Processing math: 100%

Your data matches 12 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St001488
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
St001488: Skew partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1,0]
 => [[1],[]]
 => 1
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [[2],[]]
 => 2
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [[1,1],[]]
 => 2
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [[3],[]]
 => 2
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [[2,2],[1]]
 => 3
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [[2,2],[]]
 => 2
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [[2,1],[]]
 => 3
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [[1,1,1],[]]
 => 2
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [[4],[]]
 => 2
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [[3,3],[2]]
 => 3
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [[3,3],[1]]
 => 3
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [[3,2],[1]]
 => 4
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [[2,2,2],[1,1]]
 => 3
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [[3,2],[]]
 => 3
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [[2,2,2],[1]]
 => 3
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [[3,1],[]]
 => 3
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [[2,2,1],[1]]
 => 4
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [[2,2,1],[]]
 => 3
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [[2,1,1],[]]
 => 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [[1,1,1,1],[]]
 => 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [[5],[]]
 => 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [[4,4],[3]]
 => 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [[4,3],[2]]
 => 4
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [[3,3,3],[2,2]]
 => 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [[4,2],[1]]
 => 4
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [[3,3,2],[2,1]]
 => 5
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [[3,2,2],[1,1]]
 => 4
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [[2,2,2,2],[1,1,1]]
 => 3
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [[4,1],[]]
 => 3
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [[3,3,1],[2]]
 => 4
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [[3,2,1],[1]]
 => 5
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [[2,2,2,1],[1,1]]
 => 4
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [[3,1,1],[]]
 => 3
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [[2,2,1,1],[1]]
 => 4
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [[2,1,1,1],[]]
 => 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [[1,1,1,1,1],[]]
 => 2
Description
The number of corners of a skew partition. This is also known as the number of removable cells of the skew partition.
Matching statistic: St000619
(load all 2 compositions to match this statistic)
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
St000619: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 97%●distinct values known / distinct values provided: 80%
Values
{{1}}
 => [1] => [1] => [1] => ? = 1 - 1
{{1,2}}
 => [2,1] => [2,1] => [2,1] => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,2] => [1,2] => 1 = 2 - 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => [2,3,1] => 1 = 2 - 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => [2,1,3] => 2 = 3 - 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => [3,1,2] => 1 = 2 - 1
{{1},{2,3}}
 => [1,3,2] => [2,3,1] => [3,2,1] => 2 = 3 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => [1,2,3] => 1 = 2 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => [2,3,4,1] => 1 = 2 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => [2,3,1,4] => 2 = 3 - 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,3,1] => [4,3,1,2] => 2 = 3 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [3,2,4,1] => [4,3,2,1] => 3 = 4 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => [2,1,3,4] => 2 = 3 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [4,2,1,3] => [3,1,4,2] => 2 = 3 - 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,2,1,4] => [3,1,2,4] => 2 = 3 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [2,4,1,3] => [3,2,4,1] => 2 = 3 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [2,3,1,4] => [3,2,1,4] => 3 = 4 - 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [3,4,2,1] => [4,1,3,2] => 2 = 3 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [2,3,4,1] => [4,2,3,1] => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => [1,2,3,4] => 1 = 2 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => [2,3,4,5,1] => 1 = 2 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => [2,3,4,1,5] => 2 = 3 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [4,2,3,5,1] => [5,3,4,2,1] => 3 = 4 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => [2,3,1,4,5] => 2 = 3 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [3,2,5,1,4] => [4,3,2,5,1] => 3 = 4 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [3,2,4,1,5] => [4,3,2,1,5] => 4 = 5 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [3,2,4,5,1] => [5,3,2,4,1] => 3 = 4 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => [2,1,3,4,5] => 2 = 3 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [2,5,1,3,4] => [3,2,4,5,1] => 2 = 3 - 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [2,4,1,3,5] => [3,2,4,1,5] => 3 = 4 - 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [3,4,2,5,1] => [5,4,3,2,1] => 4 = 5 - 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [2,3,1,4,5] => [3,2,1,4,5] => 3 = 4 - 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [2,3,5,1,4] => [4,2,3,5,1] => 2 = 3 - 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => 3 = 4 - 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [2,3,4,5,1] => [5,2,3,4,1] => 2 = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => 1 = 2 - 1
Description
The number of cyclic descents of a permutation. For a permutation π of {1,…,n}, this is given by the number of indices 1≤i≤n such that π(i)>π(i+1) where we set π(n+1)=π(1).
