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Your data matches 15 different statistics following compositions of up to 3 maps.
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Matching statistic: St000229
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
St000229: Set partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> 1
{{1,2}}
=> 2
{{1},{2}}
=> 2
{{1,2,3}}
=> 3
{{1,2},{3}}
=> 3
{{1,3},{2}}
=> 4
{{1},{2,3}}
=> 3
{{1},{2},{3}}
=> 3
{{1,2,3,4}}
=> 4
{{1,2,3},{4}}
=> 4
{{1,2,4},{3}}
=> 5
{{1,2},{3,4}}
=> 4
{{1,2},{3},{4}}
=> 4
{{1,3,4},{2}}
=> 5
{{1,3},{2,4}}
=> 6
{{1,3},{2},{4}}
=> 5
{{1,4},{2,3}}
=> 6
{{1},{2,3,4}}
=> 4
{{1},{2,3},{4}}
=> 4
{{1,4},{2},{3}}
=> 6
{{1},{2,4},{3}}
=> 5
{{1},{2},{3,4}}
=> 4
{{1},{2},{3},{4}}
=> 4
{{1,2,3,4,5}}
=> 5
{{1,2,3,4},{5}}
=> 5
{{1,2,3,5},{4}}
=> 6
{{1,2,3},{4,5}}
=> 5
{{1,2,3},{4},{5}}
=> 5
{{1,2,4,5},{3}}
=> 6
{{1,2,4},{3,5}}
=> 7
{{1,2,4},{3},{5}}
=> 6
{{1,2,5},{3,4}}
=> 7
{{1,2},{3,4,5}}
=> 5
{{1,2},{3,4},{5}}
=> 5
{{1,2,5},{3},{4}}
=> 7
{{1,2},{3,5},{4}}
=> 6
{{1,2},{3},{4,5}}
=> 5
{{1,2},{3},{4},{5}}
=> 5
Description
Sum of the difference between the maximal and the minimal elements of the blocks plus the number of blocks of a set partition.
This is, for a set partition $P = \{B_1,\ldots,B_k\}$ of $\{1,\ldots,n\}$, the statistic is
$$d(P) = \sum_i \big(\operatorname{max}(B_i)-\operatorname{min}(B_i)+1\big).$$
This statistic is called ''dimension index'' in [2]
Matching statistic: St001034
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001034: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> 2
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> 3
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> 4
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> 3
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 4
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 5
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 4
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 5
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 6
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 5
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> 6
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> 6
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 5
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> 6
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> 5
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> 6
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> 7
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 7
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> 5
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> 5
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> 7
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 6
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> 5
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> 5
Description
The area of the parallelogram polyomino associated with the Dyck path.
The (bivariate) generating function is given in [1].
Matching statistic: St000395
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00222: Dyck paths —peaks-to-valleys⟶ Dyck paths
St000395: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1,0]
=> [1,0]
=> 1
{{1,2}}
=> [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 2
{{1},{2}}
=> [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> 2
{{1,2,3}}
=> [2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 3
{{1,2},{3}}
=> [2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> 3
{{1,3},{2}}
=> [3,2,1] => [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 4
{{1},{2,3}}
=> [1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
{{1},{2},{3}}
=> [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
{{1,2,3,4}}
=> [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 4
{{1,2,3},{4}}
=> [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> 4
{{1,2,4},{3}}
=> [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0]
=> 5
{{1,2},{3,4}}
=> [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 4
{{1,3,4},{2}}
=> [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 5
{{1,3},{2,4}}
=> [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 6
{{1,3},{2},{4}}
=> [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 5
{{1,4},{2,3}}
=> [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6
{{1},{2,3,4}}
=> [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 6
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 5
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [1,1,0,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 6
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> 5
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> 6
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [1,1,0,1,1,0,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 7
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 7
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> 5
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 5
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 7
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 6
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> 5
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 5
Description
The sum of the heights of the peaks of a Dyck path.
