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Your data matches 106 different statistics following compositions of up to 3 maps.
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Matching statistic: St001604
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1,2,3,4}}
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [2,2,1]
=> [2,1]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,3,5},{4,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [3,3]
=> [3]
=> 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [3,3]
=> [3]
=> 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2},{3,4,5,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [2,2,2]
=> [2,2]
=> 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [3,3]
=> [3]
=> 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [2,2,2]
=> [2,2]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [2,2,2]
=> [2,2]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [3,3]
=> [3]
=> 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 0
{{1,3,4,5},{2,6}}
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 0
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001803
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [[1,3,5,7],[2,4,6,8]]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 0
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 0
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 0
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,8],[2,4,6,9,10]]
=> ? = 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [[1,2,3,6],[4,5,7,8]]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,3,5,7,9,11],[2,4,6,8,10,12]]
=> ? = 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> ? = 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> ? = 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> ? = 0
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> ? = 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> ? = 0
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,3,4,6},{2},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,3,5,6},{2,4}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,3},{2,4,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,3,5,6},{2},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,4,5,6},{2,3}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,4},{2,3,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,5},{2,3,4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1,6},{2,3,4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[1,3,5,6,7],[2,4,8,9,10]]
=> ? = 0
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[1,3,5,7,9,10],[2,4,6,8,11,12]]
=> ? = 0
{{1},{2,3,4,5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1},{2,3,4,6},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1},{2,3,5,6},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [[1,3,5,7,8,10],[2,4,6,9,11,12]]
=> ? = 0
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [[1,3,5,6,8,10],[2,4,7,9,11,12]]
=> ? = 1
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [[1,2,3,4],[5,6,7,8]]
=> 1
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau $T$ is the skew row-strict tableau obtained by gluing two copies of $T$ such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals $\max_C\big(2\ell(T) - \ell(C)\big)$, where $\ell$ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals $0$, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
Matching statistic: St001424
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001424: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001424: Binary words ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 10101010 => 3 = 0 + 3
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => ? = 0 + 3
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0 + 3
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0 + 3
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0 + 3
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0 + 3
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => ? = 0 + 3
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 11100100 => 3 = 0 + 3
{{1,2,3,4,5,6}}
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 101010101010 => ? = 1 + 3
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0 + 3
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0 + 3
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0 + 3
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0 + 3
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0 + 3
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,3,4,6},{2},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,3,5,6},{2,4}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,3},{2,4,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,3,5,6},{2},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,4,5,6},{2,3}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,4},{2,3,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,5},{2,3,4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1,6},{2,3,4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => ? = 0 + 3
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 101010101100 => ? = 0 + 3
{{1},{2,3,4,5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1},{2,3,4,6},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1},{2,3,5,6},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 101010110100 => ? = 0 + 3
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 101011010100 => ? = 1 + 3
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 11110000 => 4 = 1 + 3
Description
The number of distinct squares in a binary word.
A factor of a word is a sequence of consecutive letters. This statistic records the number of distinct non-empty words $u$ such that $uu$ is a factor of the word.
Note that every word of length at least four contains a square.
Matching statistic: St001515
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001515: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
{{1,2,3,4}}
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4 = 0 + 4
{{1,2,3,4,5}}
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 0 + 4
{{1,2,3,4},{5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,2,3,5},{4}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,2,4,5},{3}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,3,4,5},{2}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1},{2,3,4,5}}
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 4 = 0 + 4
{{1,2,3,4,5,6}}
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1 + 4
{{1,2,3,4,5},{6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,2,3,4,6},{5}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,2,3,4},{5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,2,3,5,6},{4}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,2,3,5},{4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,2,3,6},{4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,2,4,5,6},{3}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,2,4,5},{3,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,2,4,6},{3,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,2,5,6},{3,4}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,2},{3,4,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,3,4,5,6},{2}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1,3,4,5},{2,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,3,4,6},{2,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,3,4,6},{2},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,3,5,6},{2,4}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,3},{2,4,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,3,5,6},{2},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,4,5,6},{2,3}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,4},{2,3,5,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,5},{2,3,4,6}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1,6},{2,3,4,5}}
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> ? = 0 + 4
{{1},{2,3,4,5,6}}
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,0,0]
=> ? = 0 + 4
{{1},{2,3,4,5},{6}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1},{2,3,4,6},{5}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1},{2,3,5,6},{4}}
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 0 + 4
{{1},{2,3,5},{4},{6}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,6},{2,3},{4,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1},{2,3,6},{4},{5}}
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 1 + 4
{{1,4},{2,5},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,4},{2,6},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,5},{2,4},{3,6}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,6},{2,4},{3,5}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,5},{2,6},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
{{1,6},{2,5},{3,4}}
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 5 = 1 + 4
Description
The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule).
