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Your data matches 175 different statistics following compositions of up to 3 maps.
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Matching statistic: St000667
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
St000667: 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
St000667: Integer partitions ā¶ ā¤Result quality: 100% āvalues known / values provided: 100%ādistinct values known / distinct values provided: 100%
Values
[1,1,1]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [1]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,1,1]
=> [1,1]
=> [1]
=> 1
[2,2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,1,1]
=> [1,1]
=> [1]
=> 1
[3,2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,2,2]
=> [2,2]
=> [2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[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
[5,1,1]
=> [1,1]
=> [1]
=> 1
[4,2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[3,3,1]
=> [3,1]
=> [1]
=> 1
[3,2,2]
=> [2,2]
=> [2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[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]
=> [1,1,1,1,1]
=> 1
[6,1,1]
=> [1,1]
=> [1]
=> 1
[5,2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,3,1]
=> [3,1]
=> [1]
=> 1
[4,2,2]
=> [2,2]
=> [2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[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]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1
[7,1,1]
=> [1,1]
=> [1]
=> 1
[6,2,1]
=> [2,1]
=> [1]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,3,1]
=> [3,1]
=> [1]
=> 1
[5,2,2]
=> [2,2]
=> [2]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St001075
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St001075: Set partitions ā¶ ā¤Result quality: 27% āvalues known / values provided: 27%ādistinct values known / distinct values provided: 38%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00284: Standard tableaux ārowsā¶ Set partitions
St001075: Set partitions ā¶ ā¤Result quality: 27% āvalues known / values provided: 27%ādistinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{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]]
=> {{1},{2},{3},{4},{5},{6}}
=> 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> {{1,2,5},{3,4}}
=> 2
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> {{1,4,5},{2},{3}}
=> 1
[3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> {{1,3},{2,5},{4}}
=> 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> {{1,5},{2},{3},{4}}
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> {{1,4},{2,6},{3},{5}}
=> 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> {{1,6},{2},{3},{4},{5}}
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> {{1},{2},{3},{4},{5},{6}}
=> 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> {{1},{2},{3},{4},{5},{6},{7}}
=> 1
[7,1,1]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[6,2,1]
=> [2,1]
=> [[1,3],[2]]
=> {{1,3},{2}}
=> 1
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[5,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> {{1,3,4},{2}}
=> 1
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2
[5,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> {{1,4},{2},{3}}
=> 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[4,4,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> {{1,3,4,5},{2}}
=> 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]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 2
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> ? = 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 1
[3,3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> ? = 1
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> {{1,5,8},{2,7},{3},{4},{6}}
=> ? = 1
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 2
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> ? = 1
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> {{1,3},{2,5},{4,7},{6,9},{8}}
=> ? = 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> {{1,5},{2,7},{3,9},{4},{6},{8}}
=> ? = 1
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> {{1,7},{2,9},{3},{4},{5},{6},{8}}
=> ? = 1
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{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],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9},{10}}
=> ? = 1
[4,4,4]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> {{1,2,3,4},{5,6,7,8}}
=> ? = 4
[4,4,3,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> {{1,3,4,8},{2,6,7},{5}}
=> ? = 1
[4,4,2,2]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> {{1,2,7,8},{3,4},{5,6}}
=> ? = 2
[4,4,2,1,1]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> {{1,4,7,8},{2,6},{3},{5}}
=> ? = 1
[4,4,1,1,1,1]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> {{1,6,7,8},{2},{3},{4},{5}}
=> ? = 1
[4,3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> {{1,2,5},{3,4,8},{6,7}}
=> ? = 1
[4,3,3,1,1]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> {{1,4,5},{2,7,8},{3},{6}}
=> ? = 1
[4,3,2,2,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> {{1,3,8},{2,5},{4,7},{6}}
=> ? = 1
[4,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> {{1,5,8},{2,7},{3},{4},{6}}
=> ? = 1
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> {{1,7,8},{2},{3},{4},{5},{6}}
=> ? = 1
[4,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> ? = 2
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> {{1,4},{2,6},{3,8},{5},{7}}
=> ? = 1
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> {{1,6},{2,8},{3},{4},{5},{7}}
=> ? = 1
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> {{1,8},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> {{1},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> {{1,2,3},{4,5,6},{7,8,9}}
=> ? = 3
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> {{1,3,6},{2,5,9},{4,8},{7}}
=> ? = 1
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> {{1,5,6},{2,8,9},{3},{4},{7}}
=> ? = 1
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> {{1,2,9},{3,4},{5,6},{7,8}}
=> ? = 2
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> {{1,4,9},{2,6},{3,8},{5},{7}}
=> ? = 1
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> {{1,6,9},{2,8},{3},{4},{5},{7}}
=> ? = 1
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> {{1,8,9},{2},{3},{4},{5},{6},{7}}
=> ? = 1
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> {{1,3},{2,5},{4,7},{6,9},{8}}
=> ? = 1
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> {{1,5},{2,7},{3,9},{4},{6},{8}}
=> ? = 1
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> {{1,7},{2,9},{3},{4},{5},{6},{8}}
=> ? = 1
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> {{1,9},{2},{3},{4},{5},{6},{7},{8}}
=> ? = 1
[3,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> {{1},{2},{3},{4},{5},{6},{7},{8},{9}}
=> ? = 1
Description
The minimal size of a block of a set partition.
