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Your data matches 72 different statistics following compositions of up to 3 maps.
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Matching statistic: St000777
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Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00073: Permutations —major-index to inversion-number bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ 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] => ([(0,1)],2)
=> 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 4
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [4,3,1,5,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [4,1,5,3,2] => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 5
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 6
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [4,1,2,6,5,3] => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> 6
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [5,4,3,1,6,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [5,4,1,6,3,2] => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [5,1,6,4,3,2] => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 7
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [5,4,1,6,2,7,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 7
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [5,1,6,4,2,7,3] => ([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [5,4,1,2,7,6,3] => ([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [5,1,6,2,7,4,3] => ([(0,6),(1,4),(1,5),(2,3),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 7
[[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => [5,1,2,7,6,4,3] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 7
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [6,5,4,3,1,7,2] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [6,5,4,1,7,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [6,5,1,7,4,3,2] => ([(0,5),(0,6),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [6,1,7,5,4,3,2] => ([(0,6),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
Matching statistic: St001388
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001388: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Mp00235: Permutations —descent views to invisible inversion bottoms⟶ Permutations
Mp00223: Permutations —runsort⟶ Permutations
St001388: Permutations ⟶ ℤResult quality: 52% ●values known / values provided: 52%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => [1] => ? = 1 - 2
[[1],[2]]
=> [2,1] => [2,1] => [1,2] => 0 = 2 - 2
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [1,2,3] => 0 = 2 - 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [1,3,2,4] => 2 = 4 - 2
[[1],[2],[3],[4]]
=> [4,3,2,1] => [2,3,4,1] => [1,2,3,4] => 0 = 2 - 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [2,3,5,1,4] => [1,4,2,3,5] => 3 = 5 - 2
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,4,1,5,3] => [1,5,2,4,3] => 3 = 5 - 2
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [2,3,4,5,1] => [1,2,3,4,5] => 0 = 2 - 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,1,6,3,5] => [1,6,2,4,3,5] => 4 = 6 - 2
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [2,5,1,3,6,4] => [1,3,6,2,5,4] => 4 = 6 - 2
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [2,3,4,6,1,5] => [1,5,2,3,4,6] => 3 = 5 - 2
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [2,3,5,1,6,4] => [1,6,2,3,5,4] => 3 = 5 - 2
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [2,4,1,5,6,3] => [1,5,6,2,4,3] => 3 = 5 - 2
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [2,3,4,5,6,1] => [1,2,3,4,5,6] => 0 = 2 - 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,5,1,3,7,4,6] => [1,3,7,2,5,4,6] => ? = 7 - 2
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [2,3,5,1,7,4,6] => [1,7,2,3,5,4,6] => ? = 7 - 2
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,4,1,5,7,3,6] => [1,5,7,2,4,3,6] => ? = 7 - 2
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [2,3,6,1,4,7,5] => [1,4,7,2,3,6,5] => ? = 7 - 2
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,1,6,3,7,5] => [1,6,2,4,3,7,5] => ? = 7 - 2
[[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => [2,5,1,3,6,7,4] => [1,3,6,7,2,5,4] => ? = 7 - 2
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [2,3,4,5,7,1,6] => [1,6,2,3,4,5,7] => ? = 5 - 2
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [2,3,4,6,1,7,5] => [1,7,2,3,4,6,5] => ? = 5 - 2
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [2,3,5,1,6,7,4] => [1,6,7,2,3,5,4] => ? = 5 - 2
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [2,4,1,5,6,7,3] => [1,5,6,7,2,4,3] => ? = 5 - 2
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [2,3,4,5,6,7,1] => [1,2,3,4,5,6,7] => ? = 2 - 2
Description
The number of non-attacking neighbors of a permutation.
For a permutation $\sigma$, the indices $i$ and $i+1$ are attacking if $|\sigma(i)-\sigma(i+1)| = 1$.
Visually, this is, for $\sigma$ considered as a placement of kings on a chessboard, if the kings placed in columns $i$ and $i+1$ are non-attacking.