Matching statistic: St000754
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000754: Perfect matchings ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000754: Perfect matchings ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 29%
Values
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> [(1,8),(2,7),(3,6),(4,5)]
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6)]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10)]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0
{{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,10)]
=> 0
{{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0
{{1,5},{2},{3,4}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [(1,4),(2,3),(5,6),(7,10),(8,9)]
=> 0
{{1},{2,5},{3,4}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [(1,2),(3,6),(4,5),(7,10),(8,9)]
=> 0
{{1,2,3,4,5,6}}
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> [(1,12),(2,11),(3,10),(4,9),(5,8),(6,7)]
=> ? = 1
{{1,2,3,4,5},{6}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 0
{{1,2,3,4,6},{5}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 0
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 0
{{1,2,3,5},{4,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 0
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,2,4,6},{3,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,2,4,6},{3},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,2,5,6},{3,4}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,2},{3,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,2,5,6},{3},{4}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [(1,10),(2,9),(3,8),(4,7),(5,6),(11,12)]
=> ? = 0
{{1,3,4,5},{2,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,3,4,6},{2},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,3,5,6},{2,4}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,3},{2,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,3,5,6},{2},{4}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,10),(11,12)]
=> ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8),(9,10),(11,12)]
=> ? = 1
{{1,4,5,6},{2,3}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [(1,8),(2,7),(3,6),(4,5),(9,12),(10,11)]
=> ? = 0
{{1,4},{2,3,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0
{{1,4},{2,3},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1,5},{2,3,4,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0
{{1,6},{2,3,4,5}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [(1,4),(2,3),(5,12),(6,11),(7,10),(8,9)]
=> ? = 0
{{1},{2,3,4,5,6}}
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> [(1,2),(3,12),(4,11),(5,10),(6,9),(7,8)]
=> ? = 0
{{1},{2,3,4,5},{6}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> ? = 0
{{1},{2,3,4,6},{5}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> ? = 0
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [(1,2),(3,8),(4,7),(5,6),(9,10),(11,12)]
=> ? = 1
{{1,5},{2,3},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7),(9,12),(10,11)]
=> ? = 1
{{1},{2,3,5,6},{4}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7),(11,12)]
=> ? = 0
{{1},{2},{3,4,5,6}}
=> [1,1,4] => [1,0,1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,4),(5,12),(6,11),(7,10),(8,9)]
=> 0
{{1},{2},{3,4,5},{6}}
=> [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> 1
{{1},{2},{3,4,6},{5}}
=> [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> 1
{{1},{2},{3,5,6},{4}}
=> [1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [(1,2),(3,4),(5,10),(6,9),(7,8),(11,12)]
=> 1
{{1},{2},{3},{4,5,6}}
=> [1,1,1,3] => [1,0,1,0,1,0,1,1,1,0,0,0]
=> [(1,2),(3,4),(5,6),(7,12),(8,11),(9,10)]
=> 1
{{1},{2},{3},{4},{5},{6}}
=> [1,1,1,1,1,1] => [1,0,1,0,1,0,1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6),(7,8),(9,10),(11,12)]
=> 0
Description
The Grundy value for the game of removing nestings in a perfect matching.
A move consists of choosing a nesting, that is two pairs $(a,d)$ and $(b,c)$ with $a < b < c < d$ and replacing them with the two pairs $(a,b)$ and $(c,d)$. The player facing a non-nesting matching looses.
