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Your data matches 229 different statistics following compositions of up to 3 maps.
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Matching statistic: St001493
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Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St000232
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000232: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
Description
The number of crossings of a set partition.
This is given by the number of $i < i' < j < j'$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000233
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000233: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
Description
The number of nestings of a set partition.
This is given by the number of $i < i' < j' < j$ such that $i,j$ are two consecutive entries on one block, and $i',j'$ are consecutive entries in another block.
Matching statistic: St000496
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000496: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1 - 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
Description
The rcs statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''rcs''' (right-closer-smaller) of $S$ is given by a pair $i > j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a < b$.
Matching statistic: St000793
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000793: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 2 = 1 + 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2 = 1 + 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 1 + 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 2 = 1 + 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 2 = 1 + 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 2 = 1 + 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 2 = 1 + 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 2 = 1 + 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1 + 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 + 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 + 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 2 = 1 + 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 2 = 1 + 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 + 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 2 = 1 + 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 2 = 1 + 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 2 = 1 + 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 2 = 1 + 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 2 = 1 + 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 2 = 1 + 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 2 = 1 + 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 2 = 1 + 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 2 = 1 + 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 2 = 1 + 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 2 = 1 + 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 2 = 1 + 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 2 = 1 + 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 2 = 1 + 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1 + 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1 + 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2 + 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2 + 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1 + 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1 + 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1 + 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1 + 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2 + 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2 + 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1 + 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1 + 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1 + 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1 + 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1 + 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 + 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2 + 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2 + 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1 + 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1 + 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 2 = 1 + 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 + 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 2 = 1 + 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 2 = 1 + 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 2 = 1 + 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1 + 1
[1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> {{1,2},{3,4},{5,6},{7,8}}
=> 2 = 1 + 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 2 = 1 + 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 2 = 1 + 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 2 = 1 + 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 2 = 1 + 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 2 = 1 + 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 2 = 1 + 1
Description
The length of the longest partition in the vacillating tableau corresponding to a set partition.
To a set partition $\pi$ of $\{1,\dots,r\}$ with at most $n$ blocks we associate a vacillating tableau, following [1], as follows: create a triangular growth diagram by labelling the columns of a triangular grid with row lengths $r-1, \dots, 0$ from left to right $1$ to $r$, and the rows from the shortest to the longest $1$ to $r$. For each arc $(i,j)$ in the standard representation of $\pi$, place a cross into the cell in column $i$ and row $j$.
Next we label the corners of the first column beginning with the corners of the shortest row. The first corner is labelled with the partition $(n)$. If there is a cross in the row separating this corner from the next, label the next corner with the same partition, otherwise with the partition smaller by one. Do the same with the corners of the first row.
Finally, apply Fomin's local rules, to obtain the partitions along the diagonal. These will alternate in size between $n$ and $n-1$.
This statistic is the length of the longest partition on the diagonal of the diagram.
Matching statistic: St000695
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000695: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000695: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 1
Description
The number of blocks in the first part of the atomic decomposition of a set partition.
Let $\pi=(b_1,\dots,b_k)$ be a set partition with $k$ blocks, such that $\min b_i < \min b_{i+1}$. Then this statistic is the smallest number $\ell$ such that the union of the first $\ell$ blocks $b_1\cup\dots\cup b_\ell$ is an interval $\{1,\dots,m\}$.
The analogue for the decomposition of permutations is [[St000501]].
Matching statistic: St000491
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000491: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
Description
The number of inversions of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1], see also [2,3], an inversion of $S$ is given by a pair $i > j$ such that $j = \operatorname{min} B_b$ and $i \in B_a$ for $a < b$.
This statistic is called '''ros''' in [1, Definition 3] for "right, opener, smaller".
This is also the number of occurrences of the pattern {{1, 3}, {2}} such that 1 and 2 are minimal elements of blocks.
Matching statistic: St000497
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000497: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
Description
The lcb statistic of a set partition.
Let $S = B_1,\ldots,B_k$ be a set partition with ordered blocks $B_i$ and with $\operatorname{min} B_a < \operatorname{min} B_b$ for $a < b$.
According to [1, Definition 3], a '''lcb''' (left-closer-bigger) of $S$ is given by a pair $i < j$ such that $j = \operatorname{max} B_b$ and $i \in B_a$ for $a > b$.
Matching statistic: St000555
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000555: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000555: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
Description
The number of occurrences of the pattern {{1,3},{2}} in a set partition.