Matching statistic: St000314
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000314: Permutations ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 38%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000314: Permutations ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[7,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[6,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[5,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[5,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[4,4,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[4,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 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]]
=> [8,7,6,5,4,3,2,1] => ? = 1
[3,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1
[3,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 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]]
=> [9,8,7,6,5,4,3,2,1] => ? = 1
[4,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 3
[4,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[4,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1
[4,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1
[4,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
[4,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[4,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ? = 1
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 1
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ? = 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 1
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 1
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ? = 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => ? = 1
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => ? = 1
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,1,2] => ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,4,3,2,1] => ? = 1
[5,5,2]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 2
[5,5,1,1]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1
[5,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 3
Description
The number of left-to-right-maxima of a permutation.
An integer $\sigma_i$ in the one-line notation of a permutation $\sigma$ is a '''left-to-right-maximum''' if there does not exist a $j < i$ such that $\sigma_j > \sigma_i$.
This is also the number of weak exceedences of a permutation that are not mid-points of a decreasing subsequence of length 3, see [1] for more on the later description.
Matching statistic: St000654
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000654: Permutations ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 38%
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000654: Permutations ā¶ ā¤Result quality: 18% āvalues known / values provided: 18%ādistinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[7,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[6,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[5,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[5,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[4,4,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
[4,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 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]]
=> [8,7,6,5,4,3,2,1] => ? = 1
[3,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1
[3,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[3,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 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]]
=> [9,8,7,6,5,4,3,2,1] => ? = 1
[4,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 3
[4,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? = 1
[4,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? = 1
[4,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? = 1
[4,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? = 2
[4,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? = 1
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? = 1
[4,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? = 1
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? = 1
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? = 1
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? = 1
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? = 1
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ? = 1
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? = 1
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? = 1
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ? = 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 2
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? = 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? = 1
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? = 1
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? = 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [9,7,8,5,6,3,4,1,2] => ? = 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [9,8,7,5,6,3,4,1,2] => ? = 1
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,3,4,1,2] => ? = 1
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,1,2] => ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [10,9,8,7,6,5,4,3,2,1] => ? = 1
[5,5,2]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> [6,7,1,2,3,4,5] => ? = 2
[5,5,1,1]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => ? = 1
[5,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? = 3
Description
The first descent of a permutation.
For a permutation $\pi$ of $\{1,\ldots,n\}$, this is the smallest index $0 < i \leq n$ such that $\pi(i) > \pi(i+1)$ where one considers $\pi(n+1)=0$.
Matching statistic: St001232
Mp00202: Integer partitions āfirst row removalā¶ Integer partitions
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
St001232: Dyck paths ā¶ ā¤Result quality: 11% āvalues known / values provided: 11%ādistinct values known / distinct values provided: 75%
Mp00230: Integer partitions āparallelogram polyominoā¶ Dyck paths
Mp00227: Dyck paths āDelest-Viennot-inverseā¶ Dyck paths
St001232: Dyck paths ā¶ ā¤Result quality: 11% āvalues known / values provided: 11%ādistinct values known / distinct values provided: 75%
Values
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1]
=> [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]
=> ? = 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[2,1,1,1,1,1]
=> [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]
=> ? = 1
[1,1,1,1,1,1,1]
=> [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]
=> ? = 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[3,1,1,1,1,1]
=> [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]
=> ? = 1
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 2
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[2,2,1,1,1,1]
=> [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]
=> ? = 1
[2,1,1,1,1,1,1]
=> [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]
=> ? = 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[6,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[5,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[4,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[4,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[4,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[4,1,1,1,1,1]
=> [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]
=> ? = 1
[3,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> ? = 1
[3,3,1,1,1]
=> [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]
=> ? = 1
[3,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? = 2
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? = 1
[3,2,1,1,1,1]
=> [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]
=> ? = 1
[3,1,1,1,1,1,1]
=> [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]
=> ? = 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? = 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> ? = 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 1
[8,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[7,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[7,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> ? = 1
[6,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[6,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> ? = 1