Matching statistic: St001499
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00296: Dyck paths —Knuth-Krattenthaler⟶ Dyck paths
St001499: Dyck paths ⟶ ℤResult quality: 80% ●values known / values provided: 97%●distinct values known / distinct values provided: 80%
Values
{{1}}
 => [1] => [1,0]
 => [1,0]
 => ? = 1 - 1
{{1,2}}
 => [2,1] => [1,1,0,0]
 => [1,0,1,0]
 => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,0,1,0]
 => [1,1,0,0]
 => 1 = 2 - 1
{{1,2,3}}
 => [2,3,1] => [1,1,0,1,0,0]
 => [1,0,1,1,0,0]
 => 1 = 2 - 1
{{1,2},{3}}
 => [2,1,3] => [1,1,0,0,1,0]
 => [1,0,1,0,1,0]
 => 2 = 3 - 1
{{1,3},{2}}
 => [3,2,1] => [1,1,1,0,0,0]
 => [1,1,0,0,1,0]
 => 1 = 2 - 1
{{1},{2,3}}
 => [1,3,2] => [1,0,1,1,0,0]
 => [1,1,0,1,0,0]
 => 2 = 3 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 1 = 2 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [1,1,0,1,0,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 1 = 2 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [1,1,0,1,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 2 = 3 - 1
{{1,2,4},{3}}
 => [2,4,3,1] => [1,1,0,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2 = 3 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [1,1,0,0,1,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 3 = 4 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [1,1,0,0,1,0,1,0]
 => [1,0,1,1,0,0,1,0]
 => 2 = 3 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [1,1,1,0,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 2 = 3 - 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [1,1,1,0,0,0,1,0]
 => [1,1,0,1,0,0,1,0]
 => 2 = 3 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,0,1,1,0,1,0,0]
 => [1,1,0,1,1,0,0,0]
 => 2 = 3 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
 => [1,0,1,1,1,1,0,0,0,0]
 => 1 = 2 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2 = 3 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3 = 4 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
 => [1,0,1,1,1,0,0,1,0,0]
 => 2 = 3 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
 => [1,0,1,0,1,1,0,0,1,0]
 => 3 = 4 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0]
 => 4 = 5 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
 => [1,0,1,1,0,1,0,0,1,0]
 => 3 = 4 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
 => [1,0,1,1,1,0,0,0,1,0]
 => 2 = 3 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,0,1,1,1,0,0,0,0]
 => 2 = 3 - 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
 => [1,1,0,1,1,0,1,0,0,0]
 => 3 = 4 - 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 4 = 5 - 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 3 = 4 - 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
 => [1,1,1,0,1,1,0,0,0,0]
 => 2 = 3 - 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
 => [1,1,1,0,1,0,1,0,0,0]
 => 3 = 4 - 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
 => [1,1,1,1,0,1,0,0,0,0]
 => 2 = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
 => [1,1,1,1,1,0,0,0,0,0]
 => 1 = 2 - 1
Description
The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. We use the bijection in the code by Christian Stump to have a bijection to Dyck paths.