Matching statistic: St000689
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St000689: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1,5},{2},{3,4}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1},{2,5},{3,4}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1,2,3,4,5,6}}
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
{{1,2,3,4,5},{6}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,4,6},{5}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,5},{4,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,4,6},{3,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,4,6},{3},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,5,6},{3,4}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2},{3,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,2,5,6},{3},{4}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,3,4,5},{2,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,3,4,6},{2},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,3,5,6},{2,4}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,3},{2,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,3,5,6},{2},{4}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,4,5,6},{2,3}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,4},{2,3,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1,4},{2,3},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,3,4,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1,6},{2,3,4,5}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1},{2,3,4,5,6}}
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
{{1},{2,3,4,5},{6}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3,4,6},{5}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
{{1,5},{2,3},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1},{2,3,5,6},{4}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
Description
The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid.
The correspondence between LNakayama algebras and Dyck paths is explained in [[St000684]]. A module $M$ is $n$-rigid, if $\operatorname{Ext}^i(M,M)=0$ for $1\leq i\leq n$.
This statistic gives the maximal $n$ such that the minimal generator-cogenerator module $A \oplus D(A)$ of the LNakayama algebra $A$ corresponding to a Dyck path is $n$-rigid.
An application is to check for maximal $n$-orthogonal objects in the module category in the sense of [2].
Matching statistic: St001314
(load all 25 compositions to match this statistic)
(load all 25 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001314: Dyck paths ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
{{1,2,3,4}}
=> [4] => [1,1,1,1,0,0,0,0]
=> 0
{{1,2,3,4,5}}
=> [5] => [1,1,1,1,1,0,0,0,0,0]
=> 0
{{1,2,3,4},{5}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,3,5},{4}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2,4,5},{3}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,2},{3},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,3,4,5},{2}}
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,3},{2},{4,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1},{2,3,4,5}}
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1},{2,3},{4,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1,4},{2},{3,5}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 0
{{1},{2,4},{3,5}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1,5},{2},{3,4}}
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 0
{{1},{2,5},{3,4}}
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 0
{{1,2,3,4,5,6}}
=> [6] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
{{1,2,3,4,5},{6}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,4,6},{5}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,4},{5,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,3,5},{4,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,3,6},{4},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,2,4,5},{3,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,4,6},{3,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2,4,6},{3},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2,5,6},{3,4}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,2},{3,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,2,5,6},{3},{4}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> ? = 0
{{1,3,4,5},{2,6}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,3,4,6},{2},{5}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,3,5,6},{2,4}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,3},{2,4,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,3,5,6},{2},{4}}
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,3},{2,6},{4,5}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> ? = 1
{{1,4,5,6},{2,3}}
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> ? = 0
{{1,4},{2,3,5,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1,4},{2,3},{5,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1,5},{2,3,4,6}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1,6},{2,3,4,5}}
=> [2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> ? = 0
{{1},{2,3,4,5,6}}
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 0
{{1},{2,3,4,5},{6}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3,4,6},{5}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
{{1},{2,3,4},{5},{6}}
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> ? = 1
{{1,5},{2,3},{4,6}}
=> [2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> ? = 1
{{1},{2,3,5,6},{4}}
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> ? = 0
Description
The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra.