[6,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> ? = 1
[5,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[5,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> ? = 1
[5,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? = 1
[5,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? = 1
[4,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[4,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[9,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[8,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[7,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[7,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[6,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[6,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[5,5,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[5,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[5,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,4,3]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 3
[10,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 1
[9,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[8,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[8,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[7,4,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[7,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[6,5,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1
[6,4,2]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[6,3,3]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000297
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00084: Standard tableaux āconjugateā¶ Standard tableaux
Mp00134: Standard tableaux ādescent wordā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Mp00084: Standard tableaux āconjugateā¶ Standard tableaux
Mp00134: Standard tableaux ādescent wordā¶ Binary words
St000297: Binary words ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 00 => 0 = 1 - 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 001 => 0 = 1 - 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 000 => 0 = 1 - 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 0011 => 0 = 1 - 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 0101 => 0 = 1 - 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 0001 => 0 = 1 - 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0000 => 0 = 1 - 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 00111 => 0 = 1 - 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 01011 => 0 = 1 - 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 00011 => 0 = 1 - 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 10101 => 1 = 2 - 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 00101 => 0 = 1 - 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 00001 => 0 = 1 - 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 00000 => 0 = 1 - 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7]]
=> 001111 => 0 = 1 - 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7]]
=> 010111 => 0 = 1 - 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7]]
=> 000111 => 0 = 1 - 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7]]
=> 011011 => 0 = 1 - 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 101011 => 1 = 2 - 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> 001011 => 0 = 1 - 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> 000011 => 0 = 1 - 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7]]
=> 010101 => 0 = 1 - 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7]]
=> 000101 => 0 = 1 - 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> 000001 => 0 = 1 - 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 000000 => 0 = 1 - 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> 0011111 => 0 = 1 - 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7],[8]]
=> 0101111 => 0 = 1 - 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> 0001111 => 0 = 1 - 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8]]
=> 0110111 => 0 = 1 - 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 1010111 => 1 = 2 - 1
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7],[8]]
=> 0010111 => 0 = 1 - 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8]]
=> 0000111 => 0 = 1 - 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [[1,3,6],[2,4,7],[5,8]]
=> 1011011 => 1 = 2 - 1
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [[1,2,3,6],[4,7],[5,8]]
=> 0011011 => 0 = 1 - 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7],[8]]
=> 0101011 => 0 = 1 - 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7],[8]]
=> 0001011 => 0 = 1 - 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7],[8]]
=> 0000011 => 0 = 1 - 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 1010101 => 1 = 2 - 1
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [[1,2,3,5,7],[4,6,8]]
=> 0010101 => 0 = 1 - 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6,8]]
=> 0000101 => 0 = 1 - 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7],[8]]
=> 0000001 => 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> 0000000 => 0 = 1 - 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> 00111111 => 0 = 1 - 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7],[8],[9]]
=> 01011111 => 0 = 1 - 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> 00011111 => 0 = 1 - 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8],[9]]
=> 01101111 => 0 = 1 - 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 10101111 => 1 = 2 - 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7],[8],[9]]
=> 00101111 => 0 = 1 - 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> 00001111 => 0 = 1 - 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [[1,2,6],[3,7],[4,8],[5,9]]
=> 01110111 => 0 = 1 - 1
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ?
=> ? => ? = 1 - 1
[8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ?
=> ? => ? = 1 - 1
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ?
=> ? => ? = 1 - 1
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ?
=> ? => ? = 1 - 1
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
=> ? => ? = 2 - 1
[7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ?
=> ? => ? = 1 - 1
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ?
=> ? => ? = 1 - 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ?
=> ? => ? = 1 - 1
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ?
=> ? => ? = 2 - 1
[6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ?
=> ? => ? = 1 - 1
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ?
=> ? => ? = 1 - 1
[6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7],[8],[9],[10],[11]]
=> ? => ? = 1 - 1
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ?
=> ? => ? = 1 - 1
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> [[1,2,7],[3,8],[4,9],[5,10],[6,11]]
=> ? => ? = 1 - 1
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> 1011101111 => ? = 2 - 1
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> 0011101111 => ? = 1 - 1
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
=> 1101101111 => ? = 3 - 1
[5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
=> 0101101111 => ? = 1 - 1
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> 1010101111 => ? = 2 - 1
[5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6,8],[9],[10],[11]]
=> ? => ? = 1 - 1
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ?
=> ? => ? = 1 - 1
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> 1101110111 => ? = 3 - 1
[4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> 0101110111 => ? = 1 - 1
[4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> 0110110111 => ? = 1 - 1
[4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> 1010110111 => ? = 2 - 1
[4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> 0010110111 => ? = 1 - 1
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ?
=> ? => ? = 1 - 1
[4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9],[10],[11]]
=> 0101010111 => ? = 1 - 1
[4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ?
=> ? => ? = 1 - 1
[4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ?
=> ? => ? = 1 - 1
[4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ?