Matching statistic: St001894
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
St001894: Signed permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
{{1}}
=> [1] => [1] => [-1] => 1
{{1,2}}
=> [2,1] => [2,1] => [1,-2] => 2
{{1},{2}}
=> [1,2] => [1,2] => [2,-1] => 2
{{1,2,3}}
=> [2,3,1] => [2,3,1] => [1,2,-3] => 3
{{1,2},{3}}
=> [2,1,3] => [2,1,3] => [1,3,-2] => 3
{{1,3},{2}}
=> [3,2,1] => [3,2,1] => [2,1,-3] => 4
{{1},{2,3}}
=> [1,3,2] => [1,3,2] => [3,2,-1] => 3
{{1},{2},{3}}
=> [1,2,3] => [1,2,3] => [2,3,-1] => 3
{{1,2,3,4}}
=> [2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => 4
{{1,2,3},{4}}
=> [2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => 4
{{1,2,4},{3}}
=> [2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => 5
{{1,2},{3,4}}
=> [2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => 4
{{1,2},{3},{4}}
=> [2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => 4
{{1,3,4},{2}}
=> [3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => 5
{{1,3},{2,4}}
=> [3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => 6
{{1,3},{2},{4}}
=> [3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => 5
{{1,4},{2,3}}
=> [4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => 6
{{1},{2,3,4}}
=> [1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => 4
{{1},{2,3},{4}}
=> [1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => 4
{{1,4},{2},{3}}
=> [4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => 6
{{1},{2,4},{3}}
=> [1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => 5
{{1},{2},{3,4}}
=> [1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => 4
{{1},{2},{3},{4}}
=> [1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => 4
{{1,2,3,4,5}}
=> [2,3,4,5,1] => [2,3,4,5,1] => [1,2,3,4,-5] => 5
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [2,3,5,4,1] => [1,2,4,3,-5] => 6
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => 5
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [2,4,3,5,1] => [1,3,2,4,-5] => 6
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [2,4,5,1,3] => [1,5,2,3,-4] => 7
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [2,4,3,1,5] => [1,3,2,5,-4] => 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [2,5,4,3,1] => [1,4,3,2,-5] => 7
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => 5
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => 5
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [2,5,3,4,1] => [1,3,4,2,-5] => 7
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => 6
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => 5
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => 5
Description
The depth of a signed permutation.
The depth of a positive root is its rank in the root poset. The depth of an element of a Coxeter group is the minimal sum of depths for any representation as product of reflections.
Matching statistic: St001875
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 57%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 37% ●values known / values provided: 37%●distinct values known / distinct values provided: 57%
Values
{{1}}
=> [1] => ([],1)
=> ([(0,1)],2)
=> ? = 1 + 1
{{1,2}}
=> [2,1] => ([],2)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
{{1},{2}}
=> [1,2] => ([(0,1)],2)
=> ([(0,2),(2,1)],3)
=> 3 = 2 + 1
{{1,2,3}}
=> [2,3,1] => ([(1,2)],3)
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 4 = 3 + 1
{{1,2},{3}}
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 4 = 3 + 1
{{1,3},{2}}
=> [3,2,1] => ([],3)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4 + 1
{{1},{2,3}}
=> [1,3,2] => ([(0,1),(0,2)],3)
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 4 = 3 + 1
{{1},{2},{3}}
=> [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,3),(2,1),(3,2)],4)
=> 4 = 3 + 1
{{1,2,3,4}}
=> [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 4 + 1
{{1,2,3},{4}}
=> [2,3,1,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 5 = 4 + 1
{{1,2,4},{3}}
=> [2,4,3,1] => ([(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,6),(1,8),(2,6),(2,7),(3,5),(4,1),(4,2),(4,5),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> ? = 5 + 1
{{1,2},{3,4}}
=> [2,1,4,3] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> 5 = 4 + 1
{{1,2},{3},{4}}
=> [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> 5 = 4 + 1
{{1,3,4},{2}}
=> [3,2,4,1] => ([(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,7),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(5,9),(6,9),(8,1),(8,9),(9,7)],10)
