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Your data matches 58 different statistics following compositions of up to 3 maps.
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Matching statistic: St000456
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00252: Permutations —restriction⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[[1,2],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,3],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,3],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,3],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[3]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,4],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,4],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,4],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,4],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,4],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[3,4],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[4]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[3],[4]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3],[4]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,3],[2]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1,3],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,2,3],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2,2,3],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 3
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[[1,5],[2]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,5],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,5],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1,5],[5]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,5],[3]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,5],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[2,5],[5]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[3,5],[4]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[3,5],[5]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[4,5],[5]]
=> [2,1,3] => [2,1] => ([(0,1)],2)
=> 1
[[1],[2],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[3],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1],[4],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[3],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[2],[4],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[3],[4],[5]]
=> [3,2,1] => [2,1] => ([(0,1)],2)
=> 1
[[1,1,4],[2]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1,4],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,1,4],[4]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[[1,2,4],[4]]
=> [3,1,2,4] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001582
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 31%●distinct values known / distinct values provided: 20%
Mp00062: Permutations —Lehmer-code to major-code bijection⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
St001582: Permutations ⟶ ℤResult quality: 20% ●values known / values provided: 31%●distinct values known / distinct values provided: 20%
Values
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[1,1,2],[2]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 1
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[1,1,3],[2]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,1,3],[3]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,2,3],[3]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[2,2,3],[3]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => [4,3,1,2] => 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,4,2,1] => [2,4,3,1] => 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [3,2,1,4] => [2,3,1,4] => 3
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => ? = 2
[[1,5],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,5],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,5],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1,5],[5]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[2,5],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[2,5],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[2,5],[5]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[3,5],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[3,5],[5]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[4,5],[5]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 1
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => [2,3,1] => 1
[[1,1,4],[2]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,1,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,1,4],[4]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,2,4],[4]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,3,4],[4]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[2,2,4],[3]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[2,2,4],[4]]
=> [3,1,2,4] => [2,3,1,4] => [3,2,1,4] => 1
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => ? = 2
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => ? = 2
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => ? = 2
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [1,3,4,2,5] => [1,4,3,2,5] => ? = 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => ? = 2
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => ? = 2
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [3,4,5,2,1] => [2,5,3,4,1] => ? = 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,4,2,1,5] => [2,4,3,1,5] => ? = 4
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [3,2,4,1,5] => [4,3,2,1,5] => ? = 2
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [4,2,5,3,1] => [3,5,4,2,1] => ? = 2
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,2,1,5,3] => [2,5,4,1,3] => ? = 4
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [2,4,1,5,3] => [5,2,4,1,3] => ? = 2
[[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [2,3,4,5,1,6] => [5,2,3,4,1,6] => ? = 1
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [3,4,5,1,6,2] => [6,5,3,4,1,2] => ? = 1
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [3,4,1,5,2,6] => [5,4,3,1,2,6] => ? = 3
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,1,5,2,6,3] => [6,5,4,1,2,3] => ? = 3
[[1,1,1,4],[2]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,1,4],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,1,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,2,4],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,2,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,3,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,2,2,4],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,2,2,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,2,3,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,3,3,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[2,2,2,4],[3]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[2,2,2,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[2,2,3,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[2,3,3,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[3,3,3,4],[4]]
=> [4,1,2,3,5] => [2,3,4,1,5] => [4,2,3,1,5] => ? = 1
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => ? = 2
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [3,4,1,5,2] => [5,4,3,1,2] => ? = 1
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => [4,3,1,2,5] => ? = 2
Description
The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order.