Matching statistic: St000777
Mapping
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00039: Integer compositions —complement⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 44% ●values known / values provided: 44%●distinct values known / distinct values provided: 100%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 1
{{1,2}}
 => [2] => [1,1] => ([(0,1)],2)
 => 2
{{1},{2}}
 => [1,1] => [2] => ([],2)
 => ? = 2
{{1,2,3}}
 => [3] => [1,1,1] => ([(0,1),(0,2),(1,2)],3)
 => 2
{{1,2},{3}}
 => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 3
{{1,3},{2}}
 => [2,1] => [1,2] => ([(1,2)],3)
 => ? = 2
{{1},{2,3}}
 => [1,2] => [2,1] => ([(0,2),(1,2)],3)
 => 3
{{1},{2},{3}}
 => [1,1,1] => [3] => ([],3)
 => ? = 2
{{1,2,3,4}}
 => [4] => [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
{{1,2,3},{4}}
 => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
{{1,2,4},{3}}
 => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
{{1,2},{3,4}}
 => [2,2] => [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 4
{{1,2},{3},{4}}
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3
{{1,3,4},{2}}
 => [3,1] => [1,1,2] => ([(1,2),(1,3),(2,3)],4)
 => ? = 3
{{1,3},{2},{4}}
 => [2,1,1] => [1,3] => ([(2,3)],4)
 => ? = 3
{{1},{2,3,4}}
 => [1,3] => [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 3
{{1},{2,3},{4}}
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 4
{{1},{2,4},{3}}
 => [1,2,1] => [2,2] => ([(1,3),(2,3)],4)
 => ? = 3
{{1},{2},{3,4}}
 => [1,1,2] => [3,1] => ([(0,3),(1,3),(2,3)],4)
 => 3
{{1},{2},{3},{4}}
 => [1,1,1,1] => [4] => ([],4)
 => ? = 2
{{1,2,3,4,5}}
 => [5] => [1,1,1,1,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
{{1,2,3,4},{5}}
 => [4,1] => [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1,2,3},{4,5}}
 => [3,2] => [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,2,3},{4},{5}}
 => [3,1,1] => [1,1,3] => ([(2,3),(2,4),(3,4)],5)
 => ? = 3
{{1,2},{3,4,5}}
 => [2,3] => [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,2},{3,4},{5}}
 => [2,2,1] => [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 5
{{1,2},{3},{4,5}}
 => [2,1,2] => [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
 => 4
{{1,2},{3},{4},{5}}
 => [2,1,1,1] => [1,4] => ([(3,4)],5)
 => ? = 3
{{1},{2,3,4,5}}
 => [1,4] => [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
{{1},{2,3,4},{5}}
 => [1,3,1] => [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => ? = 4
{{1},{2,3},{4,5}}
 => [1,2,2] => [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 5
{{1},{2,3},{4},{5}}
 => [1,2,1,1] => [2,3] => ([(2,4),(3,4)],5)
 => ? = 4
{{1},{2},{3,4,5}}
 => [1,1,3] => [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 3
{{1},{2},{3,4},{5}}
 => [1,1,2,1] => [3,2] => ([(1,4),(2,4),(3,4)],5)
 => ? = 4
{{1},{2},{3},{4,5}}
 => [1,1,1,2] => [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 3
{{1},{2},{3},{4},{5}}
 => [1,1,1,1,1] => [5] => ([],5)
 => ? = 2
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St000259
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 80%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 0 = 1 - 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => ? = 2 - 1
{{1,2,3}}
 => [2,3,1] => [2,1,3] => ([(1,2)],3)
 => ? = 2 - 1
{{1,2},{3}}
 => [2,1,3] => [2,3,1] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1,3},{2}}
 => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
 => 1 = 2 - 1
{{1},{2,3}}
 => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
 => 2 = 3 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => ? = 2 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [2,1,3,4] => ([(2,3)],4)
 => ? = 2 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
{{1,2,4},{3}}
 => [2,4,3,1] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 4 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [3,4,2,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 4 - 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 2 = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 2 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 2 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [2,3,1,4,5] => ([(2,4),(3,4)],5)
 => ? = 3 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 4 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [2,3,4,1,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,4,5,3] => ([(0,1),(2,4),(3,4)],5)
 => ? = 4 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 5 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 4 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? = 3 - 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 4 - 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
 => 4 = 5 - 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,4,2,5] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 - 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 2 = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 2 - 1
Description
The diameter of a connected graph. This is the greatest distance between any pair of vertices.
