}
=> {{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,3},{2,6},{4},{5}}
=> {{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,3},{2},{4,6},{5}}
=> {{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,3},{2},{4},{5,6}}
=> {{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,3},{2},{4},{5},{6}}
=> {{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1 = 0 + 1
{{1,4},{2,3},{5,6}}
=> {{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1,4},{2,3},{5},{6}}
=> {{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1},{2,3,4},{5},{6}}
=> {{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,5},{2,3},{4,6}}
=> {{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1,5},{2,3},{4},{6}}
=> {{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1},{2,3,5},{4},{6}}
=> {{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1,6},{2,3},{4,5}}
=> {{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
{{1},{2,3},{4,5},{6}}
=> {{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,6},{2,3},{4},{5}}
=> {{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1},{2,3,6},{4},{5}}
=> {{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,1,1]
=> 1 = 0 + 1
{{1},{2,3},{4,6},{5}}
=> {{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
{{1,6},{2,7},{3,8},{4,9},{5,10}}
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> ?
=> ?
=> ? = 2 + 1
{{1},{2},{3},{4,7,8},{5,6}}
=> {{1},{2},{3},{4,6},{5,7,8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4},{5,6}}
=> {{1},{2},{3,6},{4},{5,7,8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4},{5,6,7}}
=> {{1},{2},{3,6,8},{4},{5,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,5,8},{4},{6,7}}
=> {{1},{2},{3,5,7},{4},{6,8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4,6},{5}}
=> {{1},{2},{3,6},{4,7,8},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,6,7},{5}}
=> {{1},{2},{3,6,8},{4,7},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,7},{5,6}}
=> {{1},{2},{3,6},{4,7},{5,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3},{4,7},{5,6},{8}}
=> {{1},{2,3},{4,6},{5,7},{8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3},{4,8},{5,6},{7}}
=> {{1},{2,3},{4,6},{5,8},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3},{4,8},{5,7},{6}}
=> {{1},{2,3},{4,7},{5,8},{6}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,4},{3},{5,8},{6,7}}
=> {{1},{2,4},{3},{5,7},{6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3},{4},{5,6,7}}
=> {{1},{2,6,8},{3},{4},{5,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,6},{3},{4,5},{7,8}}
=> {{1},{2,5},{3},{4,6},{7,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,4,8},{3},{5},{6,7}}
=> {{1},{2,4,7},{3},{5},{6,8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3},{4,7},{5,6}}
=> {{1},{2,6},{3},{4,7},{5,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,4,8},{3},{5,6},{7}}
=> {{1},{2,4,6},{3},{5,8},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,5},{3,4},{6,8},{7}}
=> {{1},{2,4},{3,5},{6,8},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,6,7},{3,4},{5},{8}}
=> {{1},{2,4},{3,6,7},{5},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,6,8},{3,4},{5},{7}}
=> {{1},{2,4},{3,6,8},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,7},{3,4},{5,6},{8}}
=> {{1},{2,4},{3,6},{5,7},{8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3,4},{5,7},{6}}
=> {{1},{2,4},{3,7},{5,8},{6}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,7},{3,5},{4},{6},{8}}
=> {{1},{2,5},{3,7},{4},{6},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,6,7},{3,5},{4},{8}}
=> {{1},{2,5},{3,6,7},{4},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3,5},{4},{6,7}}
=> {{1},{2,5},{3,7},{4},{6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,7},{3,5,6},{4},{8}}
=> {{1},{2,5,7},{3,6},{4},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3,7},{4},{5,6},{8}}
=> {{1},{2,3,6},{4},{5,7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3,7},{4},{5,6}}
=> {{1},{2,6},{3,7},{4},{5,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3,6},{4,5},{7}}
=> {{1},{2,5},{3,6},{4,8},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3,7},{4,5},{6}}
=> {{1},{2,5},{3,7},{4,8},{6}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3,8},{4,5},{6},{7}}
=> {{1},{2,3,5},{4,8},{6},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3,6,7},{4,5},{8}}
=> {{1},{2,3,5},{4,6,7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3,4,6},{5},{7}}
=> {{1},{2,4,8},{3,6},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3,8},{4,6},{5},{7}}
=> {{1},{2,3,6},{4,8},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3,4,5,6},{7}}
=> {{1},{2,4,6},{3,5,8},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1,2},{3,8},{4,5},{6},{7}}
=> {{1,2},{3,5},{4,8},{6},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1,2},{3,6,7,8},{4,5}}
=> {{1,2},{3,5},{4,6,7,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,2},{3,8},{4,5,6,7}}
=> {{1,2},{3,5,7},{4,6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,2},{3,4,5,8},{6,7}}
=> {{1,2},{3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,3},{2},{4},{5,8},{6,7}}
=> {{1,3},{2},{4},{5,7},{6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,3},{2},{4,7},{5,6},{8}}
=> {{1,3},{2},{4,6},{5,7},{8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,7},{2},{3},{4},{5,6},{8}}