=> ? = 5 + 1
{{1,3},{2,4}}
=> [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,3),(0,4),(1,7),(2,6),(3,2),(3,5),(4,1),(4,5),(5,6),(5,7),(6,8),(7,8)],9)
=> ? = 6 + 1
{{1,3},{2},{4}}
=> [3,2,1,4] => ([(0,3),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6),(5,8),(6,8),(7,8),(8,1)],9)
=> ? = 5 + 1
{{1,4},{2,3}}
=> [4,3,2,1] => ([],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(5,14),(6,11),(6,12),(7,11),(7,13),(8,12),(8,14),(9,13),(9,14),(10,12),(10,13),(11,15),(12,15),(13,15),(14,15)],16)
=> ? = 6 + 1
{{1},{2,3,4}}
=> [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,4),(1,6),(2,5),(3,1),(3,5),(4,2),(4,3),(5,6)],7)
=> 5 = 4 + 1
{{1},{2,3},{4}}
=> [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> 5 = 4 + 1
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,5),(1,6),(2,7),(2,9),(3,7),(3,8),(4,1),(4,8),(4,9),(5,11),(6,11),(7,10),(8,5),(8,10),(9,6),(9,10),(10,11)],12)
=> ? = 6 + 1
{{1},{2,4},{3}}
=> [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([(0,4),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,1),(4,2),(4,3),(5,8),(6,8),(7,8)],9)
=> ? = 5 + 1
{{1},{2},{3,4}}
=> [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> 5 = 4 + 1
{{1},{2},{3},{4}}
=> [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5 = 4 + 1
{{1,2,3,4,5}}
=> [2,3,4,5,1] => ([(1,4),(3,2),(4,3)],5)
=> ([(0,2),(0,5),(1,7),(2,6),(3,4),(3,9),(4,1),(4,8),(5,3),(5,6),(6,9),(8,7),(9,8)],10)
=> ? = 5 + 1
{{1,2,3,4},{5}}
=> [2,3,4,1,5] => ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,3),(0,5),(2,8),(3,6),(4,2),(4,7),(5,4),(5,6),(6,7),(7,8),(8,1)],9)
=> ? = 5 + 1
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(1,9),(2,6),(2,8),(3,7),(4,5),(4,7),(5,1),(5,2),(5,10),(6,11),(7,10),(8,11),(9,11),(10,8),(10,9)],12)
=> ? = 6 + 1
{{1,2,3},{4,5}}
=> [2,3,1,5,4] => ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,1),(5,7),(7,8),(8,2),(8,3)],9)
=> ? = 5 + 1
{{1,2,3},{4},{5}}
=> [2,3,1,4,5] => ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,3),(0,5),(1,7),(3,6),(4,2),(5,1),(5,6),(6,7),(7,4)],8)
=> ? = 5 + 1
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,9),(1,10),(2,8),(2,10),(3,7),(4,6),(5,1),(5,2),(5,6),(6,8),(6,9),(8,11),(9,11),(10,3),(10,11),(11,7)],12)
=> ? = 6 + 1
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,5),(1,7),(2,8),(3,10),(4,2),(4,6),(5,4),(5,10),(6,7),(6,8),(7,9),(8,9),(10,1),(10,6)],11)
=> ? = 7 + 1
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,7),(2,9),(3,7),(3,8),(4,6),(5,2),(5,3),(5,6),(6,8),(6,9),(7,10),(8,10),(9,10),(10,1)],11)
=> ? = 6 + 1
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(1,2),(1,3),(1,4)],5)
=> ([(0,1),(0,2),(1,12),(2,3),(2,4),(2,5),(2,12),(3,8),(3,10),(3,11),(4,7),(4,9),(4,11),(5,6),(5,9),(5,10),(6,13),(6,14),(7,13),(7,15),(8,14),(8,15),(9,13),(9,16),(10,14),(10,16),(11,15),(11,16),(12,6),(12,7),(12,8),(13,17),(14,17),(15,17),(16,17)],18)
=> ? = 7 + 1
{{1,2},{3,4,5}}
=> [2,1,4,5,3] => ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,3),(0,4),(1,7),(2,6),(3,8),(4,8),(5,1),(5,6),(6,7),(8,2),(8,5)],9)
=> ? = 5 + 1
{{1,2},{3,4},{5}}
=> [2,1,4,3,5] => ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(0,4),(0,5),(1,6),(2,6),(4,7),(5,7),(6,3),(7,1),(7,2)],8)
=> ? = 5 + 1
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(1,3),(1,4),(4,2)],5)
=> ([(0,1),(0,2),(1,11),(2,3),(2,4),(2,11),(3,8),(3,10),(4,5),(4,9),(4,10),(5,6),(5,7),(6,13),(7,13),(8,12),(9,7),(9,12),(10,6),(10,12),(11,8),(11,9),(12,13)],14)
=> ? = 7 + 1
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(0,5),(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,9),(5,9),(6,10),(7,10),(8,10),(9,1),(9,2),(9,3)],11)
=> ? = 6 + 1
{{1,2},{3},{4,5}}
=> [2,1,3,5,4] => ([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(0,3),(0,4),(1,6),(2,6),(3,7),(4,7),(5,1),(5,2),(7,5)],8)
=> ? = 5 + 1
{{1,2},{3},{4},{5}}
=> [2,1,3,4,5] => ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> 6 = 5 + 1
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000080
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000080: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
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)
=> 3
{{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)
=> ? = 4
{{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)
=> ? = 3
{{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)
=> ? = 4
{{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)
=> ? = 4
{{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)