Matching statistic: St001429
(load all 50 compositions to match this statistic)
(load all 50 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001429: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001429: Signed permutations ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => 0
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => 0
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => 0
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => 0
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => 0
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => 0
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => 0
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => 0
{{1,3,4,5},{2}}
=> [3,2,4,5,1] => [3,2,4,5,1] => 0
{{1,3},{2,4},{5}}
=> [3,4,1,2,5] => [3,4,1,2,5] => 0
{{1,3},{2,5},{4}}
=> [3,5,1,4,2] => [3,5,1,4,2] => 0
{{1,3},{2},{4,5}}
=> [3,2,1,5,4] => [3,2,1,5,4] => 0
{{1,4},{2,3},{5}}
=> [4,3,2,1,5] => [4,3,2,1,5] => 0
{{1},{2,3,4,5}}
=> [1,3,4,5,2] => [1,3,4,5,2] => 0
{{1,5},{2,3},{4}}
=> [5,3,2,4,1] => [5,3,2,4,1] => 0
{{1},{2,3},{4,5}}
=> [1,3,2,5,4] => [1,3,2,5,4] => 0
{{1,4},{2,5},{3}}
=> [4,5,3,1,2] => [4,5,3,1,2] => 0
{{1,4},{2},{3,5}}
=> [4,2,5,1,3] => [4,2,5,1,3] => 0
{{1,5},{2,4},{3}}
=> [5,4,3,2,1] => [5,4,3,2,1] => 0
{{1},{2,4},{3,5}}
=> [1,4,5,2,3] => [1,4,5,2,3] => 0
{{1,5},{2},{3,4}}
=> [5,2,4,3,1] => [5,2,4,3,1] => 0
{{1},{2,5},{3,4}}
=> [1,5,4,3,2] => [1,5,4,3,2] => 0
{{1,2,3,4,5,6}}
=> [2,3,4,5,6,1] => [2,3,4,5,6,1] => ? = 1
{{1,2,3,4,5},{6}}
=> [2,3,4,5,1,6] => [2,3,4,5,1,6] => ? = 0
{{1,2,3,4,6},{5}}
=> [2,3,4,6,5,1] => [2,3,4,6,5,1] => ? = 0
{{1,2,3,4},{5,6}}
=> [2,3,4,1,6,5] => [2,3,4,1,6,5] => ? = 0
{{1,2,3,4},{5},{6}}
=> [2,3,4,1,5,6] => [2,3,4,1,5,6] => ? = 0
{{1,2,3,5,6},{4}}
=> [2,3,5,4,6,1] => [2,3,5,4,6,1] => ? = 0
{{1,2,3,5},{4,6}}
=> [2,3,5,6,1,4] => [2,3,5,6,1,4] => ? = 0
{{1,2,3,5},{4},{6}}
=> [2,3,5,4,1,6] => [2,3,5,4,1,6] => ? = 0
{{1,2,3,6},{4,5}}
=> [2,3,6,5,4,1] => [2,3,6,5,4,1] => ? = 0
{{1,2,3,6},{4},{5}}
=> [2,3,6,4,5,1] => [2,3,6,4,5,1] => ? = 0
{{1,2,3},{4},{5},{6}}
=> [2,3,1,4,5,6] => [2,3,1,4,5,6] => ? = 1
{{1,2,4,5,6},{3}}
=> [2,4,3,5,6,1] => [2,4,3,5,6,1] => ? = 0
{{1,2,4,5},{3,6}}
=> [2,4,6,5,1,3] => [2,4,6,5,1,3] => ? = 0
{{1,2,4,5},{3},{6}}
=> [2,4,3,5,1,6] => [2,4,3,5,1,6] => ? = 0
{{1,2,4,6},{3,5}}
=> [2,4,5,6,3,1] => [2,4,5,6,3,1] => ? = 0
{{1,2,4,6},{3},{5}}
=> [2,4,3,6,5,1] => [2,4,3,6,5,1] => ? = 0
{{1,2,4},{3},{5},{6}}
=> [2,4,3,1,5,6] => [2,4,3,1,5,6] => ? = 1
{{1,2,5,6},{3,4}}
=> [2,5,4,3,6,1] => [2,5,4,3,6,1] => ? = 0
{{1,2},{3,4,5,6}}
=> [2,1,4,5,6,3] => [2,1,4,5,6,3] => ? = 0
{{1,2},{3,4},{5,6}}
=> [2,1,4,3,6,5] => [2,1,4,3,6,5] => ? = 1
{{1,2,5,6},{3},{4}}
=> [2,5,3,4,6,1] => [2,5,3,4,6,1] => ? = 0
{{1,2,5},{3},{4},{6}}
=> [2,5,3,4,1,6] => [2,5,3,4,1,6] => ? = 1
{{1,2},{3,5},{4,6}}
=> [2,1,5,6,3,4] => [2,1,5,6,3,4] => ? = 1
{{1,2},{3,6},{4,5}}
=> [2,1,6,5,4,3] => [2,1,6,5,4,3] => ? = 1
{{1,2,6},{3},{4},{5}}
=> [2,6,3,4,5,1] => [2,6,3,4,5,1] => ? = 1
{{1,3,4,5,6},{2}}