=> {{1,6},{2},{3},{4},{5,7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1,8},{2},{3},{4},{5,7},{6}}
=> {{1,7},{2},{3},{4},{5,8},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1,8},{2},{3},{4},{5,6,7}}
=> {{1,6,8},{2},{3},{4},{5,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1,6,7},{2},{3},{4,5},{8}}
=> {{1,5},{2},{3},{4,6,7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1,4,8},{2},{3},{5,6},{7}}
=> {{1,4,6},{2},{3},{5,8},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1,5},{2},{3,4},{6},{7},{8}}
=> {{1,4},{2},{3,5},{6},{7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1,5},{2},{3,4},{6},{7,8}}
=> {{1,4},{2},{3,5},{6},{7,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,8},{2},{3,4},{5},{6},{7}}
=> {{1,4},{2},{3,8},{5},{6},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1,5,7},{2},{3,4},{6},{8}}
=> {{1,4},{2},{3,5,7},{6},{8}}
=> ?
=> ?
=> ? = 0 + 1
Description
The side length of the Durfee square of an integer partition.
Given a partition $\lambda = (\lambda_1,\ldots,\lambda_n)$, the Durfee square is the largest partition $(s^s)$ whose diagram fits inside the diagram of $\lambda$. In symbols, $s = \max\{ i \mid \lambda_i \geq i \}$.
This is also known as the Frobenius rank.
Matching statistic: St001031
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00115: Set partitions āKasraoui-Zengā¶ Set partitions
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001031: Dyck paths ā¶ ā¤Result quality: 29% āvalues known / values provided: 69%ādistinct values known / distinct values provided: 29%
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001031: Dyck paths ā¶ ā¤Result quality: 29% āvalues known / values provided: 69%ādistinct values known / distinct values provided: 29%
Values
{{1},{2},{3},{4}}
=> {{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> {{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> {{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> {{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> {{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> {{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3,4},{5}}
=> {{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> {{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,5},{3},{4}}
=> {{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3,5},{4}}
=> {{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3},{4,5}}
=> {{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> {{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> {{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> {{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,4},{5,6}}
=> {{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2},{3,4},{5},{6}}
=> {{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> {{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,5},{4,6}}
=> {{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4},{6}}
=> {{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,6},{4,5}}
=> {{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4,5},{6}}
=> {{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> {{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,6},{4},{5}}
=> {{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4,6},{5}}
=> {{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4},{5,6}}
=> {{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> {{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,4},{2},{5},{6}}
=> {{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,4},{5,6}}
=> {{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,4},{5},{6}}
=> {{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,5},{2},{4},{6}}
=> {{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,5},{4,6}}
=> {{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4},{6}}
=> {{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,6},{4,5}}
=> {{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4,5},{6}}
=> {{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,6},{2},{4},{5}}
=> {{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,6},{4},{5}}
=> {{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4,6},{5}}
=> {{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4},{5,6}}
=> {{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4},{5},{6}}
=> {{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4},{2,3},{5,6}}
=> {{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,4},{2,3},{5},{6}}
=> {{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3,4},{5},{6}}
=> {{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,5},{2,3},{4,6}}
=> {{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1,5},{2,3},{4},{6}}
=> {{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3,5},{4},{6}}
=> {{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,6},{2,3},{4,5}}
=> {{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4,5},{6}}
=> {{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,6},{2,3},{4},{5}}
=> {{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3,6},{4},{5}}
=> {{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3},{4,6},{5}}
=> {{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,6},{2,7},{3,8},{4,9},{5,10}}
=> {{1,10},{2,9},{3,8},{4,7},{5,6}}
=> ?