Matching statistic: St001817
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001817: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 31%●distinct values known / distinct values provided: 20%
Mp00238: Permutations —Clarke-Steingrimsson-Zeng⟶ Permutations
Mp00170: Permutations —to signed permutation⟶ Signed permutations
St001817: Signed permutations ⟶ ℤResult quality: 20% ●values known / values provided: 31%●distinct values known / distinct values provided: 20%
Values
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,1],[2,2]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 4 = 1 + 3
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[1,1,3],[2]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,1,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[2,2,3],[3]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,1],[2,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 4 = 1 + 3
[[1,1],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 4 = 1 + 3
[[1,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 4 = 1 + 3
[[2,2],[3,3]]
=> [3,4,1,2] => [4,1,3,2] => [4,1,3,2] => 4 = 1 + 3
[[1,1],[2],[3]]
=> [4,3,1,2] => [3,1,4,2] => [3,1,4,2] => 4 = 1 + 3
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => [2,3,1,4] => 6 = 3 + 3
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 + 3
[[1,5],[2]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1,5],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1,5],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1,5],[5]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[2,5],[3]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[2,5],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[2,5],[5]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[3,5],[4]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[3,5],[5]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[4,5],[5]]
=> [2,1,3] => [2,1,3] => [2,1,3] => 4 = 1 + 3
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => [2,3,1] => 4 = 1 + 3
[[1,1,4],[2]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,1,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,1,4],[4]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,2,4],[4]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,3,4],[4]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[2,2,4],[3]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[2,2,4],[4]]
=> [3,1,2,4] => [3,1,2,4] => [3,1,2,4] => 4 = 1 + 3
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 + 3
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 + 3
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 + 3
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,2,1,3,5] => [4,2,1,3,5] => ? = 1 + 3
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 + 3
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 + 3
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,1,2,5,3] => [4,1,2,5,3] => ? = 1 + 3
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [3,1,4,2,5] => [3,1,4,2,5] => ? = 4 + 3
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,1,3,5] => [2,4,1,3,5] => ? = 2 + 3
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [4,1,5,3,2] => [4,1,5,3,2] => ? = 2 + 3
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [5,1,4,3,2] => [5,1,4,3,2] => ? = 4 + 3
[[1,2],[2,3],[3]]
=> [4,2,5,1,3] => [5,4,2,1,3] => [5,4,2,1,3] => ? = 2 + 3
[[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [5,1,2,3,4,6] => [5,1,2,3,4,6] => ? = 1 + 3
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [6,1,2,3,5,4] => [6,1,2,3,5,4] => ? = 1 + 3
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [5,1,2,4,3,6] => [5,1,2,4,3,6] => ? = 3 + 3
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [6,1,2,4,5,3] => [6,1,2,4,5,3] => ? = 3 + 3
[[1,1,1,4],[2]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,1,4],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,1,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,2,4],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,2,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,3,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,2,2,4],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,2,2,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,2,3,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,3,3,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[2,2,2,4],[3]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[2,2,2,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[2,2,3,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[2,3,3,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[3,3,3,4],[4]]
=> [4,1,2,3,5] => [4,1,2,3,5] => [4,1,2,3,5] => ? = 1 + 3
[[1,1,1],[2,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,1,1],[3,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,1,1],[4,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,1,4],[2,2]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 + 3
[[1,1,2],[3,4]]
=> [4,5,1,2,3] => [5,1,2,4,3] => [5,1,2,4,3] => ? = 1 + 3
[[1,1,4],[2,3]]
=> [3,4,1,2,5] => [4,1,3,2,5] => [4,1,3,2,5] => ? = 2 + 3
Description
The number of flag weak exceedances of a signed permutation.
This is the number of negative entries plus twice the number of weak exceedances of the signed permutation.