Matching statistic: St000260
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 60%
Values
{{1}}
 => [1] => [1] => ([],1)
 => 0 = 1 - 1
{{1,2}}
 => [2,1] => [2,1] => ([(0,1)],2)
 => 1 = 2 - 1
{{1},{2}}
 => [1,2] => [1,2] => ([],2)
 => ? = 2 - 1
{{1,2,3}}
 => [2,3,1] => [3,1,2] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
{{1,2},{3}}
 => [2,1,3] => [2,1,3] => ([(1,2)],3)
 => ? = 3 - 1
{{1,3},{2}}
 => [3,2,1] => [2,3,1] => ([(0,2),(1,2)],3)
 => 1 = 2 - 1
{{1},{2,3}}
 => [1,3,2] => [1,3,2] => ([(1,2)],3)
 => ? = 3 - 1
{{1},{2},{3}}
 => [1,2,3] => [1,2,3] => ([],3)
 => ? = 2 - 1
{{1,2,3,4}}
 => [2,3,4,1] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
 => 1 = 2 - 1
{{1,2,3},{4}}
 => [2,3,1,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
{{1,2,4},{3}}
 => [2,4,3,1] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
 => 2 = 3 - 1
{{1,2},{3,4}}
 => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
 => ? = 4 - 1
{{1,2},{3},{4}}
 => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
 => ? = 3 - 1
{{1,3,4},{2}}
 => [3,2,4,1] => [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
 => 2 = 3 - 1
{{1,3},{2},{4}}
 => [3,2,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
{{1},{2,3,4}}
 => [1,3,4,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
{{1},{2,3},{4}}
 => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
 => ? = 4 - 1
{{1},{2,4},{3}}
 => [1,4,3,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
 => ? = 3 - 1
{{1},{2},{3,4}}
 => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
 => ? = 3 - 1
{{1},{2},{3},{4}}
 => [1,2,3,4] => [1,2,3,4] => ([],4)
 => ? = 2 - 1
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
 => 1 = 2 - 1
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 - 1
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 4 - 1
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [3,1,2,4,5] => ([(2,4),(3,4)],5)
 => ? = 3 - 1
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
 => ? = 4 - 1
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [2,1,4,3,5] => ([(1,4),(2,3)],5)
 => ? = 5 - 1
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [2,1,3,5,4] => ([(1,4),(2,3)],5)
 => ? = 4 - 1
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [2,1,3,4,5] => ([(3,4)],5)
 => ? = 3 - 1
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
 => ? = 3 - 1
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
 => ? = 4 - 1
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [1,3,2,5,4] => ([(1,4),(2,3)],5)
 => ? = 5 - 1
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [1,3,2,4,5] => ([(3,4)],5)
 => ? = 4 - 1
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [1,2,5,3,4] => ([(2,4),(3,4)],5)
 => ? = 3 - 1
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [1,2,4,3,5] => ([(3,4)],5)
 => ? = 4 - 1
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [1,2,3,5,4] => ([(3,4)],5)
 => ? = 3 - 1
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
 => ? = 2 - 1
Description
The radius of a connected graph. This is the minimum eccentricity of any vertex.