=> ?
=> ? = 2 + 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1},{2},{3},{4,7,8},{5,6}}
=> {{1},{2},{3},{4,6},{5,7,8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4},{5,6}}
=> {{1},{2},{3,6},{4},{5,7,8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4},{5,6,7}}
=> {{1},{2},{3,6,8},{4},{5,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,5,8},{4},{6,7}}
=> {{1},{2},{3,5,7},{4},{6,8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,7,8},{4,6},{5}}
=> {{1},{2},{3,6},{4,7,8},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,6,7},{5}}
=> {{1},{2},{3,6,8},{4,7},{5}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2},{3,8},{4,7},{5,6}}
=> {{1},{2},{3,6},{4,7},{5,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3},{4,7},{5,6},{8}}
=> {{1},{2,3},{4,6},{5,7},{8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3},{4,8},{5,6},{7}}
=> {{1},{2,3},{4,6},{5,8},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3},{4,8},{5,7},{6}}
=> {{1},{2,3},{4,7},{5,8},{6}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,4},{3},{5,8},{6,7}}
=> {{1},{2,4},{3},{5,7},{6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3},{4},{5,6,7}}
=> {{1},{2,6,8},{3},{4},{5,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,6},{3},{4,5},{7,8}}
=> {{1},{2,5},{3},{4,6},{7,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,4,8},{3},{5},{6,7}}
=> {{1},{2,4,7},{3},{5},{6,8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3},{4,7},{5,6}}
=> {{1},{2,6},{3},{4,7},{5,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,4,8},{3},{5,6},{7}}
=> {{1},{2,4,6},{3},{5,8},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,5},{3,4},{6,8},{7}}
=> {{1},{2,4},{3,5},{6,8},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,6,7},{3,4},{5},{8}}
=> {{1},{2,4},{3,6,7},{5},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,6,8},{3,4},{5},{7}}
=> {{1},{2,4},{3,6,8},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,7},{3,4},{5,6},{8}}
=> {{1},{2,4},{3,6},{5,7},{8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3,4},{5,7},{6}}
=> {{1},{2,4},{3,7},{5,8},{6}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,7},{3,5},{4},{6},{8}}
=> {{1},{2,5},{3,7},{4},{6},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,6,7},{3,5},{4},{8}}
=> {{1},{2,5},{3,6,7},{4},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3,5},{4},{6,7}}
=> {{1},{2,5},{3,7},{4},{6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,7},{3,5,6},{4},{8}}
=> {{1},{2,5,7},{3,6},{4},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3,7},{4},{5,6},{8}}
=> {{1},{2,3,6},{4},{5,7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3,7},{4},{5,6}}
=> {{1},{2,6},{3,7},{4},{5,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3,6},{4,5},{7}}
=> {{1},{2,5},{3,6},{4,8},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,8},{3,7},{4,5},{6}}
=> {{1},{2,5},{3,7},{4,8},{6}}
=> ?
=> ?