Matching statistic: St001875
(load all 17 compositions to match this statistic)
(load all 17 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 23%●distinct values known / distinct values provided: 20%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00208: Permutations —lattice of intervals⟶ Lattices
St001875: Lattices ⟶ ℤResult quality: 20% ●values known / values provided: 23%●distinct values known / distinct values provided: 20%
Values
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1],[2],[3]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[1,1,2],[2]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,1],[2,2]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1],[2],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[1],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[2],[3],[4]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[1,1,3],[2]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,1,3],[3]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,2,3],[3]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[2,2,3],[3]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,1],[2,3]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,1],[3,3]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,2],[3,3]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[2,2],[3,3]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,3,1,2] => ([(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)
=> ? = 1 + 4
[[1,3],[2],[3]]
=> [3,2,1,4] => [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)
=> ? = 3 + 4
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 4
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [4,5,1,3,2] => ([(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)
=> ? = 1 + 4
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 2 + 4
[[1,5],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1,5],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[2,5],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[2,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[2,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[3,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[3,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[4,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> 5 = 1 + 4
[[1],[2],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[1],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[1],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[2],[3],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[2],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[3],[4],[5]]
=> [3,2,1] => [3,2,1] => ([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5),(4,6),(5,6)],7)
=> 5 = 1 + 4
[[1,1,4],[2]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,1,4],[3]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,1,4],[4]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,2,4],[3]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,2,4],[4]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,3,4],[4]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[2,2,4],[3]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[2,2,4],[4]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[2,3,4],[4]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[3,3,4],[4]]
=> [3,1,2,4] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(2,5),(3,5),(4,5)],6)
=> 5 = 1 + 4
[[1,1],[2,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,1],[3,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,1],[4,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,2],[3,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,2],[4,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,3],[4,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[2,2],[3,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[2,2],[4,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[2,3],[4,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[3,3],[4,4]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,3,1,2] => ([(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)
=> ? = 1 + 4
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => ([(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)
=> ? = 1 + 4
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => ([(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)
=> ? = 1 + 4
[[1,4],[2],[3]]
=> [3,2,1,4] => [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)
=> ? = 3 + 4
[[1,4],[2],[4]]
=> [3,2,1,4] => [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)
=> ? = 3 + 4
[[1,4],[3],[4]]
=> [3,2,1,4] => [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)
=> ? = 3 + 4
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,3,1,2] => ([(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)
=> ? = 1 + 4
[[2,4],[3],[4]]
=> [3,2,1,4] => [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)
=> ? = 3 + 4
[[1],[2],[3],[4]]
=> [4,3,2,1] => [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)
=> ? = 3 + 4
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 4
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 4
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 4
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 4
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [4,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 4
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [4,5,1,3,2] => ([(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)
=> ? = 1 + 4
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [4,5,1,3,2] => ([(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)
=> ? = 1 + 4
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 2 + 4
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,3,2] => ([(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)
=> ? = 1 + 4
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 2 + 4
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 2 + 4
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,3,2] => ([(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)
=> ? = 1 + 4
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [2,5,1,4,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 1 + 4
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,5,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 6 = 2 + 4
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [4,5,1,3,2] => ([(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)
=> ? = 1 + 4
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [5,4,1,3,2] => ([(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)
=> ? = 1 + 4
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,3,1,5,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 4 + 4
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [4,2,1,5,3] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,7),(2,7),(3,7),(4,6),(5,6),(6,7)],8)
=> ? = 2 + 4
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,3,4,1,2] => ([(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)
=> ? = 2 + 4
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,3,5,1,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,8),(2,6),(3,6),(4,7),(5,7),(6,9),(7,8),(8,9)],10)
=> ? = 4 + 4
[[1,1,1,1,2],[2]]
=> [5,1,2,3,4,6] => [5,1,6,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,10),(3,10),(4,8),(5,7),(6,7),(6,8),(7,9),(8,9),(9,10)],11)
=> ? = 1 + 4
[[1,1,1,1],[2,2]]
=> [5,6,1,2,3,4] => [5,6,1,4,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,8),(4,7),(5,7),(6,8),(6,9),(7,12),(8,11),(9,11),(10,12),(11,10)],13)
=> ? = 1 + 4
[[1,1,1,2],[2,2]]
=> [4,5,1,2,3,6] => [4,6,1,5,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,8),(2,8),(3,8),(4,8),(5,7),(6,7),(7,8)],9)
=> ? = 3 + 4
[[1,1,1],[2,2,2]]
=> [4,5,6,1,2,3] => [4,6,5,1,3,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,10),(2,9),(3,7),(4,7),(5,8),(6,8),(7,9),(8,10),(9,11),(10,11)],12)
=> ? = 3 + 4
[[1,1],[2,5]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
[[1,1],[3,5]]
=> [3,4,1,2] => [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)
=> ? = 1 + 4
Description
The number of simple modules with projective dimension at most 1.