Matching statistic: St000739
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St000739: Semistandard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
{{1}}
 => [1] => [[1]]
 => [[1]]
 => 1
{{1,2}}
 => [2,1] => [[0,1],[1,0]]
 => [[1,2],[2]]
 => 2
{{1},{2}}
 => [1,2] => [[1,0],[0,1]]
 => [[1,1],[2]]
 => 2
{{1,2,3}}
 => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ? = 2
{{1,2},{3}}
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ? = 3
{{1,3},{2}}
 => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => ? = 2
{{1},{2,3}}
 => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ? = 3
{{1},{2},{3}}
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ? = 2
{{1,2,3,4}}
 => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => ? = 2
{{1,2,3},{4}}
 => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => ? = 3
{{1,2,4},{3}}
 => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => ? = 3
{{1,2},{3,4}}
 => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => ? = 4
{{1,2},{3},{4}}
 => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ? = 3
{{1,3,4},{2}}
 => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => ? = 3
{{1,3},{2},{4}}
 => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => ? = 3
{{1},{2,3,4}}
 => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => ? = 3
{{1},{2,3},{4}}
 => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => ? = 4
{{1},{2,4},{3}}
 => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => ? = 3
{{1},{2},{3,4}}
 => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => ? = 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ? = 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ? = 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ? = 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
 => ? = 4
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => ? = 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
 => ? = 4
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
 => ? = 5
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
 => ? = 4
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
 => ? = 3
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ? = 3
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ? = 4
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
 => ? = 5
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
 => ? = 4
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
 => ? = 3
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
 => ? = 4
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
 => ? = 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
 => ? = 2
Description
The first entry in the last row of a semistandard tableau.
Matching statistic: St001401
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00063: Permutations —to alternating sign matrix⟶ Alternating sign matrices
Mp00001: Alternating sign matrices —to semistandard tableau via monotone triangles⟶ Semistandard tableaux
St001401: Semistandard tableaux ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
{{1}}
 => [1] => [[1]]
 => [[1]]
 => 1
{{1,2}}
 => [2,1] => [[0,1],[1,0]]
 => [[1,2],[2]]
 => 2
{{1},{2}}
 => [1,2] => [[1,0],[0,1]]
 => [[1,1],[2]]
 => 2
{{1,2,3}}
 => [2,3,1] => [[0,0,1],[1,0,0],[0,1,0]]
 => [[1,1,3],[2,3],[3]]
 => ? = 2
{{1,2},{3}}
 => [2,1,3] => [[0,1,0],[1,0,0],[0,0,1]]
 => [[1,1,2],[2,2],[3]]
 => ? = 3
{{1,3},{2}}
 => [3,2,1] => [[0,0,1],[0,1,0],[1,0,0]]
 => [[1,2,3],[2,3],[3]]
 => ? = 2
{{1},{2,3}}
 => [1,3,2] => [[1,0,0],[0,0,1],[0,1,0]]
 => [[1,1,1],[2,3],[3]]
 => ? = 3
{{1},{2},{3}}
 => [1,2,3] => [[1,0,0],[0,1,0],[0,0,1]]
 => [[1,1,1],[2,2],[3]]
 => ? = 2
{{1,2,3,4}}
 => [2,3,4,1] => [[0,0,0,1],[1,0,0,0],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,4],[2,2,4],[3,4],[4]]
 => ? = 2
{{1,2,3},{4}}
 => [2,3,1,4] => [[0,0,1,0],[1,0,0,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,3],[2,2,3],[3,3],[4]]
 => ? = 3
{{1,2,4},{3}}
 => [2,4,3,1] => [[0,0,0,1],[1,0,0,0],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,4],[2,3,4],[3,4],[4]]
 => ? = 3
{{1,2},{3,4}}
 => [2,1,4,3] => [[0,1,0,0],[1,0,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,2],[2,2,2],[3,4],[4]]
 => ? = 4
{{1,2},{3},{4}}
 => [2,1,3,4] => [[0,1,0,0],[1,0,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,2],[2,2,2],[3,3],[4]]
 => ? = 3
{{1,3,4},{2}}
 => [3,2,4,1] => [[0,0,0,1],[0,1,0,0],[1,0,0,0],[0,0,1,0]]
 => [[1,1,2,4],[2,2,4],[3,4],[4]]
 => ? = 3
{{1,3},{2},{4}}