=> ? = 5
{{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)
=> ? = 4
{{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)
=> ? = 5
{{1,3},{2,4}}
=> [3,4,1,2] => ([(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)
=> ? = 6
{{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)
=> ? = 5
{{1,4},{2,3}}
=> [4,3,2,1] => ([(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)
=> ? = 6
{{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)
=> ? = 4
{{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
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(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)
=> ? = 6
{{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)
=> ? = 5
{{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)
=> ? = 4
{{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)
=> ? = 4
{{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)
=> ? = 5
{{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)
=> ? = 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 6
{{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)
=> ? = 5
{{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)
=> ? = 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 6
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 7
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(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)
=> ? = 7
{{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)
=> ? = 5
{{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
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 7
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 6
{{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)
=> ? = 5
{{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)
=> ? = 5
Description
The rank of the poset.
Matching statistic: St000307
Mp00080: Set partitions —to permutation⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
Mp00208: Permutations —lattice of intervals⟶ Lattices
Mp00193: Lattices —to poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 43%
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)
=> 3
{{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)
=> ? = 4
{{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)
=> ? = 3
{{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)
=> ? = 4
{{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)
=> ? = 4
{{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)
=> ? = 5
{{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)
=> ? = 4
{{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)
=> ? = 5
{{1,3},{2,4}}
=> [3,4,1,2] => ([(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)
=> ? = 6
{{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)
=> ? = 5
{{1,4},{2,3}}
=> [4,3,2,1] => ([(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)
=> ? = 6
{{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)
=> ? = 4
{{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
{{1,4},{2},{3}}
=> [4,2,3,1] => ([(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)
=> ? = 6
{{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)
=> ? = 5
{{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)
=> ? = 4
{{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)
=> ? = 4
{{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)
=> ? = 5
{{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)
=> ? = 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(2,6),(3,7),(4,7),(5,6),(5,8),(6,10),(7,8),(8,10),(10,9)],11)
=> ? = 6
{{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)
=> ? = 5
{{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)
=> ? = 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,9),(4,8),(5,7),(6,8),(6,9),(8,10),(9,10),(10,7)],11)
=> ? = 6
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 7
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => ([(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)
=> ? = 7
{{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)
=> ? = 5
{{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
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,8),(4,9),(5,7),(6,9),(8,7),(9,8)],10)
=> ? = 7
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,7),(3,6),(4,6),(5,7),(5,8),(6,10),(7,9),(8,9),(9,10)],11)
=> ? = 6
{{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)
=> ? = 5
{{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)
=> ? = 5
Description
The number of rowmotion orbits of a poset.