Matching statistic: St000559
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000559: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00284: Standard tableaux —rows⟶ Set partitions
St000559: Set partitions ⟶ ℤResult quality: 15% ●values known / values provided: 15%●distinct values known / distinct values provided: 33%
Values
[1,0,1,0,1,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,0,1,1,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10]]
=> {{1,2,3,4},{5,6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> {{1,2,3},{4,5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> {{1,2,3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> {{1,2,3},{4,5},{6,7},{8}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> {{1,2,3,4},{5,6,7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> {{1,2,3},{4,5,6},{7},{8}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [[1,2,3,4],[5,6],[7],[8]]
=> {{1,2,3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> {{1,2,3,4},{5,6,7},{8,9}}
=> ? = 1 - 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> {{1,2,3},{4,5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> {{1,2,3,4},{5,6},{7,8}}
=> ? = 1 - 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> {{1,2},{3,4},{5,6}}
=> 0 = 1 - 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> {{1,2,3,4},{5,6,7},{8}}
=> ? = 1 - 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> {{1,2,3},{4,5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> {{1,2,3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> {{1,2,3},{4,5},{6}}
=> 0 = 1 - 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> {{1,2},{3,4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> {{1,2,3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> {{1,2,3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> {{1,2},{3},{4}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [[1],[2],[3]]
=> {{1},{2},{3}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> {{1,2,3,4},{5,6,7}}
=> 0 = 1 - 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> {{1,2,3},{4,5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> {{1,2,3,4},{5,6}}
=> 0 = 1 - 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [[1,2,3],[4,5]]
=> {{1,2,3},{4,5}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [[1,2],[3,4]]
=> {{1,2},{3,4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [[1,2,3,4],[5]]
=> {{1,2,3,4},{5}}
=> 0 = 1 - 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [[1,2,3],[4]]
=> {{1,2,3},{4}}
=> 0 = 1 - 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [[1,2],[3]]
=> {{1,2},{3}}
=> 0 = 1 - 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [[1],[2]]
=> {{1},{2}}
=> 0 = 1 - 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14},{15}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [5,3,3,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12,13},{14}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12]]
=> {{1,2,3},{4,5,6},{7,8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12,13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12,13},{14}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11,12},{13}}
=> ? = 2 - 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,2,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11,12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11,12},{13}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> {{1,2,3},{4,5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12]]
=> {{1,2,3,4,5},{6,7},{8,9},{10,11},{12}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> {{1,2,3,4},{5,6},{7,8},{9,10},{11}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> {{1,2,3},{4,5},{6,7},{8,9},{10}}
=> ? = 1 - 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> {{1,2},{3,4},{5,6},{7,8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13],[14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13},{14}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [4,4,3,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4},{5,6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [5,3,3,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8},{9,10,11},{12},{13}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [4,3,3,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7},{8,9,10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> {{1,2,3},{4,5,6},{7,8,9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11],[12],[13]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11},{12},{13}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,0,1,1,0,0]
=> [4,4,2,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4},{5,6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8],[9,10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8},{9,10},{11},{12}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> {{1,2,3,4},{5,6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,0,1,1,0,0,0]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> {{1,2,3},{4,5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,0,1,0]
=> [5,2,2,1,1]
=> [[1,2,3,4,5],[6,7],[8,9],[10],[11]]
=> {{1,2,3,4,5},{6,7},{8,9},{10},{11}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,0,1,0,0]
=> [4,2,2,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10]]
=> {{1,2,3,4},{5,6},{7,8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,0,1,0,0,0]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> {{1,2,3},{4,5},{6,7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> {{1,2},{3,4},{5,6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10],[11],[12]]
=> {{1,2,3,4,5},{6,7,8,9},{10},{11},{12}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,0,1,1,0,0]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> {{1,2,3,4},{5,6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,1,1]
=> [[1,2,3,4,5],[6,7,8],[9],[10],[11]]
=> {{1,2,3,4,5},{6,7,8},{9},{10},{11}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10]]
=> {{1,2,3,4},{5,6,7},{8},{9},{10}}
=> ? = 2 - 1
[1,0,1,1,1,0,0,1,1,0,0,0]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> {{1,2,3},{4,5,6},{7},{8},{9}}
=> ? = 2 - 1
[1,0,1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10]]
=> {{1,2,3,4,5},{6,7},{8},{9},{10}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> {{1,2,3,4},{5,6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> {{1,2,3},{4,5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> {{1,2},{3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,0,0,0,1,0]
=> [5,1,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8],[9]]
=> {{1,2,3,4,5},{6},{7},{8},{9}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> {{1,2,3,4},{5},{6},{7},{8}}
=> ? = 1 - 1
[1,0,1,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> {{1,2,3},{4},{5},{6},{7}}
=> 0 = 1 - 1
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> {{1,2},{3},{4},{5},{6}}
=> 0 = 1 - 1
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> {{1},{2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14]]
=> {{1,2,3,4,5},{6,7,8,9},{10,11,12},{13,14}}
=> ? = 1 - 1
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> {{1,2},{3,4},{5,6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> {{1,2,3},{4,5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> {{1,2},{3,4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> {{1,2,3,4},{5},{6},{7}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> {{1,2,3},{4},{5},{6}}
=> 0 = 1 - 1
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> {{1,2},{3},{4},{5}}
=> 0 = 1 - 1
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> {{1},{2},{3},{4}}
=> 0 = 1 - 1
[1,1,1,0,0,1,1,0,1,0,0,0]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> {{1,2,3},{4,5},{6,7}}
=> 0 = 1 - 1
Description
The number of occurrences of the pattern {{1,3},{2,4}} in a set partition.