Matching statistic: St000651
Mp00083: Standard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St000651: Permutations ⟶ ℤResult quality: 40% ●values known / values provided: 40%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1]
=> [1,0]
=> [2,1] => 0 = 1 - 1
[[1],[2]]
=> [1,1]
=> [1,1,0,0]
=> [2,3,1] => 1 = 2 - 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [4,3,1,2] => 1 = 2 - 1
[[1,3],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => 3 = 4 - 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [5,4,1,2,3] => 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 4 = 5 - 1
[[1,3],[2],[4],[5]]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [6,1,5,2,3,4] => 4 = 5 - 1
[[1],[2],[3],[4],[5]]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [5,6,1,2,3,4] => 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 6 - 1
[[1,3,4],[2],[5],[6]]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,1,2,6,3,4,5] => ? = 6 - 1
[[1,5],[2],[3],[4],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 5 - 1
[[1,4],[2],[3],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 5 - 1
[[1,3],[2],[4],[5],[6]]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,1,7,2,3,4,5] => ? = 5 - 1
[[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,1,2,3,4,5] => 1 = 2 - 1
[[1,3,4,6],[2],[5],[7]]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,1,2,3,7,4,5,6] => ? = 7 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 7 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 7 - 1
[[1,4,5],[2],[3],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 7 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 7 - 1
[[1,3,4],[2],[5],[6],[7]]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,1,2,8,3,4,5,6] => ? = 7 - 1
[[1,6],[2],[3],[4],[5],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ? = 5 - 1
[[1,5],[2],[3],[4],[6],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ? = 5 - 1
[[1,4],[2],[3],[5],[6],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ? = 5 - 1
[[1,3],[2],[4],[5],[6],[7]]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,1,7,2,3,4,5,6] => ? = 5 - 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,1,2,3,4,5,6] => 1 = 2 - 1
Description
The maximal size of a rise in a permutation.
This is $\max_i \sigma_{i+1}-\sigma_i$, except for the permutations without rises, where it is $0$.
Matching statistic: St000845
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Mp00209: Permutations —pattern poset⟶ Posets
St000845: Posets ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 4 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ? = 6 - 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6 - 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 5 - 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,9),(1,29),(2,13),(2,14),(2,27),(3,12),(3,16),(3,27),(4,11),(4,14),(4,15),(4,27),(5,1),(5,10),(5,15),(5,16),(5,27),(6,10),(6,11),(6,12),(6,13),(7,23),(9,23),(9,24),(10,18),(10,20),(10,28),(10,29),(11,17),(11,20),(11,28),(12,18),(12,28),(13,17),(13,28),(14,17),(14,19),(15,9),(15,19),(15,20),(15,29),(16,7),(16,18),(16,29),(17,26),(18,23),(18,25),(19,24),(19,26),(20,24),(20,25),(20,26),(21,8),(22,8),(23,22),(24,21),(24,22),(25,21),(25,22),(26,21),(27,19),(27,28),(27,29),(28,25),(28,26),(29,23),(29,24),(29,25)],30)
=> ? = 7 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,12),(1,16),(1,27),(2,11),(2,13),(2,27),(3,13),(3,15),(3,27),(4,8),(4,9),(4,10),(4,26),(5,11),(5,12),(5,14),(5,27),(6,4),(6,14),(6,15),(6,16),(6,27),(8,18),(8,23),(9,18),(9,23),(9,25),(10,23),(10,24),(11,21),(11,22),(12,17),(12,21),(13,22),(14,9),(14,17),(14,21),(14,26),(15,10),(15,22),(15,26),(16,8),(16,17),(16,26),(17,18),(17,25),(18,19),(19,7),(20,7),(21,24),(21,25),(22,24),(23,19),(23,20),(24,20),(25,19),(25,20),(26,23),(26,24),(26,25),(27,21),(27,22),(27,26)],28)
=> ? = 7 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,28),(2,12),(2,13),(2,19),(3,11),(3,13),(3,18),(4,1),(4,14),(4,15),(4,18),(4,19),(5,10),(5,11),(5,14),(5,19),(6,10),(6,12),(6,15),(6,18),(8,17),(8,25),(9,17),(9,26),(10,16),(10,22),(10,23),(11,22),(11,27),(12,23),(12,27),(13,27),(14,8),(14,16),(14,22),(14,28),(15,9),(15,16),(15,23),(15,28),(16,17),(16,25),(16,26),(17,20),(18,22),(18,27),(18,28),(19,23),(19,27),(19,28),(20,7),(21,7),(22,24),(22,25),(23,24),(23,26),(24,21),(25,20),(25,21),(26,20),(26,21),(27,24),(28,24),(28,25),(28,26)],29)
=> ? = 7 - 1
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ? = 7 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,12),(1,16),(1,27),(2,11),(2,13),(2,27),(3,13),(3,15),(3,27),(4,8),(4,9),(4,10),(4,26),(5,11),(5,12),(5,14),(5,27),(6,4),(6,14),(6,15),(6,16),(6,27),(8,18),(8,23),(9,18),(9,23),(9,25),(10,23),(10,24),(11,21),(11,22),(12,17),(12,21),(13,22),(14,9),(14,17),(14,21),(14,26),(15,10),(15,22),(15,26),(16,8),(16,17),(16,26),(17,18),(17,25),(18,19),(19,7),(20,7),(21,24),(21,25),(22,24),(23,19),(23,20),(24,20),(25,19),(25,20),(26,23),(26,24),(26,25),(27,21),(27,22),(27,26)],28)
=> ? = 7 - 1
[[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ? = 7 - 1
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 5 - 1
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 5 - 1
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 5 - 1
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 5 - 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 2 - 1
Description
The maximal number of elements covered by an element in a poset.