=> ? = 1 + 1
{{1},{2,3,8},{4,5},{6},{7}}
=> {{1},{2,3,5},{4,8},{6},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3,6,7},{4,5},{8}}
=> {{1},{2,3,5},{4,6,7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3,4,6},{5},{7}}
=> {{1},{2,4,8},{3,6},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,3,8},{4,6},{5},{7}}
=> {{1},{2,3,6},{4,8},{5},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1},{2,8},{3,4,5,6},{7}}
=> {{1},{2,4,6},{3,5,8},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1,2},{3,8},{4,5},{6},{7}}
=> {{1,2},{3,5},{4,8},{6},{7}}
=> ?
=> ?
=> ? = 1 + 1
{{1,2},{3,6,7,8},{4,5}}
=> {{1,2},{3,5},{4,6,7,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,2},{3,8},{4,5,6,7}}
=> {{1,2},{3,5,7},{4,6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,2},{3,4,5,8},{6,7}}
=> {{1,2},{3,4,5,7},{6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,3},{2},{4},{5,8},{6,7}}
=> {{1,3},{2},{4},{5,7},{6,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,3},{2},{4,7},{5,6},{8}}
=> {{1,3},{2},{4,6},{5,7},{8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,7},{2},{3},{4},{5,6},{8}}
=> {{1,6},{2},{3},{4},{5,7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1,8},{2},{3},{4},{5,7},{6}}
=> {{1,7},{2},{3},{4},{5,8},{6}}
=> ?
=> ?
=> ? = 0 + 1
{{1,8},{2},{3},{4},{5,6,7}}
=> {{1,6,8},{2},{3},{4},{5,7}}
=> ?
=> ?
=> ? = 0 + 1
{{1,6,7},{2},{3},{4,5},{8}}
=> {{1,5},{2},{3},{4,6,7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1,4,8},{2},{3},{5,6},{7}}
=> {{1,4,6},{2},{3},{5,8},{7}}
=> ?
=> ?
=> ? = 0 + 1
{{1,5},{2},{3,4},{6},{7},{8}}
=> {{1,4},{2},{3,5},{6},{7},{8}}
=> ?
=> ?
=> ? = 0 + 1
{{1,5},{2},{3,4},{6},{7,8}}
=> {{1,4},{2},{3,5},{6},{7,8}}
=> ?
=> ?
=> ? = 1 + 1
{{1,8},{2},{3,4},{5},{6},{7}}
=> {{1,4},{2},{3,8},{5},{6},{7}}
=> ?
=> ?
=> ? = 0 + 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
Matching statistic: St001556
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St001556: Permutations ā¶ ā¤Result quality: 29% āvalues known / values provided: 59%ādistinct values known / distinct values provided: 29%
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
Mp00025: Dyck paths āto 132-avoiding permutationā¶ Permutations
St001556: Permutations ā¶ ā¤Result quality: 29% āvalues known / values provided: 59%ādistinct values known / distinct values provided: 29%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,3,4,5,6,1] => ? = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => 1 = 0 + 1
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => 1 = 0 + 1
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => 2 = 1 + 1
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [3,2,4,5,6,1] => ? = 0 + 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,3,4,5,6,7,1] => ? = 0 + 1
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,2,4},{3},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,2},{3,4},{5},{6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2,5},{3},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,2},{3,5},{4},{6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2},{3},{4,5},{6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2,6},{3},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,2},{3,6},{4},{5},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2},{3},{4,6},{5},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2},{3},{4},{5,6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2,7},{3},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,2},{3,7},{4},{5},{6}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2},{3},{4,7},{5},{6}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2},{3},{4},{5,7},{6}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2},{3},{4},{5},{6,7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,2},{3},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 0 + 1
{{1,3,4},{2},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,3},{2,4},{5},{6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3,5},{2},{4},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,3},{2,5},{4},{6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3},{2},{4,5},{6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3,6},{2},{4},{5},{7}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,3},{2,6},{4},{5},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3},{2},{4,6},{5},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3},{2},{4},{5,6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3,7},{2},{4},{5},{6}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
{{1,3},{2,7},{4},{5},{6}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3},{2},{4,7},{5},{6}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3},{2},{4},{5,7},{6}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3},{2},{4},{5},{6,7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1,3},{2},{4},{5},{6},{7}}
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [3,2,4,5,6,7,1] => ? = 0 + 1
{{1,4},{2,3},{5},{6},{7}}
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [3,4,2,5,6,1] => ? = 0 + 1
{{1},{2,3,4},{5},{6},{7}}
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [4,2,3,5,6,1] => ? = 0 + 1
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation $\pi$ of length $n$,
$$\# \{3 < k \leq n \mid \pi(3) > \pi(k)\}.$$
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001195
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St001195: Dyck paths ā¶ ā¤Result quality: 14% āvalues known / values provided: 14%ādistinct values known / distinct values provided: 29%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St001195: Dyck paths ā¶ ā¤Result quality: 14% āvalues known / values provided: 14%ādistinct values known / distinct values provided: 29%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1,5},{2,4},{3},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1},{2,4,5},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
{{1},{2,4},{3,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0
{{1},{2,4,6},{3},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3,4,5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3,4,6},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1,5,6},{2},{3},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2,5,6},{3},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3,5,6},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4,5,6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 0
{{1,2,3},{4},{5,6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 0
{{1,2,3},{4},{5,7},{6}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 0