 => [3,2,1,4] => [[0,0,1,0],[0,1,0,0],[1,0,0,0],[0,0,0,1]]
 => [[1,1,2,3],[2,2,3],[3,3],[4]]
 => ? = 3
{{1},{2,3,4}}
 => [1,3,4,2] => [[1,0,0,0],[0,0,0,1],[0,1,0,0],[0,0,1,0]]
 => [[1,1,1,1],[2,2,4],[3,4],[4]]
 => ? = 3
{{1},{2,3},{4}}
 => [1,3,2,4] => [[1,0,0,0],[0,0,1,0],[0,1,0,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,3],[3,3],[4]]
 => ? = 4
{{1},{2,4},{3}}
 => [1,4,3,2] => [[1,0,0,0],[0,0,0,1],[0,0,1,0],[0,1,0,0]]
 => [[1,1,1,1],[2,3,4],[3,4],[4]]
 => ? = 3
{{1},{2},{3,4}}
 => [1,2,4,3] => [[1,0,0,0],[0,1,0,0],[0,0,0,1],[0,0,1,0]]
 => [[1,1,1,1],[2,2,2],[3,4],[4]]
 => ? = 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => [[1,0,0,0],[0,1,0,0],[0,0,1,0],[0,0,0,1]]
 => [[1,1,1,1],[2,2,2],[3,3],[4]]
 => ? = 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,5],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ? = 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ? = 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
 => ? = 4
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => [[0,0,1,0,0],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,4],[5]]
 => ? = 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
 => ? = 4
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,4],[4,4],[5]]
 => ? = 5
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,5],[5]]
 => ? = 4
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,2],[2,2,2,2],[3,3,3],[4,4],[5]]
 => ? = 3
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => [[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
 => ? = 3
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => [[1,0,0,0,0],[0,0,0,1,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,4],[3,3,4],[4,4],[5]]
 => ? = 4
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,5],[5]]
 => ? = 5
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => [[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,3],[3,3,3],[4,4],[5]]
 => ? = 4
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,0,1],[0,0,1,0,0],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,5],[4,5],[5]]
 => ? = 3
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,4],[4,4],[5]]
 => ? = 4
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,0,1],[0,0,0,1,0]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,5],[5]]
 => ? = 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => [[1,0,0,0,0],[0,1,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,0,0,0,1]]
 => [[1,1,1,1,1],[2,2,2,2],[3,3,3],[4,4],[5]]
 => ? = 2
Description
The number of distinct entries in a semistandard tableau.
Matching statistic: St001637
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001637: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
{{1}}
 => [1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
{{1,2}}
 => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
{{1},{2}}
 => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
{{1,2,3}}
 => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2
{{1,2},{3}}
 => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3
{{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
{{1},{2,3}}
 => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3
{{1},{2},{3}}
 => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 2
{{1,2,3},{4}}
 => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 3
{{1,2,4},{3}}
 => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 3
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 4
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 3
{{1,3,4},{2}}
 => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 3
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 3
{{1},{2,3,4}}
 => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 3
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 3
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 4
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 4
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 5
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ? = 4
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 3
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 3
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 4
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 5
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 4
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 3
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 4
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 2
Description
The number of (upper) dissectors of a poset.