Matching statistic: St000846
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Mp00209: Permutations —pattern poset⟶ Posets
St000846: Posets ⟶ ℤResult quality: 32% ●values known / values provided: 32%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 4 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ? = 6 - 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6 - 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 5 - 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,9),(1,29),(2,13),(2,14),(2,27),(3,12),(3,16),(3,27),(4,11),(4,14),(4,15),(4,27),(5,1),(5,10),(5,15),(5,16),(5,27),(6,10),(6,11),(6,12),(6,13),(7,23),(9,23),(9,24),(10,18),(10,20),(10,28),(10,29),(11,17),(11,20),(11,28),(12,18),(12,28),(13,17),(13,28),(14,17),(14,19),(15,9),(15,19),(15,20),(15,29),(16,7),(16,18),(16,29),(17,26),(18,23),(18,25),(19,24),(19,26),(20,24),(20,25),(20,26),(21,8),(22,8),(23,22),(24,21),(24,22),(25,21),(25,22),(26,21),(27,19),(27,28),(27,29),(28,25),(28,26),(29,23),(29,24),(29,25)],30)
=> ? = 7 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,12),(1,16),(1,27),(2,11),(2,13),(2,27),(3,13),(3,15),(3,27),(4,8),(4,9),(4,10),(4,26),(5,11),(5,12),(5,14),(5,27),(6,4),(6,14),(6,15),(6,16),(6,27),(8,18),(8,23),(9,18),(9,23),(9,25),(10,23),(10,24),(11,21),(11,22),(12,17),(12,21),(13,22),(14,9),(14,17),(14,21),(14,26),(15,10),(15,22),(15,26),(16,8),(16,17),(16,26),(17,18),(17,25),(18,19),(19,7),(20,7),(21,24),(21,25),(22,24),(23,19),(23,20),(24,20),(25,19),(25,20),(26,23),(26,24),(26,25),(27,21),(27,22),(27,26)],28)
=> ? = 7 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,28),(2,12),(2,13),(2,19),(3,11),(3,13),(3,18),(4,1),(4,14),(4,15),(4,18),(4,19),(5,10),(5,11),(5,14),(5,19),(6,10),(6,12),(6,15),(6,18),(8,17),(8,25),(9,17),(9,26),(10,16),(10,22),(10,23),(11,22),(11,27),(12,23),(12,27),(13,27),(14,8),(14,16),(14,22),(14,28),(15,9),(15,16),(15,23),(15,28),(16,17),(16,25),(16,26),(17,20),(18,22),(18,27),(18,28),(19,23),(19,27),(19,28),(20,7),(21,7),(22,24),(22,25),(23,24),(23,26),(24,21),(25,20),(25,21),(26,20),(26,21),(27,24),(28,24),(28,25),(28,26)],29)
=> ? = 7 - 1
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ? = 7 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,12),(1,16),(1,27),(2,11),(2,13),(2,27),(3,13),(3,15),(3,27),(4,8),(4,9),(4,10),(4,26),(5,11),(5,12),(5,14),(5,27),(6,4),(6,14),(6,15),(6,16),(6,27),(8,18),(8,23),(9,18),(9,23),(9,25),(10,23),(10,24),(11,21),(11,22),(12,17),(12,21),(13,22),(14,9),(14,17),(14,21),(14,26),(15,10),(15,22),(15,26),(16,8),(16,17),(16,26),(17,18),(17,25),(18,19),(19,7),(20,7),(21,24),(21,25),(22,24),(23,19),(23,20),(24,20),(25,19),(25,20),(26,23),(26,24),(26,25),(27,21),(27,22),(27,26)],28)
=> ? = 7 - 1
[[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ? = 7 - 1
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 5 - 1
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 5 - 1
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 5 - 1
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 5 - 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 2 - 1
Description
The maximal number of elements covering an element of a poset.
Matching statistic: St000307
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000307: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Mp00262: Binary words —poset of factors⟶ Posets
St000307: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Values
[[1]]
=> => ?