Rowmotion is an operation on order ideals in a poset $P$. It sends an order ideal $I$ to the order ideal generated by the minimal antichain of $P \setminus I$.
Matching statistic: St001880
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 43%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001880: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 43%
Values
{{1}}
=> [1] => [[1],[]]
=> ([],1)
=> ? = 1
{{1,2}}
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> ? = 2
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
{{1,2,3}}
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 3
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 4
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 3
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
{{1,2,3,4}}
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 4
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 6
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 6
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 4
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 6
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 5
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 5
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 5
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 7
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 6
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 7
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 5
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 7
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 6
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? = 5
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St001879
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00128: Set partitions —to composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 43%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St001879: Posets ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 43%
Values
{{1}}
=> [1] => [[1],[]]
=> ([],1)
=> ? = 1 - 1
{{1,2}}
=> [2] => [[2],[]]
=> ([(0,1)],2)
=> ? = 2 - 1
{{1},{2}}
=> [1,1] => [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2 - 1
{{1,2,3}}
=> [3] => [[3],[]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
{{1,2},{3}}
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 3 - 1
{{1,3},{2}}
=> [2,1] => [[2,2],[1]]
=> ([(0,2),(1,2)],3)
=> ? = 4 - 1
{{1},{2,3}}
=> [1,2] => [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 3 - 1
{{1},{2},{3}}
=> [1,1,1] => [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 2 = 3 - 1
{{1,2,3,4}}
=> [4] => [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
{{1,2,3},{4}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
{{1,2,4},{3}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 1
{{1,2},{3,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 4 - 1
{{1,2},{3},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 4 - 1
{{1,3,4},{2}}
=> [3,1] => [[3,3],[2]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 1
{{1,3},{2,4}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 6 - 1
{{1,3},{2},{4}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 5 - 1
{{1,4},{2,3}}
=> [2,2] => [[3,2],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 6 - 1
{{1},{2,3,4}}
=> [1,3] => [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4 - 1
{{1},{2,3},{4}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 4 - 1
{{1,4},{2},{3}}
=> [2,1,1] => [[2,2,2],[1,1]]
=> ([(0,3),(1,2),(2,3)],4)
=> ? = 6 - 1
{{1},{2,4},{3}}
=> [1,2,1] => [[2,2,1],[1]]
=> ([(0,3),(1,2),(1,3)],4)
=> ? = 5 - 1
{{1},{2},{3,4}}
=> [1,1,2] => [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4 - 1
{{1},{2},{3},{4}}
=> [1,1,1,1] => [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 3 = 4 - 1
{{1,2,3,4,5}}
=> [5] => [[5],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 4 = 5 - 1
{{1,2,3,4},{5}}
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
{{1,2,3,5},{4}}
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
{{1,2,3},{4,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 5 - 1
{{1,2,3},{4},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 5 - 1
{{1,2,4,5},{3}}
=> [4,1] => [[4,4],[3]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 6 - 1
{{1,2,4},{3,5}}
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 7 - 1
{{1,2,4},{3},{5}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 6 - 1
{{1,2,5},{3,4}}
=> [3,2] => [[4,3],[2]]
=> ([(0,3),(1,2),(1,4),(3,4)],5)
=> ? = 7 - 1
{{1,2},{3,4,5}}
=> [2,3] => [[4,2],[1]]
=> ([(0,4),(1,2),(1,4),(2,3)],5)
=> ? = 5 - 1
{{1,2},{3,4},{5}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 5 - 1
{{1,2,5},{3},{4}}
=> [3,1,1] => [[3,3,3],[2,2]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ? = 7 - 1
{{1,2},{3,5},{4}}
=> [2,2,1] => [[3,3,2],[2,1]]
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ? = 6 - 1
{{1,2},{3},{4,5}}
=> [2,1,2] => [[3,2,2],[1,1]]
=> ([(0,4),(1,2),(1,3),(3,4)],5)
=> ? = 5 - 1
{{1,2},{3},{4},{5}}
=> [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ? = 5 - 1
Description
The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice.