{{1,2,3},{4},{5},{6,7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 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]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> ? = 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]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 0
Description
The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$.
Matching statistic: St001200
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St001200: Dyck paths ā¶ ā¤Result quality: 14% āvalues known / values provided: 14%ādistinct values known / distinct values provided: 29%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00229: Dyck paths āDelest-Viennotā¶ Dyck paths
St001200: Dyck paths ā¶ ā¤Result quality: 14% āvalues known / values provided: 14%ādistinct values known / distinct values provided: 29%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{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]
=> 2 = 0 + 2
{{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]
=> 2 = 0 + 2
{{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]
=> 2 = 0 + 2
{{1,4},{2},{3},{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]
=> 2 = 0 + 2
{{1},{2,4},{3},{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]
=> 2 = 0 + 2
{{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]
=> 2 = 0 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{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]
=> 2 = 0 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1,5},{2,4},{3},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,4,5},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 3 = 1 + 2
{{1},{2,4},{3,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1,6},{2,4},{3},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2 = 0 + 2
{{1},{2,4,6},{3},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3,4,5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3,4,6},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1,5,6},{2},{3},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,5,6},{3},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3,5,6},{4}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3},{4,5,6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{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]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{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]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{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]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4},{5,6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{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]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4},{5,7},{6}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,3},{4},{5},{6,7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{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]
=> [1,1,0,1,0,1,1,1,0,1,0,0,0,0]
=> ? = 0 + 2
{{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]
=> [1,1,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 2
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0 + 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000689
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00321: Integer partitions ā2-conjugateā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St000689: Dyck paths ā¶ ā¤Result quality: 12% āvalues known / values provided: 12%ādistinct values known / distinct values provided: 14%
Mp00321: Integer partitions ā2-conjugateā¶ Integer partitions
Mp00043: Integer partitions āto Dyck pathā¶ Dyck paths
St000689: Dyck paths ā¶ ā¤Result quality: 12% āvalues known / values provided: 12%ādistinct values known / distinct values provided: 14%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 1
{{1,5},{2,4},{3},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,4,5},{3},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 0
{{1},{2,4},{3},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1},{2},{3,4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,5},{2},{3},{4},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1},{2,5},{3},{4},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1},{2},{3,5},{4},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1},{2},{3},{4,5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,6},{2},{3},{4},{5}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1},{2,6},{3},{4},{5}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1},{2},{3,6},{4},{5}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1},{2},{3},{4,6},{5}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1},{2},{3},{4},{5,6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,3},{4},{5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,3},{4},{5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,3},{4},{5},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,4},{3},{5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,4,7},{3},{5},{6}}
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
{{1,2,4},{3,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,4},{3},{5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,4},{3},{5},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,5},{3,4},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2},{3,4,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,6},{3,4},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2},{3,4,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
{{1,2,7},{3,4},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 0
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Matching statistic: St001001
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00321: Integer partitions ā2-conjugateā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001001: Dyck paths ā¶ ā¤Result quality: 7% āvalues known / values provided: 7%ādistinct values known / distinct values provided: 14%
Mp00321: Integer partitions ā2-conjugateā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
St001001: Dyck paths ā¶ ā¤Result quality: 7% āvalues known / values provided: 7%ādistinct values known / distinct values provided: 14%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 0
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 0
{{1,4},{2},{3},{5},{6}}
=> [2,1,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 0