=> [3,2,4,5,6,1] => [3,2,4,5,6,1] => ? = 0
{{1,3,4,5},{2,6}}
=> [3,6,4,5,1,2] => [3,6,4,5,1,2] => ? = 0
{{1,3,4,5},{2},{6}}
=> [3,2,4,5,1,6] => [3,2,4,5,1,6] => ? = 0
{{1,3,4,6},{2,5}}
=> [3,5,4,6,2,1] => [3,5,4,6,2,1] => ? = 0
{{1,3,4,6},{2},{5}}
=> [3,2,4,6,5,1] => [3,2,4,6,5,1] => ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,2,4,1,5,6] => [3,2,4,1,5,6] => ? = 1
{{1,3,5,6},{2,4}}
=> [3,4,5,2,6,1] => [3,4,5,2,6,1] => ? = 0
{{1,3},{2,4,5,6}}
=> [3,4,1,5,6,2] => [3,4,1,5,6,2] => ? = 0
{{1,3},{2,4},{5,6}}
=> [3,4,1,2,6,5] => [3,4,1,2,6,5] => ? = 1
{{1,3,5,6},{2},{4}}
=> [3,2,5,4,6,1] => [3,2,5,4,6,1] => ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,2,5,4,1,6] => [3,2,5,4,1,6] => ? = 1
{{1,3},{2,5},{4,6}}
=> [3,5,1,6,2,4] => [3,5,1,6,2,4] => ? = 1
{{1,3},{2,6},{4,5}}
=> [3,6,1,5,4,2] => [3,6,1,5,4,2] => ? = 1
{{1,3,6},{2},{4},{5}}
=> [3,2,6,4,5,1] => [3,2,6,4,5,1] => ? = 1
{{1,4,5,6},{2,3}}
=> [4,3,2,5,6,1] => [4,3,2,5,6,1] => ? = 0
{{1,4},{2,3,5,6}}
=> [4,3,5,1,6,2] => [4,3,5,1,6,2] => ? = 0
{{1,4},{2,3},{5,6}}
=> [4,3,2,1,6,5] => [4,3,2,1,6,5] => ? = 1
{{1,5},{2,3,4,6}}
=> [5,3,4,6,1,2] => [5,3,4,6,1,2] => ? = 0
{{1,6},{2,3,4,5}}
=> [6,3,4,5,2,1] => [6,3,4,5,2,1] => ? = 0
{{1},{2,3,4,5,6}}
=> [1,3,4,5,6,2] => [1,3,4,5,6,2] => ? = 0
{{1},{2,3,4,5},{6}}
=> [1,3,4,5,2,6] => [1,3,4,5,2,6] => ? = 0
{{1},{2,3,4,6},{5}}
=> [1,3,4,6,5,2] => [1,3,4,6,5,2] => ? = 0
{{1},{2,3,4},{5},{6}}
=> [1,3,4,2,5,6] => [1,3,4,2,5,6] => ? = 1
{{1,5},{2,3},{4,6}}
=> [5,3,2,6,1,4] => [5,3,2,6,1,4] => ? = 1
{{1},{2,3,5,6},{4}}
=> [1,3,5,4,6,2] => [1,3,5,4,6,2] => ? = 0
Description
The number of negative entries in a signed permutation.
Matching statistic: St001435
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
{{1,2,3,4}}
=> [4]
=> [[4],[]]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [[5],[]]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> [[6],[]]
=> ? = 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,2,3,5},{4,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2},{3,4,5,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,3,4,5},{2,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,3,4,6},{2},{5}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,3,5,6},{2,4}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,3},{2,4,5,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,3,5,6},{2},{4}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,4,5,6},{2,3}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,4},{2,3,5,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,5},{2,3,4,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,6},{2,3,4,5}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1},{2,3,4,5,6}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1},{2,3,4,5},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1},{2,3,4,6},{5}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1},{2,3,5,6},{4}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
Description
The number of missing boxes in the first row.