=> ? = 1 - 1
[[1],[2]]
=> 1 => ([(0,1)],2)
=> 1 = 2 - 1
[[1],[2],[3]]
=> 11 => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 3 = 4 - 1
[[1],[2],[3],[4]]
=> 111 => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 5 - 1
[[1,3],[2],[4],[5]]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 5 - 1
[[1],[2],[3],[4],[5]]
=> 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
=> 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 6 - 1
[[1,3,4],[2],[5],[6]]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 6 - 1
[[1,5],[2],[3],[4],[6]]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[[1,4],[2],[3],[5],[6]]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 5 - 1
[[1,3],[2],[4],[5],[6]]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 1
[[1],[2],[3],[4],[5],[6]]
=> 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[[1,3,4,6],[2],[5],[7]]
=> 100101 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 7 - 1
[[1,4,6],[2],[3],[5],[7]]
=> 110101 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 7 - 1
[[1,3,6],[2],[4],[5],[7]]
=> 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 7 - 1
[[1,4,5],[2],[3],[6],[7]]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 7 - 1
[[1,3,5],[2],[4],[6],[7]]
=> 101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 7 - 1
[[1,3,4],[2],[5],[6],[7]]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 7 - 1
[[1,6],[2],[3],[4],[5],[7]]
=> 111101 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 5 - 1
[[1,5],[2],[3],[4],[6],[7]]
=> 111011 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 5 - 1
[[1,4],[2],[3],[5],[6],[7]]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 5 - 1
[[1,3],[2],[4],[5],[6],[7]]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 5 - 1
[[1],[2],[3],[4],[5],[6],[7]]
=> 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 1 = 2 - 1
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: St001942
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001942: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00209: Permutations —pattern poset⟶ Posets
St001942: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[[1]]
=> [1] => ([],1)
=> 0 = 1 - 1
[[1],[2]]
=> [2,1] => ([(0,1)],2)
=> 1 = 2 - 1
[[1],[2],[3]]
=> [3,2,1] => ([(0,2),(2,1)],3)
=> 1 = 2 - 1
[[1,3],[2],[4]]
=> [4,2,1,3] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> 3 = 4 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => ([(0,3),(2,1),(3,2)],4)
=> 1 = 2 - 1
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => ([(0,2),(0,3),(0,4),(1,7),(1,9),(2,8),(3,5),(3,8),(4,1),(4,5),(4,8),(5,7),(5,9),(7,6),(8,9),(9,6)],10)
=> ? = 5 - 1
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 2 - 1
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(1,19),(2,9),(2,12),(2,19),(3,8),(3,12),(3,19),(4,6),(4,8),(4,10),(4,19),(5,6),(5,9),(5,11),(5,19),(6,13),(6,17),(6,18),(8,16),(8,17),(9,16),(9,18),(10,13),(10,17),(11,13),(11,18),(12,16),(13,15),(14,7),(15,7),(16,14),(17,14),(17,15),(18,14),(18,15),(19,16),(19,17),(19,18)],20)
=> ? = 6 - 1
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => ([(0,3),(0,4),(0,5),(1,13),(2,6),(2,8),(3,9),(3,10),(4,2),(4,10),(4,11),(5,1),(5,9),(5,11),(6,14),(6,15),(8,14),(9,12),(9,13),(10,8),(10,12),(11,6),(11,12),(11,13),(12,14),(12,15),(13,15),(14,7),(15,7)],16)
=> ? = 6 - 1
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => ([(0,3),(0,4),(0,5),(1,9),(1,13),(2,8),(2,13),(3,11),(4,2),(4,6),(4,11),(5,1),(5,6),(5,11),(6,8),(6,9),(6,13),(8,10),(8,12),(9,10),(9,12),(10,7),(11,13),(12,7),(13,12)],14)
=> ? = 5 - 1
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => ([(0,2),(0,3),(0,5),(1,8),(1,12),(2,10),(3,6),(3,10),(4,1),(4,9),(4,11),(5,4),(5,6),(5,10),(6,9),(6,11),(8,7),(9,8),(9,12),(10,11),(11,12),(12,7)],13)
=> ? = 5 - 1
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 1 = 2 - 1