Matching statistic: St001668
Mapping
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St001668: Posets ⟶ ℤResult quality: 8% ●values known / values provided: 8%●distinct values known / distinct values provided: 40%
Values
{{1}}
 => [1] => ([(0,1)],2)
 => ([(0,1)],2)
 => 1
{{1,2}}
 => [2,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
{{1},{2}}
 => [1,2] => ([(0,1),(0,2),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,3),(2,3)],4)
 => 2
{{1,2,3}}
 => [2,3,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 2
{{1,2},{3}}
 => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3
{{1,3},{2}}
 => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
{{1},{2,3}}
 => [1,3,2] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
 => ? = 3
{{1},{2},{3}}
 => [1,2,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
 => ? = 2
{{1,2,3,4}}
 => [2,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 2
{{1,2,3},{4}}
 => [2,3,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 3
{{1,2,4},{3}}
 => [2,4,3,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 3
{{1,2},{3,4}}
 => [2,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,6),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
 => ? = 4
{{1,2},{3},{4}}
 => [2,1,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 3
{{1,3,4},{2}}
 => [3,2,4,1] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 3
{{1,3},{2},{4}}
 => [3,2,1,4] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 3
{{1},{2,3,4}}
 => [1,3,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,6),(4,7),(5,7),(7,6)],8)
 => ? = 3
{{1},{2,3},{4}}
 => [1,3,2,4] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,7),(4,6),(5,6),(5,7),(6,8),(7,8)],9)
 => ? = 4
{{1},{2,4},{3}}
 => [1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,7),(2,6),(3,5),(4,5),(4,6),(5,8),(6,8),(8,7)],9)
 => ? = 3
{{1},{2},{3,4}}
 => [1,2,4,3] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(4,7),(5,8),(6,7),(7,8)],9)
 => ? = 3
{{1},{2},{3},{4}}
 => [1,2,3,4] => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(1,9),(2,8),(3,8),(3,10),(4,9),(4,10),(6,5),(7,5),(8,6),(9,7),(10,6),(10,7)],11)
 => ? = 2
{{1,2,3,4,5}}
 => [2,3,4,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,10),(3,6),(4,10),(4,12),(5,11),(5,12),(7,9),(8,9),(9,6),(10,7),(11,8),(12,7),(12,8)],13)
 => ? = 2
{{1,2,3,4},{5}}
 => [2,3,4,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 3
{{1,2,3},{4,5}}
 => [2,3,1,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 4
{{1,2,3},{4},{5}}
 => [2,3,1,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 3
{{1,2},{3,4,5}}
 => [2,1,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
 => ? = 4
{{1,2},{3,4},{5}}
 => [2,1,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 5
{{1,2},{3},{4,5}}
 => [2,1,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(1,9),(2,7),(3,7),(4,6),(5,6),(6,9),(7,8),(8,10),(9,10)],11)
 => ? = 4
{{1,2},{3},{4},{5}}
 => [2,1,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 3
{{1},{2,3,4,5}}
 => [1,3,4,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,8),(3,7),(4,6),(5,6),(5,7),(6,10),(7,10),(8,9),(10,8)],11)
 => ? = 3
{{1},{2,3,4},{5}}
 => [1,3,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,7),(5,9),(6,9),(7,10),(8,10),(9,7),(9,8)],11)
 => ? = 4
{{1},{2,3},{4,5}}
 => [1,3,2,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(7,9),(8,10),(9,10)],11)
 => ? = 5
{{1},{2,3},{4},{5}}
 => [1,3,2,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 4
{{1},{2},{3,4,5}}
 => [1,2,4,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,7),(4,7),(5,6),(5,9),(6,10),(7,8),(8,9),(9,10)],11)
 => ? = 3
{{1},{2},{3,4},{5}}
 => [1,2,4,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(2,12),(3,12),(4,9),(5,10),(5,11),(7,6),(8,6),(9,8),(10,7),(11,7),(11,8),(12,9),(12,11)],13)
 => ? = 4
{{1},{2},{3},{4,5}}
 => [1,2,3,5,4] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,11),(2,11),(3,10),(4,9),(4,12),(5,10),(5,12),(7,6),(8,6),(9,7),(10,8),(11,9),(12,7),(12,8)],13)
 => ? = 3
{{1},{2},{3},{4},{5}}
 => [1,2,3,4,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,12),(2,11),(3,11),(3,14),(4,12),(4,15),(5,14),(5,15),(7,9),(8,10),(9,6),(10,6),(11,7),(12,8),(13,9),(13,10),(14,7),(14,13),(15,8),(15,13)],16)
 => ? = 2
Description
The number of points of the poset minus the width of the poset.
The following 2 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000101The cocharge of a semistandard tableau. St000454The largest eigenvalue of a graph if it is integral.