Matching statistic: St001438
(load all 20 compositions to match this statistic)
(load all 20 compositions to match this statistic)
Mp00079: Set partitions —shape⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 1% ●values known / values provided: 1%●distinct values known / distinct values provided: 14%
Values
{{1,2,3,4}}
=> [4]
=> [[4],[]]
=> 0
{{1,2,3,4,5}}
=> [5]
=> [[5],[]]
=> 0
{{1,2,3,4},{5}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,2,3,5},{4}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,2,4,5},{3}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,2},{3,4},{5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,2},{3,5},{4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,2},{3},{4,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,3,4,5},{2}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,3},{2,4},{5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,3},{2,5},{4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,3},{2},{4,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,4},{2,3},{5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1},{2,3,4,5}}
=> [4,1]
=> [[4,1],[]]
=> 0
{{1,5},{2,3},{4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1},{2,3},{4,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,4},{2,5},{3}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,4},{2},{3,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,5},{2,4},{3}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1},{2,4},{3,5}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,5},{2},{3,4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1},{2,5},{3,4}}
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
{{1,2,3,4,5,6}}
=> [6]
=> [[6],[]]
=> ? = 1
{{1,2,3,4,5},{6}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,2,3,4,6},{5}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,2,3,4},{5,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,3,4},{5},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,3,5,6},{4}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,2,3,5},{4,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,3,5},{4},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,3,6},{4,5}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,3,6},{4},{5}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,3},{4},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,2,4,5,6},{3}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,2,4,5},{3,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,4,5},{3},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,4,6},{3,5}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2,4,6},{3},{5}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,4},{3},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,2,5,6},{3,4}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2},{3,4,5,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,2},{3,4},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,2,5,6},{3},{4}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,2,5},{3},{4},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,2},{3,5},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,2},{3,6},{4,5}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,2,6},{3},{4},{5}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,3,4,5,6},{2}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1,3,4,5},{2,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,3,4,5},{2},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,3,4,6},{2,5}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,3,4,6},{2},{5}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,3,4},{2},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,3,5,6},{2,4}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,3},{2,4,5,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,3},{2,4},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,3,5,6},{2},{4}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1,3,5},{2},{4},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,3},{2,5},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,3},{2,6},{4,5}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,3,6},{2},{4},{5}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,4,5,6},{2,3}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,4},{2,3,5,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,4},{2,3},{5,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1,5},{2,3,4,6}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1,6},{2,3,4,5}}
=> [4,2]
=> [[4,2],[]]
=> ? = 0
{{1},{2,3,4,5,6}}
=> [5,1]
=> [[5,1],[]]
=> ? = 0
{{1},{2,3,4,5},{6}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1},{2,3,4,6},{5}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
{{1},{2,3,4},{5},{6}}
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 1
{{1,5},{2,3},{4,6}}
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 1
{{1},{2,3,5,6},{4}}
=> [4,1,1]
=> [[4,1,1],[]]
=> ? = 0
Description
The number of missing boxes of a skew partition.
The following 96 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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 $K[x]/(x^n)$. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001569The maximal modular displacement of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000221The number of strong fixed points of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000315The number of isolated vertices of a graph. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length $3$. St000406The number of occurrences of the pattern 3241 in a permutation. St000488The number of cycles of a permutation of length at most 2. St000489The number of cycles of a permutation of length at most 3. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000666The number of right tethers of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001061The number of indices that are both descents and recoils of a permutation. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001204Call 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 special CNakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001381The fertility of a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001430The number of positive entries in a signed permutation. St001434The number of negative sum pairs of a signed 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. St001513The number of nested exceedences of a permutation. St001520The number of strict 3-descents. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. 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. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001831The multiplicity of the non-nesting perfect matching in the chord expansion of a perfect matching. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000056The decomposition (or block) number of a permutation. St000069The number of maximal elements of a poset. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000181The number of connected components of the Hasse diagram for the poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000486The number of cycles of length at least 3 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. St000694The number of affine bounded permutations that project to a given permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000968We 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 $c_{n−1}$. St001081The number of minimal length factorizations of a permutation into star transpositions. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001590The crossing number of a perfect matching. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001665The number of pure excedances of a permutation. St001761The maximal multiplicity of a letter in a reduced word of a permutation. 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. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000542The number of left-to-right-minima of a permutation. St001152The number of pairs with even minimum in a perfect matching. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001734The lettericity of a graph. St001741The largest integer such that all patterns of this size are contained in the permutation. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001488The number of corners of a skew partition. St001738The minimal order of a graph which is not an induced subgraph of the given graph.
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