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,9),(1,29),(2,13),(2,14),(2,27),(3,12),(3,16),(3,27),(4,11),(4,14),(4,15),(4,27),(5,1),(5,10),(5,15),(5,16),(5,27),(6,10),(6,11),(6,12),(6,13),(7,23),(9,23),(9,24),(10,18),(10,20),(10,28),(10,29),(11,17),(11,20),(11,28),(12,18),(12,28),(13,17),(13,28),(14,17),(14,19),(15,9),(15,19),(15,20),(15,29),(16,7),(16,18),(16,29),(17,26),(18,23),(18,25),(19,24),(19,26),(20,24),(20,25),(20,26),(21,8),(22,8),(23,22),(24,21),(24,22),(25,21),(25,22),(26,21),(27,19),(27,28),(27,29),(28,25),(28,26),(29,23),(29,24),(29,25)],30)
=> ? = 7 - 1
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,12),(1,16),(1,27),(2,11),(2,13),(2,27),(3,13),(3,15),(3,27),(4,8),(4,9),(4,10),(4,26),(5,11),(5,12),(5,14),(5,27),(6,4),(6,14),(6,15),(6,16),(6,27),(8,18),(8,23),(9,18),(9,23),(9,25),(10,23),(10,24),(11,21),(11,22),(12,17),(12,21),(13,22),(14,9),(14,17),(14,21),(14,26),(15,10),(15,22),(15,26),(16,8),(16,17),(16,26),(17,18),(17,25),(18,19),(19,7),(20,7),(21,24),(21,25),(22,24),(23,19),(23,20),(24,20),(25,19),(25,20),(26,23),(26,24),(26,25),(27,21),(27,22),(27,26)],28)
=> ? = 7 - 1
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(1,9),(1,28),(2,12),(2,13),(2,19),(3,11),(3,13),(3,18),(4,1),(4,14),(4,15),(4,18),(4,19),(5,10),(5,11),(5,14),(5,19),(6,10),(6,12),(6,15),(6,18),(8,17),(8,25),(9,17),(9,26),(10,16),(10,22),(10,23),(11,22),(11,27),(12,23),(12,27),(13,27),(14,8),(14,16),(14,22),(14,28),(15,9),(15,16),(15,23),(15,28),(16,17),(16,25),(16,26),(17,20),(18,22),(18,27),(18,28),(19,23),(19,27),(19,28),(20,7),(21,7),(22,24),(22,25),(23,24),(23,26),(24,21),(25,20),(25,21),(26,20),(26,21),(27,24),(28,24),(28,25),(28,26)],29)
=> ? = 7 - 1
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => ([(0,4),(0,5),(0,6),(1,17),(2,10),(2,12),(3,9),(3,11),(4,3),(4,13),(4,15),(5,2),(5,14),(5,15),(6,1),(6,13),(6,14),(8,20),(9,18),(9,22),(10,19),(10,22),(11,8),(11,18),(12,8),(12,19),(13,9),(13,16),(13,17),(14,10),(14,16),(14,17),(15,11),(15,12),(15,16),(16,18),(16,19),(16,22),(17,22),(18,20),(18,21),(19,20),(19,21),(20,7),(21,7),(22,21)],23)
=> ? = 7 - 1
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => ([(0,1),(0,2),(0,3),(0,5),(0,6),(1,12),(1,16),(1,27),(2,11),(2,13),(2,27),(3,13),(3,15),(3,27),(4,8),(4,9),(4,10),(4,26),(5,11),(5,12),(5,14),(5,27),(6,4),(6,14),(6,15),(6,16),(6,27),(8,18),(8,23),(9,18),(9,23),(9,25),(10,23),(10,24),(11,21),(11,22),(12,17),(12,21),(13,22),(14,9),(14,17),(14,21),(14,26),(15,10),(15,22),(15,26),(16,8),(16,17),(16,26),(17,18),(17,25),(18,19),(19,7),(20,7),(21,24),(21,25),(22,24),(23,19),(23,20),(24,20),(25,19),(25,20),(26,23),(26,24),(26,25),(27,21),(27,22),(27,26)],28)
=> ? = 7 - 1
[[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => ([(0,4),(0,5),(0,6),(1,16),(2,8),(2,9),(3,2),(3,11),(3,12),(4,10),(4,13),(5,3),(5,13),(5,14),(6,1),(6,10),(6,14),(8,19),(9,19),(9,20),(10,15),(10,16),(11,9),(11,17),(11,18),(12,8),(12,17),(13,12),(13,15),(14,11),(14,15),(14,16),(15,17),(15,18),(16,18),(17,19),(17,20),(18,20),(19,7),(20,7)],21)
=> ? = 7 - 1
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 5 - 1
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 5 - 1
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => ([(0,1),(0,5),(0,6),(1,14),(2,11),(2,17),(3,4),(3,13),(3,17),(4,10),(4,16),(5,2),(5,12),(5,14),(6,3),(6,12),(6,14),(8,7),(9,8),(9,15),(10,8),(10,15),(11,9),(11,16),(12,11),(12,13),(12,17),(13,9),(13,10),(13,16),(14,17),(15,7),(16,15),(17,16)],18)
=> ? = 5 - 1
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => ([(0,2),(0,3),(0,6),(1,10),(1,15),(2,12),(3,7),(3,12),(4,5),(4,11),(4,13),(5,1),(5,9),(5,14),(6,4),(6,7),(6,12),(7,11),(7,13),(9,10),(9,15),(10,8),(11,9),(11,14),(12,13),(13,14),(14,15),(15,8)],16)
=> ? = 5 - 1
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ? = 2 - 1
Description
The number of loops of the quiver corresponding to the reduced incidence algebra of a poset.