=> ? => ? = 1 - 1
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> 1011011011 => ? = 1 - 1
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> 0011011011 => ? = 1 - 1
[3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> 0101011011 => ? = 1 - 1
[3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? => ? = 1 - 1
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> [[1,2,3,4,5,6,9],[7,10],[8,11]]
=> ? => ? = 1 - 1
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? => ? = 2 - 1
[3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? => ? = 1 - 1
[3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ?
=> ? => ? = 1 - 1
[3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ?
=> ? => ? = 1 - 1
[3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? => ? = 1 - 1
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? => ? = 1 - 1
[2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ?
=> ? => ? = 1 - 1
[2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> [[1,2,3,4,5,6,8,10],[7,9,11]]
=> ? => ? = 1 - 1
[2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ?
=> ? => ? = 1 - 1
[2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? => ? = 1 - 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1 - 1
[10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ?
=> ? => ? = 1 - 1
[9,2,1]
=> [[1,3,6,7,8,9,10,11,12],[2,5],[4]]
=> ?
=> ? => ? = 1 - 1
[9,1,1,1]
=> [[1,5,6,7,8,9,10,11,12],[2],[3],[4]]
=> ?
=> ? => ? = 1 - 1
Description
The number of leading ones in a binary word.
Matching statistic: St000733
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00084: Standard tableaux āconjugateā¶ Standard tableaux
St000733: Standard tableaux ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Mp00084: Standard tableaux āconjugateā¶ Standard tableaux
St000733: Standard tableaux ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [[1,6,7],[2],[3],[4],[5]]
=> 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [[1,5,7],[2,6],[3],[4]]
=> 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [[1,5,6,7],[2],[3],[4]]
=> 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [[1,4,7],[2,5],[3,6]]
=> 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [[1,4,6],[2,5,7],[3]]
=> 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [[1,4,6,7],[2,5],[3]]
=> 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [[1,4,5,6,7],[2],[3]]
=> 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [[1,6,8],[2,7],[3],[4],[5]]
=> 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [[1,5,8],[2,6],[3,7],[4]]
=> 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [[1,5,7],[2,6,8],[3],[4]]
=> 2
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [[1,5,7,8],[2,6],[3],[4]]
=> 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [[1,5,6,7,8],[2],[3],[4]]
=> 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [[1,4,7],[2,5,8],[3,6]]
=> 2
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [[1,4,7,8],[2,5],[3,6]]
=> 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [[1,4,6,8],[2,5,7],[3]]
=> 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [[1,4,6,7,8],[2,5],[3]]
=> 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [[1,4,5,6,7,8],[2],[3]]
=> 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [[1,3,5,7,8],[2,4,6]]
=> 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [[1,3,5,6,7,8],[2,4]]
=> 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [[1,3,4,5,6,7,8],[2]]
=> 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [[1,7,9],[2,8],[3],[4],[5],[6]]
=> 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [[1,6,9],[2,7],[3,8],[4],[5]]
=> 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [[1,6,8],[2,7,9],[3],[4],[5]]
=> 2
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [[1,6,8,9],[2,7],[3],[4],[5]]
=> 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> 1
[9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1
[8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> [[1,9,11],[2,10],[3],[4],[5],[6],[7],[8]]
=> ? = 1
[8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1
[7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> [[1,8,11],[2,9],[3,10],[4],[5],[6],[7]]
=> ? = 1
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> [[1,8,10],[2,9,11],[3],[4],[5],[6],[7]]
=> ? = 2
[7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> [[1,8,10,11],[2,9],[3],[4],[5],[6],[7]]
=> ? = 1
[7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ?
=> ? = 1
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> [[1,7,10],[2,8,11],[3,9],[4],[5],[6]]
=> ? = 2
[6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> [[1,7,10,11],[2,8],[3,9],[4],[5],[6]]
=> ? = 1
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> [[1,7,9,11],[2,8,10],[3],[4],[5],[6]]
=> ? = 1
[6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> [[1,7,9,10,11],[2,8],[3],[4],[5],[6]]
=> ? = 1
[6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> [[1,6,11],[2,7],[3,8],[4,9],[5,10]]
=> ? = 1
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [[1,6,10],[2,7,11],[3,8],[4,9],[5]]
=> ? = 2
[5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [[1,6,10,11],[2,7],[3,8],[4,9],[5]]
=> ? = 1
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [[1,6,9],[2,7,10],[3,8,11],[4],[5]]
=> ? = 3
[5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [[1,6,9,11],[2,7,10],[3,8],[4],[5]]
=> ? = 1
[5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [[1,6,9,10,11],[2,7],[3,8],[4],[5]]
=> ? = 1
[5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [[1,6,8,10],[2,7,9,11],[3],[4],[5]]
=> ? = 2
[5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [[1,6,8,10,11],[2,7,9],[3],[4],[5]]
=> ? = 1
[5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> [[1,6,8,9,10,11],[2,7],[3],[4],[5]]
=> ? = 1
[5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8]]
=> ? = 3
[4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [[1,5,9,11],[2,6,10],[3,7],[4,8]]
=> ? = 1
[4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [[1,5,9,10,11],[2,6],[3,7],[4,8]]
=> ? = 1
[4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [[1,5,8,11],[2,6,9],[3,7,10],[4]]
=> ? = 1
[4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [[1,5,8,10],[2,6,9,11],[3,7],[4]]
=> ? = 2
[4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [[1,5,8,10,11],[2,6,9],[3,7],[4]]
=> ? = 1
[4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [[1,5,7,9,11],[2,6,8,10],[3],[4]]
=> ? = 1
[4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ?