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,3},{4},{5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,3},{4},{5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,3},{4},{5},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3},{5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3},{5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,4},{3},{5},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,5},{3,4},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3,4,5},{6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,6},{3,4},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3,4,6},{5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,7},{3,4},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3,4,7},{5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,5},{3,6},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,5},{3},{4,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,5},{3,7},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,5},{3},{4,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,5},{3},{4},{6,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,6},{3,5},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3,5,6},{4},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,7},{3,5},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3,5,7},{4},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,6},{3},{4,5},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3},{4,5,6},{7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,7},{3},{4,5},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3},{4,5,7},{6}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,6},{3,7},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,6},{3},{4,7},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,6},{3},{4},{5,7}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,7},{3,6},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3,6,7},{4},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2,7},{3},{4,6},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
{{1,2},{3},{4,6,7},{5}}
=> [3,2,1,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 0
Description
The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001232
Mp00079: Set partitions āshapeā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St001232: Dyck paths ā¶ ā¤Result quality: 5% āvalues known / values provided: 5%ādistinct values known / distinct values provided: 14%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00222: Dyck paths āpeaks-to-valleysā¶ Dyck paths
St001232: Dyck paths ā¶ ā¤Result quality: 5% āvalues known / values provided: 5%ādistinct values known / distinct values provided: 14%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,2},{3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,3},{2},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,3},{4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,4},{2},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,4},{3},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2},{3,4},{5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,5},{2},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,5},{3},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2},{3,5},{4}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2},{3},{4,5}}
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,2},{3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,2},{3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,3},{2},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,3},{2},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1},{2,3},{4,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,3},{4,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,3},{4},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1},{2,3},{4},{5},{6}}
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,4,5},{2},{3},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,4},{2,5},{3},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,4},{2},{3,5},{6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,4,6},{2},{3},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 0 + 2
{{1,4},{2,6},{3},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,4},{2},{3,6},{5}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,4},{2},{3},{5,6}}
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 0 + 2
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,3},{4,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,4},{3,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,4},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,4},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,4},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,4},{5,6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,6},{4,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,5},{3,7},{4,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,5},{4,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,5,6},{4,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,5},{4,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,5,7},{4,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,5},{4,6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,6},{3,7},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2,7},{3,6},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,6,7},{4,5}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,6},{4,5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,2},{3,7},{4,5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,4},{2,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,4},{2,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,4},{2,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,5},{2,4},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4,5},{6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,6},{2,4},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4,6},{5,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3,7},{2,4},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4,7},{5,6}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
{{1,3},{2,4},{5,6,7}}
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3 = 1 + 2
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000655
Mp00128: Set partitions āto compositionā¶ Integer compositions
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St000655: Dyck paths ā¶ ā¤Result quality: 5% āvalues known / values provided: 5%ādistinct values known / distinct values provided: 29%
Mp00231: Integer compositions ābounce pathā¶ Dyck paths
Mp00199: Dyck paths āprime Dyck pathā¶ Dyck paths
St000655: Dyck paths ā¶ ā¤Result quality: 5% āvalues known / values provided: 5%ādistinct values known / distinct values provided: 29%
Values
{{1},{2},{3},{4}}
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3},{4},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4},{2},{3},{5}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,4},{3},{5}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3,4},{5}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,5},{2},{3},{4}}