Matching statistic: St000782
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Values
[[1,2],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,3],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[2,3],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1],[2],[3]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[1,1,2],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,4],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,4],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[2,4],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[2,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[3,4],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1],[2],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[1],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[2],[3],[4]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[1,1,3],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,1,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[2,2,3],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,1],[2,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,1],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[2,2],[3,3]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,1],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[[1,3],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3
[[1,1,1,2],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1
[[1,1,1],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[[1,1,2],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[[1,5],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,5],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,5],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,5],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[2,5],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[2,5],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[2,5],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[3,5],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[3,5],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[4,5],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1],[2],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[1],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[1],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[2],[3],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[2],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[3],[4],[5]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [(1,6),(2,3),(4,5)]
=> 1
[[1,1,4],[2]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,1,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,1,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,2,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,3,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[2,2,4],[3]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[2,2,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[2,3,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[3,3,4],[4]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,8),(6,7)]
=> ? = 1
[[1,1],[2,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,1],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,1],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[2,2],[3,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[2,2],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[2,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[3,3],[4,4]]
=> [2,2]
=> [1,1,1,0,0,0]
=> [(1,6),(2,5),(3,4)]
=> 1
[[1,1],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[[1,1],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[[1,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[[1,4],[2],[3]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3
[[1,4],[2],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3
[[1,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3
[[2,2],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 1
[[2,4],[3],[4]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [(1,8),(2,3),(4,5),(6,7)]
=> ? = 3
[[1,1,1,3],[2]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1
[[1,1,1,3],[3]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1
[[1,1,2,3],[3]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1
[[1,2,2,3],[3]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1
[[2,2,2,3],[3]]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [(1,2),(3,4),(5,6),(7,10),(8,9)]
=> ? = 1
[[1,1,1],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[[1,1,1],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[[1,1,3],[2,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[[1,1,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[[1,1,3],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[[1,1,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[[1,2,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[[1,2,3],[2,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[[1,2,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[[2,2,2],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 1
[[2,2,3],[3,3]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 2
[[1,1,1],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 4
[[1,2,3],[2],[3]]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [(1,2),(3,4),(5,10),(6,7),(8,9)]
=> ? = 2
[[1,1],[2,2],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> ? = 2
[[1,1],[2,3],[3]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [(1,8),(2,5),(3,4),(6,7)]
=> ? = 4
[[1,6],[2]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,6],[3]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,6],[4]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,6],[5]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
[[1,6],[6]]
=> [2,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1
Description
The indicator function of whether a given perfect matching is an L & P matching.
An L&P matching is built inductively as follows:
starting with either a single edge, or a hairpin $([1,3],[2,4])$, insert a noncrossing matching or inflate an edge by a ladder, that is, a number of nested edges.
The number of L&P matchings is (see [thm. 1, 2])
$$\frac{1}{2} \cdot 4^{n} + \frac{1}{n + 1}{2 \, n \choose n} - {2 \, n + 1 \choose n} + {2 \, n - 1 \choose n - 1}$$
Matching statistic: St000307
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St000307: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Values
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,3],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[2,2,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,5],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,5],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,5],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[4,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,4],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[2,2,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[2,2,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[2,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[3,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 4 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 4 + 1