=> ? = 1
[4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9]]
=> ? = 1
[3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [[1,4,7,10,11],[2,5,8],[3,6,9]]
=> ? = 1
[3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [[1,4,7,9,11],[2,5,8,10],[3,6]]
=> ? = 1
[3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ?
=> ? = 1
[3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> [[1,4,7,8,9,10,11],[2,5],[3,6]]
=> ? = 1
[3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> [[1,4,6,8,10],[2,5,7,9,11],[3]]
=> ? = 2
[3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ?
=> ? = 1
[3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> [[1,3,5,7,9,11],[2,4,6,8,10]]
=> ? = 1
[2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ?
=> ? = 1
[2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> [[1,3,5,7,8,9,10,11],[2,4,6]]
=> ? = 1
[2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00084: Standard tableaux āconjugateā¶ Standard tableaux
St000745: Standard tableaux ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Mp00084: Standard tableaux āconjugateā¶ Standard tableaux
St000745: Standard tableaux ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7]]
=> 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7]]
=> 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7]]
=> 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7]]
=> 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7]]
=> 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7]]
=> 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7],[8]]
=> 1
[5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> 1
[4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8]]
=> 1
[4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 2
[4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7],[8]]
=> 1
[4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8]]
=> 1
[3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [[1,3,6],[2,4,7],[5,8]]
=> 2
[3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [[1,2,3,6],[4,7],[5,8]]
=> 1
[3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7],[8]]
=> 1
[3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7],[8]]
=> 1
[3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7],[8]]
=> 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
[2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [[1,2,3,5,7],[4,6,8]]
=> 1
[2,2,1,1,1,1]
=> [[1,6],[2,8],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6,8]]
=> 1
[2,1,1,1,1,1,1]
=> [[1,8],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7],[8]]
=> 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [[1,2,3,4,5,6,7,8]]
=> 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7],[8],[9]]
=> 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8],[9]]
=> 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 2
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7],[8],[9]]
=> 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [[1,2,6],[3,7],[4,8],[5,9]]
=> 1
[9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ?
=> ? = 1
[8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ?
=> ? = 1
[8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ?
=> ? = 1
[7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ?
=> ? = 1
[7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
=> ? = 2
[7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ?
=> ? = 1
[7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ?
=> ? = 1
[6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ?
=> ? = 1
[6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ?
=> ? = 2
[6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ?
=> ? = 1
[6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ?
=> ? = 1
[6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7],[8],[9],[10],[11]]
=> ? = 1
[6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ?
=> ? = 1
[5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> [[1,2,7],[3,8],[4,9],[5,10],[6,11]]
=> ? = 1
[5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 2
[5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> ? = 1
[5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
=> ? = 3
[5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
=> ? = 1
[5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [[1,2,3,4,7],[5,8],[6,9],[10],[11]]
=> ? = 1
[5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 2
[5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [[1,2,3,5,7],[4,6,8],[9],[10],[11]]
=> ? = 1
[5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6,8],[9],[10],[11]]
=> ? = 1
[5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ?
=> ? = 1
[4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 3
[4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> ? = 1
[4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [[1,2,3,4,8],[5,9],[6,10],[7,11]]
=> ? = 1
[4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? = 1
[4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? = 2
[4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> ? = 1
[4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ?
=> ? = 1
[4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9],[10],[11]]
=> ? = 1
[4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ?
=> ? = 1
[4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ?
=> ? = 1
[4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ?
=> ? = 1
[3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 1
[3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? = 1
[3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 1
[3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? = 1
[3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> [[1,2,3,4,5,6,9],[7,10],[8,11]]
=> ? = 1
[3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 2
[3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? = 1
[3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ?
=> ? = 1
[3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ?
=> ? = 1
[3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? = 1
[2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? = 1
[2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ?
=> ? = 1
[2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> [[1,2,3,4,5,6,8,10],[7,9,11]]
=> ? = 1
[2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ?
=> ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000996
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00042: Integer partitions āinitial tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000996: Permutations ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000996: Permutations ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
[2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
[3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
[2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
[2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
[4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
[3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
[3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
[2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
[2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
[5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> [7,6,1,2,3,4,5] => 1
[4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => 1
[4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => 1
[3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => 1
[3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => 2
[3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => 1
[3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => 1
[2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => 1
[2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => 1
[2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => 1
[6,1,1]
=> [[1,2,3,4,5,6],[7],[8]]
=> [8,7,1,2,3,4,5,6] => 1
[5,2,1]
=> [[1,2,3,4,5],[6,7],[8]]
=> [8,6,7,1,2,3,4,5] => 1
[5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> [8,7,6,1,2,3,4,5] => 1
[4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> [8,5,6,7,1,2,3,4] => 1
[4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> [7,8,5,6,1,2,3,4] => 2
[4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> [8,7,5,6,1,2,3,4] => 1
[4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> [8,7,6,5,1,2,3,4] => 1
[3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => 2
[3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => 1
[3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => 1
[3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => 1
[3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => 1
[2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => 2
[2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => 1
[2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => 1
[2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => 1
[1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => 1
[7,1,1]
=> [[1,2,3,4,5,6,7],[8],[9]]
=> [9,8,1,2,3,4,5,6,7] => 1
[6,2,1]
=> [[1,2,3,4,5,6],[7,8],[9]]
=> [9,7,8,1,2,3,4,5,6] => 1
[6,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9]]
=> [9,8,7,1,2,3,4,5,6] => 1
[5,3,1]
=> [[1,2,3,4,5],[6,7,8],[9]]
=> [9,6,7,8,1,2,3,4,5] => 1
[5,2,2]
=> [[1,2,3,4,5],[6,7],[8,9]]
=> [8,9,6,7,1,2,3,4,5] => 2
[5,2,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9]]
=> [9,8,6,7,1,2,3,4,5] => 1
[5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> [9,8,7,6,1,2,3,4,5] => 1
[4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [9,5,6,7,8,1,2,3,4] => 1
[9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? = 1
[8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? => ? = 1
[8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 1
[7,3,1]
=> [[1,2,3,4,5,6,7],[8,9,10],[11]]
=> ? => ? = 1
[7,2,2]
=> [[1,2,3,4,5,6,7],[8,9],[10,11]]
=> ? => ? = 2
[7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? => ? = 1
[7,1,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10],[11]]
=> ? => ? = 1
[6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? => ? = 1
[6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? => ? = 2
[6,3,1,1]
=> [[1,2,3,4,5,6],[7,8,9],[10],[11]]
=> ? => ? = 1
[6,2,2,1]
=> [[1,2,3,4,5,6],[7,8],[9,10],[11]]
=> ? => ? = 1
[6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? => ? = 1
[6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? => ? = 1
[5,4,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11]]
=> [10,11,6,7,8,9,1,2,3,4,5] => ? = 2
[5,4,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11]]
=> [11,10,6,7,8,9,1,2,3,4,5] => ? = 1
[5,3,3]
=> [[1,2,3,4,5],[6,7,8],[9,10,11]]
=> [9,10,11,6,7,8,1,2,3,4,5] => ? = 3
[5,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11]]
=> [11,9,10,6,7,8,1,2,3,4,5] => ? = 1
[5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> [11,10,9,6,7,8,1,2,3,4,5] => ? = 1
[5,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11]]
=> [10,11,8,9,6,7,1,2,3,4,5] => ? = 2
[5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> [11,10,8,9,6,7,1,2,3,4,5] => ? = 1
[5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? = 1
[5,1,1,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? = 3
[4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [11,9,10,5,6,7,8,1,2,3,4] => ? = 1
[4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [11,10,9,5,6,7,8,1,2,3,4] => ? = 1
[4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? = 1
[4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [10,11,8,9,5,6,7,1,2,3,4] => ? = 2
[4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? = 1
[4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? = 1
[4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,5,6,1,2,3,4] => ? = 1
[4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? = 1
[4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? = 1
[3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? = 1
[3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? = 1
[3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? = 1
[3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? = 2
[3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? = 1
[3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? = 1
[3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? = 1
[2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? = 1
[2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000007
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00044: Integer partitions āconjugateā¶ Integer partitions
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000007: Permutations ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Mp00045: Integer partitions āreading tableauā¶ Standard tableaux
Mp00081: Standard tableaux āreading word permutationā¶ Permutations
St000007: Permutations ā¶ ā¤Result quality: 8% āvalues known / values provided: 8%ādistinct values known / distinct values provided: 38%
Values
[1,1,1]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 1
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1
[3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => 1
[2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => 1
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> [1,2,3,4,5] => 1
[4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => 1