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,5},{3},{4}}
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3,5},{4}}
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1},{2},{3},{4,5}}
=> [1,1,1,2] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1},{2},{3},{4},{5}}
=> [1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3,4},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3,5},{4},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,2},{3},{4,5},{6}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3,6},{4},{5}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4,6},{5}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4},{5,6}}
=> [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1,2},{3},{4},{5},{6}}
=> [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,4},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2,5},{4},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,3},{2},{4,5},{6}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2,6},{4},{5}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4,6},{5}}
=> [2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4},{5,6}}
=> [2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
{{1,3},{2},{4},{5},{6}}
=> [2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,4},{2,3},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,4},{2,3},{5},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,5},{2,3},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1,5},{2,3},{4},{6}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3,5},{4},{6}}
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1,6},{2,3},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
{{1},{2,3},{4,5},{6}}
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,6},{2,3},{4},{5}}
=> [2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3,6},{4},{5}}
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 0 + 1
{{1},{2,3},{4,6},{5}}
=> [1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
{{1,2,3,4},{5},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,5},{4},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,3},{4,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,6},{4},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,3},{4,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,3},{4},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3,7},{4},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,3},{4},{5},{6,7}}
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,3},{4},{5},{6},{7}}
=> [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,4,5},{3},{6},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3,5},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,4},{3,5},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,4,6},{3},{5},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3,6},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,4},{3,6},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3,7},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,4},{3},{5,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,4,7},{3},{5},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3,7},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3},{5,7},{6}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,4},{3},{5},{6,7}}
=> [3,1,1,2] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 1
{{1,2,4},{3},{5},{6},{7}}
=> [3,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,5},{3,4},{6,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,5},{3,4},{6},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2},{3,4,5},{6,7}}
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2},{3,4,5},{6},{7}}
=> [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,6},{3,4},{5,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,6},{3,4},{5},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2},{3,4,6},{5,7}}
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2},{3,4,6},{5},{7}}
=> [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,7},{3,4},{5,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2},{3,4,7},{5,6}}
=> [2,3,2] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2},{3,4},{5,6,7}}
=> [2,2,3] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,1,0,0,0,0]
=> ? = 1 + 1
{{1,2},{3,4},{5,6},{7}}
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
{{1,2,7},{3,4},{5},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2},{3,4,7},{5},{6}}
=> [2,3,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2},{3,4},{5,7},{6}}
=> [2,2,2,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
{{1,2},{3,4},{5},{6,7}}
=> [2,2,1,2] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,5,6},{3},{4},{7}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,5},{3,6},{4,7}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,5},{3,6},{4},{7}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,5},{3,7},{4,6}}
=> [3,2,2] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 1 + 1
{{1,2,5},{3},{4,6},{7}}
=> [3,1,2,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 0 + 1
{{1,2,5,7},{3},{4},{6}}
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 0 + 1
{{1,2,5},{3,7},{4},{6}}
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 0 + 1
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
The following 32 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000487The length of the shortest cycle of a permutation. St000455The second largest eigenvalue of a graph if it is integral. St000654The first descent of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St000274The number of perfect matchings of a graph. St000666The number of right tethers of a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001386The number of prime labellings of a graph. St001549The number of restricted non-inversions between exceedances. St001810The number of fixed points of a permutation smaller than its largest moved point. St001871The number of triconnected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000667The greatest common divisor of the parts of the partition. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{nā1}]$ by adding $c_0$ to $c_{nā1}$. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001111The weak 2-dynamic chromatic number of a graph. St000842The breadth of a permutation. St001260The permanent of an alternating sign matrix. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001964The interval resolution global dimension of a poset. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001645The pebbling number of a connected graph.
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