The following 219 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. 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. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000058The order of a permutation. St000091The descent variation of a composition. St000234The number of global ascents of a permutation. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000022The number of fixed points of a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St000788The number of nesting-similar perfect matchings of a perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000455The second largest eigenvalue of a graph if it is integral. St000056The decomposition (or block) number of a permutation. 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 $K[x]/(x^n)$. 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 $K[x]/(x^n)$. St001256Number of simple reflexive modules that are 2-stable reflexive. St001344The neighbouring number of a permutation. 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. St001590The crossing number of a perfect matching. St001722The number of minimal chains with small intervals between a binary word and the top element. 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. St000221The number of strong fixed points of 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 $3$. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. 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. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. 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. 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. 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. St001850The number of Hecke atoms of a permutation. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St000255The number of reduced Kogan faces with the permutation as type. St000078The number of alternating sign matrices whose left key is the permutation. St000570The Edelman-Greene number of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001162The minimum jump of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000485The length of the longest cycle of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000516The number of stretching pairs of a permutation. St000623The number of occurrences of the pattern 52341 in a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001330The hat guessing number of a graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000993The multiplicity of the largest part of an integer partition. St001884The number of borders of a binary word. St000295The length of the border of a binary word. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000929The constant term of the character polynomial of an integer partition. St001371The length of the longest Yamanouchi prefix of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001396Number of triples of incomparable elements in a finite poset. St001625The Möbius invariant of a lattice. St000617The number of global maxima of a Dyck path. St001621The number of atoms of a lattice. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000741The Colin de Verdière graph invariant. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001846The number of elements which do not have a complement in the lattice. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001498The normalised height of a Nakayama algebra with magnitude 1. St000306The bounce count of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St001845The number of join irreducibles minus the rank of a lattice. St000068The number of minimal elements in a poset. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St000065The number of entries equal to -1 in an alternating sign matrix. St001434The number of negative sum pairs of a signed permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000627The exponent of a binary word. St000878The number of ones minus the number of zeros of a binary word. St000124The cardinality of the preimage of the Simion-Schmidt map. St000297The number of leading ones in a binary word. St000895The number of ones on the main diagonal of an alternating sign matrix. St001964The interval resolution global dimension of a poset. St000052The number of valleys of a Dyck path not on the x-axis. St000629The defect of a binary word. St001771The number of occurrences of the signed pattern 1-2 in a signed 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. 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. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St000153The number of adjacent cycles of a permutation. St000210Minimum over maximum difference of elements in cycles. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001728The number of invisible descents of a permutation. St001260The permanent of an alternating sign matrix. St001429The number of negative entries in a signed permutation. St000352The Elizalde-Pak rank of a permutation. St000546The number of global descents of a permutation. St000842The breadth of a permutation. St001060The distinguishing index of a graph. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000007The number of saliances of the permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. 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 $eA$. St000383The last part of an integer composition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000381The largest part of an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St000264The girth of a graph, which is not a tree. St000782The indicator function of whether a given perfect matching is an L & P matching. St000237The number of small exceedances. St000648The number of 2-excedences of a permutation. St000731The number of double exceedences of a permutation. St000862The number of parts of the shifted shape of a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St001555The order of a signed permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000392The length of the longest run of ones in a binary word. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000807The sum of the heights of the valleys of the associated bargraph. St001134The largest label in the subtree rooted at the sister of 1 in the leaf labelled binary unordered tree associated with the perfect matching. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001862The number of crossings of a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001889The size of the connectivity set of a signed permutation. St000298The order dimension or Dushnik-Miller dimension of a poset. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St000717The number of ordinal summands of a poset. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets.
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