[3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => 1
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [3,2,1,4,5,6] => 1
[2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 2
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => 1
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [2,1,3,4,5,6] => 1
[1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => 1
[5,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7] => 1
[4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => 1
[4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [4,3,2,1,5,6,7] => 1
[3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [5,6,3,4,1,2,7] => 1
[3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [5,2,6,7,1,3,4] => 2
[3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => 1
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [3,2,1,4,5,6,7] => 1
[2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => 1
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [3,4,1,2,5,6,7] => 1
[2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => 1
[1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [1,2,3,4,5,6,7] => 1
[6,1,1]
=> [3,1,1,1,1,1]
=> [[1,7,8],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8] => 1
[5,2,1]
=> [3,2,1,1,1]
=> [[1,5,8],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8] => 1
[5,1,1,1]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8] => 1
[4,3,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3,8] => 1
[4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [6,3,2,7,8,1,4,5] => 2
[4,2,1,1]
=> [4,2,1,1]
=> [[1,4,7,8],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7,8] => 1
[4,1,1,1,1]
=> [5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> [4,3,2,1,5,6,7,8] => 1
[3,3,2]
=> [3,3,2]
=> [[1,2,5],[3,4,8],[6,7]]
=> [6,7,3,4,8,1,2,5] => 2
[3,3,1,1]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [5,6,3,4,1,2,7,8] => 1
[3,2,2,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [5,2,6,7,1,3,4,8] => 1
[3,2,1,1,1]
=> [5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [4,2,5,1,3,6,7,8] => 1
[3,1,1,1,1,1]
=> [6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [3,2,1,4,5,6,7,8] => 1
[2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [5,6,7,8,1,2,3,4] => 2
[2,2,2,1,1]
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => 1
[2,2,1,1,1,1]
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => 1
[2,1,1,1,1,1,1]
=> [7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [2,1,3,4,5,6,7,8] => 1
[1,1,1,1,1,1,1,1]
=> [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => 1
[7,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1,8,9] => 1
[6,2,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [7,5,4,3,2,8,1,6,9] => 1
[6,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7,8,9] => 1
[5,3,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9] => 1
[5,2,2]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6] => 2
[5,2,1,1]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5,8,9] => 1
[5,1,1,1,1]
=> [5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [5,4,3,2,1,6,7,8,9] => 1
[4,4,1]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9] => 1
[9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? => ? = 1
[8,2,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? => ? = 1
[8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? => ? = 1
[7,3,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ? => ? = 1
[7,2,2]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? => ? = 2
[7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ? => ? = 1
[7,1,1,1,1]
=> [5,1,1,1,1,1,1]
=> [[1,8,9,10,11],[2],[3],[4],[5],[6],[7]]
=> ? => ? = 1
[6,4,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ? => ? = 1
[6,3,2]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ? => ? = 2
[6,3,1,1]
=> [4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ? => ? = 1
[6,2,2,1]
=> [4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ? => ? = 1
[6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? => ? = 1
[6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? = 1
[5,5,1]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? => ? = 1
[5,4,2]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [9,6,10,4,7,2,5,11,1,3,8] => ? = 2
[5,4,1,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [8,6,9,4,7,2,5,1,3,10,11] => ? = 1
[5,3,3]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [9,6,3,10,11,2,7,8,1,4,5] => ? = 3
[5,3,2,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [8,5,3,9,2,6,10,1,4,7,11] => ? = 1
[5,3,1,1,1]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [7,5,3,8,2,6,1,4,9,10,11] => ? = 1
[5,2,2,2]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [8,4,3,2,9,10,11,1,5,6,7] => ? = 2
[5,2,2,1,1]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [7,4,3,2,8,9,1,5,6,10,11] => ? = 1
[5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? => ? = 1
[5,1,1,1,1,1,1]
=> [7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2],[3],[4],[5]]
=> ? => ? = 1
[4,4,3]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [9,10,6,7,11,3,4,8,1,2,5] => ? = 3
[4,4,2,1]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [8,9,5,6,3,4,10,1,2,7,11] => ? = 1
[4,4,1,1,1]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2,9,10,11] => ? = 1
[4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [8,5,9,10,2,6,7,1,3,4,11] => ? = 1
[4,3,2,2]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [8,4,9,2,5,10,11,1,3,6,7] => ? = 2
[4,3,2,1,1]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [7,4,8,2,5,9,1,3,6,10,11] => ? = 1
[4,3,1,1,1,1]
=> [6,2,2,1]
=> [[1,3,8,9,10,11],[2,5],[4,7],[6]]
=> ? => ? = 1
[4,2,2,2,1]
=> [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [7,3,2,8,9,10,1,4,5,6,11] => ? = 1
[4,2,2,1,1,1]
=> [6,3,1,1]
=> [[1,4,5,9,10,11],[2,7,8],[3],[6]]
=> ? => ? = 1
[4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? => ? = 1
[4,1,1,1,1,1,1,1]
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? = 1
[3,3,3,2]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [8,9,10,4,5,6,11,1,2,3,7] => ? = 1
[3,3,3,1,1]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [7,8,9,4,5,6,1,2,3,10,11] => ? = 1
[3,3,2,2,1]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [7,8,3,4,9,10,1,2,5,6,11] => ? = 1
[3,3,2,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? => ? = 1
[3,3,1,1,1,1,1]
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> ? => ? = 1
[3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? => ? = 2
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? => ? = 1
[3,2,2,1,1,1,1]
=> [7,3,1]
=> [[1,3,4,8,9,10,11],[2,6,7],[5]]
=> ? => ? = 1
[3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? => ? = 1
[3,1,1,1,1,1,1,1,1]
=> [9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? = 1
[2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? => ? = 1
[2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? => ? = 1
[2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> ? => ? = 1
