Your data matches 66 different statistics following compositions of up to 3 maps.
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Matching statistic: St000147
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
St000147: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
Description
The largest part of an integer partition.
Matching statistic: St000473
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00044: Integer partitions ā€”conjugateāŸ¶ Integer partitions
St000473: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
Description
The number of parts of a partition that are strictly bigger than the number of ones. This is part of the definition of Dyson's crank of a partition, see [[St000474]].
Matching statistic: St000668
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
St000668: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 2
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St001280
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00044: Integer partitions ā€”conjugateāŸ¶ Integer partitions
St001280: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [4]
 => 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2,2]
 => 2
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [3]
 => 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [3,1]
 => 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000319
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
St000319: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
Description
The spin of an integer partition. The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape. The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$ The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross. This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
St000320: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
Description
The dinv adjustment of an integer partition. The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$. The dinv adjustment is then defined by $$\sum_{j:n_j > 0}(\lambda_1-1-j).$$ The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions $$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$ and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$. The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St001918
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
St001918: Integer partitions āŸ¶ ā„¤Result quality: 100% ā—values known / values provided: 100%ā—distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,1,1]
 => 0 = 1 - 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [2]
 => 1 = 2 - 1
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,1]
 => 0 = 1 - 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,1]
 => 0 = 1 - 1
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition. Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$. The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is $$ \sum_{p\in\lambda} [p]_{q^{N/p}}, $$ where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer. This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals $$ \left(1 - \frac{1}{\lambda_1}\right) N, $$ where $\lambda_1$ is the largest part of $\lambda$. The statistic is undefined for the empty partition.
Matching statistic: St000120
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00043: Integer partitions ā€”to Dyck pathāŸ¶ Dyck paths
St000120: Dyck paths āŸ¶ ā„¤Result quality: 92% ā—values known / values provided: 92%ā—distinct values known / distinct values provided: 100%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [1,1,0,0,1,1,0,0]
 => 2
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 1
{{1},{2},{3},{4},{5},{6},{7}}
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8}}
 => [1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7,8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6,8},{7}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5,6},{7},{8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5,7},{6},{8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5,8},{6},{7}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4,8},{5},{6},{7}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3,8},{4},{5},{6},{7}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2,8},{3},{4},{5},{6},{7}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1,4},{2},{3},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1,5},{2},{3},{4},{6},{7},{8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1,6},{2},{3},{4},{5},{7},{8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1,7},{2},{3},{4},{5},{6},{8}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1,8},{2},{3},{4},{5},{6},{7}}
 => [2,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8,9}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7,9},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6,9},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5,9},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4,9},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3,9},{4},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2,9},{3},{4},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7,8,9}}
 => [3,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8,10},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6,9},{7,8}}
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7,10},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5,9},{6,7},{8}}
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6,10},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4,9},{5,6},{7},{8}}
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4},{5,10},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2},{3,9},{4,5},{6},{7},{8}}
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
{{1},{2},{3},{4,10},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1},{2,9},{3,4},{5},{6},{7},{8}}
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
{{1},{2},{3,10},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1,9},{2,3},{4},{5},{6},{7},{8}}
 => [2,2,1,1,1,1,1]
 => [2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => ? = 1
{{1},{2,10},{3},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1,4},{2},{3},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1,5},{2},{3},{4},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1,6},{2},{3},{4},{5},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1,7},{2},{3},{4},{5},{6},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
{{1,8},{2},{3},{4},{5},{6},{7},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => ? = 1
Description
The number of left tunnels of a Dyck path. A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b
Matching statistic: St001597
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00202: Integer partitions ā€”first row removalāŸ¶ Integer partitions
Mp00179: Integer partitions ā€”to skew partitionāŸ¶ Skew partitions