Matching statistic: St000739
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: 29%
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: 29%
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]]
=> ? = 3
{{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]]
=> ? = 4
{{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]]
=> ? = 3
{{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]]
=> ? = 4
{{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]]
=> ? = 4
{{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]]
=> ? = 5
{{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]]
=> ? = 4
{{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]]
=> ? = 5
{{1,3},{2,4}}
=> [3,4,1,2] => [[0,0,1,0],[0,0,0,1],[1,0,0,0],[0,1,0,0]]
=> [[1,1,3,3],[2,3,4],[3,4],[4]]
=> ? = 6
{{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]]
=> ? = 5
{{1,4},{2,3}}
=> [4,3,2,1] => [[0,0,0,1],[0,0,1,0],[0,1,0,0],[1,0,0,0]]
=> [[1,2,3,4],[2,3,4],[3,4],[4]]
=> ? = 6
{{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]]
=> ? = 4
{{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
{{1,4},{2},{3}}
=> [4,2,3,1] => [[0,0,0,1],[0,1,0,0],[0,0,1,0],[1,0,0,0]]
=> [[1,2,2,4],[2,3,4],[3,4],[4]]
=> ? = 6
{{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]]
=> ? = 5
{{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]]
=> ? = 4
{{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]]
=> ? = 4
{{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]]
=> ? = 5
{{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]]
=> ? = 5
{{1,2,3,5},{4}}
=> [2,3,5,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,1,0,0,0],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,5],[2,2,2,5],[3,4,5],[4,5],[5]]
=> ? = 6
{{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]]
=> ? = 5
{{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]]
=> ? = 5
{{1,2,4,5},{3}}
=> [2,4,3,5,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,1,0]]
=> [[1,1,1,1,5],[2,2,3,5],[3,3,5],[4,5],[5]]
=> ? = 6
{{1,2,4},{3,5}}
=> [2,4,5,1,3] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,0,0,1],[0,1,0,0,0],[0,0,1,0,0]]
=> [[1,1,1,1,4],[2,2,4,4],[3,4,5],[4,5],[5]]
=> ? = 7
{{1,2,4},{3},{5}}
=> [2,4,3,1,5] => [[0,0,0,1,0],[1,0,0,0,0],[0,0,1,0,0],[0,1,0,0,0],[0,0,0,0,1]]
=> [[1,1,1,1,4],[2,2,3,4],[3,3,4],[4,4],[5]]
=> ? = 6
{{1,2,5},{3,4}}
=> [2,5,4,3,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,0,1,0],[0,0,1,0,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,4,5],[3,4,5],[4,5],[5]]
=> ? = 7
{{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]]
=> ? = 5
{{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
{{1,2,5},{3},{4}}
=> [2,5,3,4,1] => [[0,0,0,0,1],[1,0,0,0,0],[0,0,1,0,0],[0,0,0,1,0],[0,1,0,0,0]]
=> [[1,1,1,1,5],[2,3,3,5],[3,4,5],[4,5],[5]]
=> ? = 7
{{1,2},{3,5},{4}}
=> [2,1,5,4,3] => [[0,1,0,0,0],[1,0,0,0,0],[0,0,0,0,1],[0,0,0,1,0],[0,0,1,0,0]]
=> [[1,1,1,1,2],[2,2,2,2],[3,4,5],[4,5],[5]]
=> ? = 6
{{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]]
=> ? = 5
{{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]]
=> ? = 5
Description
The first entry in the last row of a semistandard tableau.
The following 5 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
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