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? => ? = 1
[2,1,1,1,1,1,1,1,1,1]
=> [10,1]
=> [[1,3,4,5,6,7,8,9,10,11],[2]]
=> ? => ? = 1
[1,1,1,1,1,1,1,1,1,1,1]
=> [11]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? => ? = 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
The following 165 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000883The number of longest increasing subsequences of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St001267The length of the Lyndon factorization of the binary word. St001437The flex of a binary word. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St000445The number of rises of length 1 of a Dyck path. St000645The sum of the areas of the rectangles formed by two consecutive peaks and the valley in between. St000617The number of global maxima of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000306The bounce count of a Dyck path. St000657The smallest part of an integer composition. St000765The number of weak records in an integer composition. St000383The last part of an integer composition. St000245The number of ascents of a permutation. St000834The number of right outer peaks of a permutation. St000871The number of very big ascents of a permutation. St000359The number of occurrences of the pattern 23-1. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000742The number of big ascents of a permutation after prepending zero. St000842The breadth of a permutation. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000788The number of nesting-similar perfect matchings of a perfect matching. St000022The number of fixed points of a permutation. St000153The number of adjacent cycles of a permutation. St000352The Elizalde-Pak rank of a permutation. St000787The number of flips required to make a perfect matching noncrossing. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000054The first entry of the permutation. St000793The length of the longest partition in the vacillating tableau corresponding to a set partition. St000505The biggest entry in the block containing the 1. St000701The protection number of a binary tree. St000971The smallest closer of a set partition. St000990The first ascent of a permutation. St001737The number of descents of type 2 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St000023The number of inner peaks of a permutation. St000056The decomposition (or block) number of a permutation. St000221The number of strong fixed points of a permutation. St000646The number of big ascents of a permutation. St000663The number of right floats of a permutation. St000729The minimal arc length of a set partition. St000782The indicator function of whether a given perfect matching is an L & P matching. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001722The number of minimal chains with small intervals between a binary word and the top element. St000099The number of valleys of a permutation, including the boundary. St000546The number of global descents of a permutation. St000648The number of 2-excedences of a permutation. St000740The last entry of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000991The number of right-to-left minima of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{nā1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001948The number of augmented double ascents of a permutation. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St000031The number of cycles in the cycle decomposition of a permutation. St000823The number of unsplittable factors of the set partition. St001693The excess length of a longest path consisting of elements and blocks of a set partition. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000260The radius of a connected graph. St000486The number of cycles of length at least 3 of a permutation. St000504The cardinality of the first block of a set partition. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra $A/I$ when $I$ is the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000565The major index of a set partition. St000623The number of occurrences of the pattern 52341 in a permutation. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000273The domination number of a graph. St000374The number of exclusive right-to-left minima of a permutation. St000420The number of Dyck paths that are weakly above a Dyck path. St000544The cop number of a graph. St000678The number of up steps after the last double rise of a Dyck path. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000916The packing number of a graph. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n-1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a Dyck path as follows:
St001316The domatic number of a graph. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001471The magnitude of a Dyck path. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001672The restrained domination number of a graph. St001733The number of weak left to right maxima of a Dyck path. St001784The minimum of the smallest closer and the second element of the block containing 1 in a set partition. St001808The box weight or horizontal decoration of a Dyck path. St001829The common independence number of a graph. St000053The number of valleys of the Dyck path. St000090The variation of a composition. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000671The maximin edge-connectivity for choosing a subgraph. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000918The 2-limited packing number of a graph. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n-1}]$ by adding $c_0$ to $c_{n-1}$. St000970Number of peaks minus the dominant dimension of the corresponding LNakayama algebra. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001197The global dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001271The competition number of a graph. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001395The number of strictly unfriendly partitions of a graph. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001842The major index of a set partition. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000210Minimum over maximum difference of elements in cycles. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000534The number of 2-rises of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000642The size of the smallest orbit of antichains under Panyushev complementation. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St000516The number of stretching pairs of a permutation. St000989The number of final rises of a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001330The hat guessing number of a graph. St001513The number of nested exceedences of a permutation. St001537The number of cyclic crossings of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000862The number of parts of the shifted shape of a permutation.
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