Matching statistic: St000632
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St000632: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Mp00262: Binary words —poset of factors⟶ Posets
St000632: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Values
[[1]]
=> => ?
=> ? = 1 - 2
[[1],[2]]
=> 1 => ([(0,1)],2)
=> 0 = 2 - 2
[[1],[2],[3]]
=> 11 => ([(0,2),(2,1)],3)
=> 0 = 2 - 2
[[1,3],[2],[4]]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 4 - 2
[[1],[2],[3],[4]]
=> 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 2 - 2
[[1,4],[2],[3],[5]]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 5 - 2
[[1,3],[2],[4],[5]]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 5 - 2
[[1],[2],[3],[4],[5]]
=> 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 2 - 2
[[1,3,5],[2],[4],[6]]
=> 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 6 - 2
[[1,3,4],[2],[5],[6]]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 6 - 2
[[1,5],[2],[3],[4],[6]]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 2
[[1,4],[2],[3],[5],[6]]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 5 - 2
[[1,3],[2],[4],[5],[6]]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 2
[[1],[2],[3],[4],[5],[6]]
=> 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 2 - 2
[[1,3,4,6],[2],[5],[7]]
=> 100101 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 7 - 2
[[1,4,6],[2],[3],[5],[7]]
=> 110101 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 7 - 2
[[1,3,6],[2],[4],[5],[7]]
=> 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 7 - 2
[[1,4,5],[2],[3],[6],[7]]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 7 - 2
[[1,3,5],[2],[4],[6],[7]]
=> 101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 7 - 2
[[1,3,4],[2],[5],[6],[7]]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 7 - 2
[[1,6],[2],[3],[4],[5],[7]]
=> 111101 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 5 - 2
[[1,5],[2],[3],[4],[6],[7]]
=> 111011 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 5 - 2
[[1,4],[2],[3],[5],[6],[7]]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 5 - 2
[[1,3],[2],[4],[5],[6],[7]]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 5 - 2
[[1],[2],[3],[4],[5],[6],[7]]
=> 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 2 - 2
Description
The jump number of the poset.
A jump in a linear extension $e_1, \dots, e_n$ of a poset $P$ is a pair $(e_i, e_{i+1})$ so that $e_{i+1}$ does not cover $e_i$ in $P$. The jump number of a poset is the minimal number of jumps in linear extensions of a poset.
Matching statistic: St001633
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00134: Standard tableaux —descent word⟶ Binary words
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Mp00262: Binary words —poset of factors⟶ Posets
St001633: Posets ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 33%
Values
[[1]]
=> => ?
=> ? = 1 - 2
[[1],[2]]
=> 1 => ([(0,1)],2)
=> 0 = 2 - 2
[[1],[2],[3]]
=> 11 => ([(0,2),(2,1)],3)
=> 0 = 2 - 2
[[1,3],[2],[4]]
=> 101 => ([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> 2 = 4 - 2
[[1],[2],[3],[4]]
=> 111 => ([(0,3),(2,1),(3,2)],4)
=> 0 = 2 - 2
[[1,4],[2],[3],[5]]
=> 1101 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 5 - 2
[[1,3],[2],[4],[5]]