St001597: Skew partitions āŸ¶ ā„¤Result quality: 67% ā—values known / values provided: 90%ā—distinct values known / distinct values provided: 67%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,1,1,1]
 => [[1,1,1,1],[]]
 => 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [2,2]
 => [[2,2],[]]
 => 2
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,1,1]
 => [[1,1,1],[]]
 => 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [2,1,1]
 => [[2,1,1],[]]
 => 1
{{1,6},{2,7},{3,8},{4,9},{5,10}}
 => [2,2,2,2,2]
 => [2,2,2,2]
 => [[2,2,2,2],[]]
 => ? = 2
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8,10},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7,10},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2},{3},{4},{5},{6,10},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2},{3},{4},{5,10},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2},{3},{4,10},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2},{3,10},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1},{2,10},{3},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,4},{2},{3},{5},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,5},{2},{3},{4},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,6},{2},{3},{4},{5},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,7},{2},{3},{4},{5},{6},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,8},{2},{3},{4},{5},{6},{7},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,9},{2},{3},{4},{5},{6},{7},{8},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,2,4,8},{3,6,12},{5,10},{7},{9},{11}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10,11}}
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,9},{2,8},{3,7},{4,6},{5},{10}}
 => [2,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1},{2,10},{3,9},{4,8},{5,7},{6}}
 => [2,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,7},{2,8},{3,9},{4,10},{5,11},{6,12}}
 => [2,2,2,2,2,2]
 => [2,2,2,2,2]
 => [[2,2,2,2,2],[]]
 => ? = 2
{{1},{2},{3},{4},{5},{6,9},{7},{8},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10},{11}}
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,10},{2,3},{4},{5},{6},{7},{8},{9}}
 => [2,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1],[]]
 => ? = 1
{{1,11},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1,1]
 => [1,1,1,1,1,1,1,1,1]
 => [[1,1,1,1,1,1,1,1,1],[]]
 => ? = 1
{{1,10},{2,9},{3},{4},{5},{6},{7},{8}}
 => [2,2,1,1,1,1,1,1]
 => [2,1,1,1,1,1,1]
 => [[2,1,1,1,1,1,1],[]]
 => ? = 1
{{1,2,4,8},{3,6,12},{5,11},{7},{9},{10}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,4,9},{3,6,12},{5,10},{7},{8},{11}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,4,9},{3,6,12},{5,11},{7},{8},{10}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,4,10},{3,6,12},{5,11},{7},{8},{9}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,4,8},{3,7},{5,10},{6,12},{9},{11}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,4,8},{3,7},{5,11},{6,12},{9},{10}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,4,9},{3,7},{5,10},{6,12},{8},{11}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,4,9},{3,7},{5,11},{6,12},{8},{10}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,4,10},{3,7},{5,11},{6,12},{8},{9}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,4,9},{3,8},{5,10},{6,12},{7},{11}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,4,9},{3,8},{5,11},{6,12},{7},{10}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,4,10},{3,8},{5,11},{6,12},{7},{9}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,4,10},{3,9},{5,11},{6,12},{7},{8}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,5,10},{3,6,12},{4,8},{7},{9},{11}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,5,11},{3,6,12},{4,8},{7},{9},{10}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,5,10},{3,6,12},{4,9},{7},{8},{11}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,5,11},{3,6,12},{4,9},{7},{8},{10}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,5,11},{3,6,12},{4,10},{7},{8},{9}}
 => [4,3,2,1,1,1]
 => [3,2,1,1,1]
 => [[3,2,1,1,1],[]]
 => ? = 2
{{1,2,5,10},{3,7},{4,8},{6,12},{9},{11}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
{{1,2,5,11},{3,7},{4,8},{6,12},{9},{10}}
 => [4,2,2,2,1,1]
 => [2,2,2,1,1]
 => [[2,2,2,1,1],[]]
 => ? = 2
Description
The Frobenius rank of a skew partition. This is the minimal number of border strips in a border strip decomposition of the skew partition.
Matching statistic: St001031
(load all 7 compositions to match this statistic)
Mapping
Mp00079: Set partitions ā€”shapeāŸ¶ Integer partitions
Mp00230: Integer partitions ā€”parallelogram polyominoāŸ¶ Dyck paths
Mp00121: Dyck paths ā€”Cori-Le Borgne involutionāŸ¶ Dyck paths
St001031: Dyck paths āŸ¶ ā„¤Result quality: 67% ā—values known / values provided: 84%ā—distinct values known / distinct values provided: 67%
Values
{{1},{2},{3},{4}}
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0]
 => 1
{{1,2},{3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,3},{2},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1},{2,3},{4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,4},{2},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1},{2,4},{3},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1},{2},{3,4},{5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,5},{2},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1},{2,5},{3},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1},{2},{3,5},{4}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1},{2},{3},{4,5}}
 => [2,1,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 1
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,0]
 => 1
{{1,2,3},{4},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,2,4},{3},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,2},{3,4},{5,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,2},{3,4},{5},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2,5},{3},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,2},{3,5},{4,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,2},{3,5},{4},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2},{3,6},{4,5}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,2},{3},{4,5},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2,6},{3},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,2},{3,6},{4},{5}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2},{3},{4,6},{5}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2},{3},{4},{5,6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,2},{3},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,3,4},{2},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,3},{2,4},{5,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,3},{2,4},{5},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,3,5},{2},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,3},{2,5},{4,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,3},{2,5},{4},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,3},{2,6},{4,5}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,3},{2},{4,5},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,3,6},{2},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,3},{2,6},{4},{5}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,3},{2},{4,6},{5}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,3},{2},{4},{5,6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,3},{2},{4},{5},{6}}