=> 1011 => ([(0,2),(0,3),(1,6),(2,7),(2,8),(3,1),(3,7),(3,8),(5,4),(6,4),(7,5),(8,5),(8,6)],9)
=> ? = 5 - 2
[[1],[2],[3],[4],[5]]
=> 1111 => ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0 = 2 - 2
[[1,3,5],[2],[4],[6]]
=> 10101 => ([(0,1),(0,2),(1,8),(1,9),(2,8),(2,9),(4,3),(5,3),(6,4),(6,5),(7,4),(7,5),(8,6),(8,7),(9,6),(9,7)],10)
=> ? = 6 - 2
[[1,3,4],[2],[5],[6]]
=> 10011 => ([(0,3),(0,4),(1,9),(2,6),(2,11),(3,2),(3,10),(3,12),(4,1),(4,10),(4,12),(6,7),(7,5),(8,5),(9,8),(10,6),(11,7),(11,8),(12,9),(12,11)],13)
=> ? = 6 - 2
[[1,5],[2],[3],[4],[6]]
=> 11101 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 2
[[1,4],[2],[3],[5],[6]]
=> 11011 => ([(0,2),(0,3),(1,5),(1,6),(2,10),(2,11),(3,1),(3,10),(3,11),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9)],12)
=> ? = 5 - 2
[[1,3],[2],[4],[5],[6]]
=> 10111 => ([(0,3),(0,4),(1,2),(1,11),(2,8),(3,9),(3,10),(4,1),(4,9),(4,10),(6,7),(7,5),(8,5),(9,6),(10,6),(10,11),(11,7),(11,8)],12)
=> ? = 5 - 2
[[1],[2],[3],[4],[5],[6]]
=> 11111 => ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> 0 = 2 - 2
[[1,3,4,6],[2],[5],[7]]
=> 100101 => ([(0,2),(0,3),(1,6),(1,11),(2,14),(2,15),(3,1),(3,14),(3,15),(5,8),(6,7),(7,9),(8,10),(9,4),(10,4),(11,7),(11,13),(12,8),(12,13),(13,9),(13,10),(14,5),(14,6),(14,12),(15,5),(15,11),(15,12)],16)
=> ? = 7 - 2
[[1,4,6],[2],[3],[5],[7]]
=> 110101 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 7 - 2
[[1,3,6],[2],[4],[5],[7]]
=> 101101 => ([(0,2),(0,3),(1,10),(1,11),(2,13),(2,14),(3,1),(3,13),(3,14),(5,8),(6,7),(7,4),(8,4),(9,7),(9,8),(10,6),(10,9),(11,5),(11,9),(12,5),(12,6),(13,10),(13,12),(14,11),(14,12)],15)
=> ? = 7 - 2
[[1,4,5],[2],[3],[6],[7]]
=> 110011 => ([(0,3),(0,4),(1,15),(1,16),(2,10),(2,11),(3,1),(3,13),(3,14),(4,2),(4,13),(4,14),(6,9),(7,8),(8,5),(9,5),(10,7),(11,6),(12,8),(12,9),(13,10),(13,15),(14,11),(14,16),(15,7),(15,12),(16,6),(16,12)],17)
=> ? = 7 - 2
[[1,3,5],[2],[4],[6],[7]]
=> 101011 => ([(0,2),(0,3),(1,9),(2,12),(2,14),(3,1),(3,12),(3,14),(5,7),(6,8),(7,4),(8,4),(9,5),(10,6),(10,11),(11,7),(11,8),(12,10),(12,13),(13,5),(13,6),(13,11),(14,9),(14,10),(14,13)],15)
=> ? = 7 - 2
[[1,3,4],[2],[5],[6],[7]]
=> 100111 => ([(0,4),(0,5),(1,11),(2,1),(2,13),(3,7),(3,14),(4,2),(4,12),(4,16),(5,3),(5,12),(5,16),(7,8),(8,9),(9,6),(10,6),(11,10),(12,7),(13,11),(13,15),(14,8),(14,15),(15,9),(15,10),(16,13),(16,14)],17)
=> ? = 7 - 2
[[1,6],[2],[3],[4],[5],[7]]
=> 111101 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 5 - 2
[[1,5],[2],[3],[4],[6],[7]]
=> 111011 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 5 - 2
[[1,4],[2],[3],[5],[6],[7]]
=> 110111 => ([(0,3),(0,4),(1,10),(2,1),(2,6),(2,12),(3,14),(3,15),(4,2),(4,14),(4,15),(6,7),(7,8),(8,5),(9,5),(10,9),(11,7),(11,13),(12,10),(12,13),(13,8),(13,9),(14,6),(14,11),(15,11),(15,12)],16)
=> ? = 5 - 2
[[1,3],[2],[4],[5],[6],[7]]
=> 101111 => ([(0,4),(0,5),(1,3),(1,12),(2,11),(3,2),(3,14),(4,10),(4,13),(5,1),(5,10),(5,13),(7,8),(8,9),(9,6),(10,7),(11,6),(12,8),(12,14),(13,7),(13,12),(14,9),(14,11)],15)
=> ? = 5 - 2
[[1],[2],[3],[4],[5],[6],[7]]
=> 111111 => ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> 0 = 2 - 2
Description
The number of simple modules with projective dimension two in the incidence algebra of the poset.
Matching statistic: St001555
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(load all 4 compositions to match this statistic)
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Mp00067: Permutations —Foata bijection⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001555: Signed permutations ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 67%
Values
[[1]]
=> [1] => [1] => [1] => 1
[[1],[2]]
=> [2,1] => [2,1] => [2,1] => 2
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [3,2,1] => 2
[[1,3],[2],[4]]
=> [4,2,1,3] => [2,4,1,3] => [2,4,1,3] => 4
[[1],[2],[3],[4]]
=> [4,3,2,1] => [4,3,2,1] => [4,3,2,1] => 2
[[1,4],[2],[3],[5]]
=> [5,3,2,1,4] => [3,5,2,1,4] => [3,5,2,1,4] => 5
[[1,3],[2],[4],[5]]
=> [5,4,2,1,3] => [2,5,4,1,3] => [2,5,4,1,3] => 5
[[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => [5,4,3,2,1] => [5,4,3,2,1] => ? = 2
[[1,3,5],[2],[4],[6]]
=> [6,4,2,1,3,5] => [2,4,6,1,3,5] => [2,4,6,1,3,5] => ? = 6
[[1,3,4],[2],[5],[6]]
=> [6,5,2,1,3,4] => [2,1,6,5,3,4] => [2,1,6,5,3,4] => ? = 6
[[1,5],[2],[3],[4],[6]]
=> [6,4,3,2,1,5] => [4,6,3,2,1,5] => [4,6,3,2,1,5] => ? = 5
[[1,4],[2],[3],[5],[6]]
=> [6,5,3,2,1,4] => [3,6,5,2,1,4] => [3,6,5,2,1,4] => ? = 5
[[1,3],[2],[4],[5],[6]]
=> [6,5,4,2,1,3] => [2,6,5,4,1,3] => [2,6,5,4,1,3] => ? = 5
[[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [6,5,4,3,2,1] => ? = 2
[[1,3,4,6],[2],[5],[7]]
=> [7,5,2,1,3,4,6] => [2,1,5,7,3,4,6] => [2,1,5,7,3,4,6] => ? = 7
[[1,4,6],[2],[3],[5],[7]]
=> [7,5,3,2,1,4,6] => [3,5,7,2,1,4,6] => [3,5,7,2,1,4,6] => ? = 7
[[1,3,6],[2],[4],[5],[7]]
=> [7,5,4,2,1,3,6] => [2,5,7,4,1,3,6] => [2,5,7,4,1,3,6] => ? = 7
[[1,4,5],[2],[3],[6],[7]]
=> [7,6,3,2,1,4,5] => [3,2,7,6,1,4,5] => [3,2,7,6,1,4,5] => ? = 7
[[1,3,5],[2],[4],[6],[7]]
=> [7,6,4,2,1,3,5] => [2,4,7,6,1,3,5] => [2,4,7,6,1,3,5] => ? = 7
[[1,3,4],[2],[5],[6],[7]]
=> [7,6,5,2,1,3,4] => [2,1,7,6,5,3,4] => [2,1,7,6,5,3,4] => ? = 7
[[1,6],[2],[3],[4],[5],[7]]
=> [7,5,4,3,2,1,6] => [5,7,4,3,2,1,6] => [5,7,4,3,2,1,6] => ? = 5
[[1,5],[2],[3],[4],[6],[7]]
=> [7,6,4,3,2,1,5] => [4,7,6,3,2,1,5] => [4,7,6,3,2,1,5] => ? = 5
[[1,4],[2],[3],[5],[6],[7]]
=> [7,6,5,3,2,1,4] => [3,7,6,5,2,1,4] => [3,7,6,5,2,1,4] => ? = 5
[[1,3],[2],[4],[5],[6],[7]]
=> [7,6,5,4,2,1,3] => [2,7,6,5,4,1,3] => [2,7,6,5,4,1,3] => ? = 5
[[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => [7,6,5,4,3,2,1] => ? = 2
Description
The order of a signed permutation.
The following 62 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000259The diameter of a connected graph. St000260The radius of a connected graph. St001557The number of inversions of the second entry of a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000100The number of linear extensions of a poset. St000526The number of posets with combinatorially isomorphic order polytopes. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St000455The second largest eigenvalue of a graph if it is integral. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001964The interval resolution global dimension of a poset. St000907The number of maximal antichains of minimal length in a poset. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000524The number of posets with the same order polynomial. St000525The number of posets with the same zeta polynomial. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001171The vector space dimension of $Ext_A^1(I_o,A)$ when $I_o$ is the tilting module corresponding to the permutation $o$ in the Auslander algebra $A$ of $K[x]/(x^n)$. St001413Half the length of the longest even length palindromic prefix of a binary word. St001645The pebbling number of a connected graph. St001773The number of minimal elements in Bruhat order not less than the signed permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St001686The order of promotion on a Gelfand-Tsetlin pattern. St000181The number of connected components of the Hasse diagram for the poset. St000635The number of strictly order preserving maps of a poset into itself. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001890The maximum magnitude of the Möbius function of a poset. St001207The Lowey length of the algebra $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001424The number of distinct squares in a binary word. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001927Sparre Andersen's number of positives of a signed permutation. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001624The breadth of a lattice. St000075The orbit size of a standard tableau under promotion. St000298The order dimension or Dushnik-Miller dimension of a poset. St000679The pruning number of an ordered tree. St001330The hat guessing number of a graph. St001399The distinguishing number of a poset. St001488The number of corners of a skew partition. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001621The number of atoms of a lattice. St000717The number of ordinal summands of a poset. St001498The normalised height of a Nakayama algebra with magnitude 1. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001816Eigenvalues of the top-to-random operator acting on a simple module.
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