 => [2,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,0]
 => 1
{{1,4},{2,3},{5,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,4},{2,3},{5},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1},{2,3,4},{5},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,5},{2,3},{4,6}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1,5},{2,3},{4},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1},{2,3,5},{4},{6}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1,6},{2,3},{4,5}}
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,1,1,0,0,0,0]
 => 2
{{1},{2,3},{4,5},{6}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1,6},{2,3},{4},{5}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1},{2,3,6},{4},{5}}
 => [3,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,0]
 => 1
{{1},{2,3},{4,6},{5}}
 => [2,2,1,1]
 => [1,1,1,0,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0]
 => 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9}}
 => [1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8,9}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7,9},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6,9},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5,9},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4,9},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3,9},{4},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2,9},{3},{4},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,9},{2},{3},{4},{5},{6},{7},{8}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => [1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8},{9,10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7,8,9}}
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7},{8,10},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6,9},{7,8}}
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6},{7,10},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4},{5,9},{6,7},{8}}
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
{{1},{2},{3},{4},{5},{6,10},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3},{4,9},{5,6},{7},{8}}
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
{{1},{2},{3},{4},{5,10},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2},{3,9},{4,5},{6},{7},{8}}
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
{{1},{2},{3},{4,10},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1},{2,9},{3,4},{5},{6},{7},{8}}
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
{{1},{2},{3,10},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,9},{2,3},{4},{5},{6},{7},{8}}
 => [2,2,1,1,1,1,1]
 => [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
 => ? = 1
{{1},{2,10},{3},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,10},{2},{3},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,4},{2},{3},{5},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,5},{2},{3},{4},{6},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,6},{2},{3},{4},{5},{7},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,7},{2},{3},{4},{5},{6},{8},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,8},{2},{3},{4},{5},{6},{7},{9}}
 => [2,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,2},{3},{4},{5},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,2,3},{4},{5},{6},{7},{8},{9}}
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1
{{1,3},{2},{4},{5},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,3,4},{2},{5},{6},{7},{8},{9}}
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1
{{1,4},{2},{3},{5},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,5},{2},{3},{4},{6},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,6},{2},{3},{4},{5},{7},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,7},{2},{3},{4},{5},{6},{8},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,8},{2},{3},{4},{5},{6},{7},{9},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,8,9},{2},{3},{4},{5},{6},{7}}
 => [3,1,1,1,1,1,1]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => ? = 1
{{1,9},{2},{3},{4},{5},{6},{7},{8},{10}}
 => [2,1,1,1,1,1,1,1,1]
 => [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 1
{{1,2,3,4,5,6},{7},{8},{9}}
 => [6,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,2,3,4,5},{6},{7},{8},{9}}
 => [5,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,2,3,4},{5},{6},{7},{8},{9}}
 => [4,1,1,1,1,1]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,2,3,4,5,6,7},{8},{9},{10}}
 => [7,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,2,3,4,5,6},{7},{8},{9},{10}}
 => [6,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
{{1,2,3,4,5},{6},{7},{8},{9},{10}}
 => [5,1,1,1,1,1]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
 => ? = 1
Description
The height of the bicoloured Motzkin path associated with the Dyck path.
The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000660The number of rises of length at least 3 of a Dyck path. St000703The number of deficiencies of a permutation. St000994The number of cycle peaks and the number of cycle valleys of a permutation. St000710The number of big deficiencies of a permutation. St001471The magnitude of a Dyck path. St001269The sum of the minimum of the number of exceedances and deficiencies in each cycle of a permutation. St001665The number of pure excedances of a permutation. St001729The number of visible descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000183The side length of the Durfee square of an integer partition. St001556The number of inversions of the third entry of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000655The length of the minimal rise of a Dyck path. 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. 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) $[c_0,c_1,...,c_{nāˆ’1}]$ by adding $c_0$ to $c_{nāˆ’1}$. 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. St000274The number of perfect matchings of a graph. St000666The number of right tethers of a permutation. St001111The weak 2-dynamic chromatic number of a graph. 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. 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. 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. St000787The number of flips required to make a perfect matching noncrossing. 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. St001845The number of join irreducibles minus the rank of a lattice. St001645The pebbling number of a connected graph.