[[1,6],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,6],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,6],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,6],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,6],[6]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 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: St001491
Mp00077: Semistandard tableaux —shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00268: Binary words —zeros to flag zeros⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Values
[[1,2],[2]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,3],[2]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,3],[3]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[2,3],[3]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1],[2],[3]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,1],[2,2]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,4],[2]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,4],[3]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,4],[4]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[2,4],[3]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[2,4],[4]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[3,4],[4]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1],[2],[4]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[1],[3],[4]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[2],[3],[4]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[1,1,3],[2]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,1,3],[3]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,2,3],[3]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[2,2,3],[3]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,1],[2,3]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,1],[3,3]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,2],[3,3]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[2,2],[3,3]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,1],[2],[3]]
=> [2,1,1]
=> 10110 => 10001 => ? = 1 + 1
[[1,3],[2],[3]]
=> [2,1,1]
=> 10110 => 10001 => ? = 3 + 1
[[1,1,1,2],[2]]
=> [4,1]
=> 100010 => 100101 => ? = 1 + 1
[[1,1,1],[2,2]]
=> [3,2]
=> 10100 => 01001 => ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,2]
=> 10100 => 01001 => ? = 2 + 1
[[1,5],[2]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,5],[3]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,5],[4]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,5],[5]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[2,5],[3]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[2,5],[4]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[2,5],[5]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[3,5],[4]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[3,5],[5]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[4,5],[5]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1],[2],[5]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[1],[3],[5]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[1],[4],[5]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[2],[3],[5]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[2],[4],[5]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[3],[4],[5]]
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
[[1,1,4],[2]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,1,4],[3]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,1,4],[4]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,2,4],[3]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,2,4],[4]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,3,4],[4]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[2,2,4],[3]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[2,2,4],[4]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[2,3,4],[4]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[3,3,4],[4]]
=> [3,1]
=> 10010 => 01101 => ? = 1 + 1
[[1,1],[2,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,1],[3,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,1],[4,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,2],[3,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,2],[4,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,3],[4,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[2,2],[3,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[2,2],[4,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[2,3],[4,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[3,3],[4,4]]
=> [2,2]
=> 1100 => 1011 => 2 = 1 + 1
[[1,1],[2],[4]]
=> [2,1,1]
=> 10110 => 10001 => ? = 1 + 1
[[1,1],[3],[4]]
=> [2,1,1]
=> 10110 => 10001 => ? = 1 + 1
[[1,2],[3],[4]]
=> [2,1,1]
=> 10110 => 10001 => ? = 1 + 1
[[1,4],[2],[3]]
=> [2,1,1]
=> 10110 => 10001 => ? = 3 + 1
[[1,4],[2],[4]]
=> [2,1,1]
=> 10110 => 10001 => ? = 3 + 1
[[1,4],[3],[4]]
=> [2,1,1]
=> 10110 => 10001 => ? = 3 + 1
[[2,2],[3],[4]]
=> [2,1,1]
=> 10110 => 10001 => ? = 1 + 1
[[2,4],[3],[4]]
=> [2,1,1]
=> 10110 => 10001 => ? = 3 + 1
[[1],[2],[3],[4]]
=> [1,1,1,1]
=> 11110 => 01111 => ? = 3 + 1
[[1,1,1,3],[2]]
=> [4,1]
=> 100010 => 100101 => ? = 1 + 1
[[1,1,1,3],[3]]
=> [4,1]
=> 100010 => 100101 => ? = 1 + 1
[[1,1,2,3],[3]]
=> [4,1]
=> 100010 => 100101 => ? = 1 + 1
[[1,2,2,3],[3]]
=> [4,1]
=> 100010 => 100101 => ? = 1 + 1
[[2,2,2,3],[3]]
=> [4,1]
=> 100010 => 100101 => ? = 1 + 1
[[1,1,1],[2,3]]
=> [3,2]
=> 10100 => 01001 => ? = 1 + 1
[[1,1,1],[3,3]]
=> [3,2]
=> 10100 => 01001 => ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,2]
=> 10100 => 01001 => ? = 2 + 1
[[1,1,2],[3,3]]
=> [3,2]
=> 10100 => 01001 => ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,2]
=> 10100 => 01001 => ? = 2 + 1
[[1,1,3],[3,3]]
=> [3,2]
=> 10100 => 01001 => ? = 2 + 1
[[1,2,2],[3,3]]
=> [3,2]
=> 10100 => 01001 => ? = 1 + 1
[[1,2,3],[2,3]]
=> [3,2]
=> 10100 => 01001 => ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,2]
=> 10100 => 01001 => ? = 2 + 1
[[2,2,2],[3,3]]
=> [3,2]
=> 10100 => 01001 => ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,2]
=> 10100 => 01001 => ? = 2 + 1
[[1,1,1],[2],[3]]
=> [3,1,1]
=> 100110 => 011101 => ? = 1 + 1
[[1,1,3],[2],[3]]
=> [3,1,1]
=> 100110 => 011101 => ? = 4 + 1
[[1,2,3],[2],[3]]
=> [3,1,1]
=> 100110 => 011101 => ? = 2 + 1
[[1,1],[2,2],[3]]
=> [2,2,1]
=> 11010 => 10011 => ? = 2 + 1
[[1,1],[2,3],[3]]
=> [2,2,1]
=> 11010 => 10011 => ? = 4 + 1
[[1,6],[2]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,6],[3]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,6],[4]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,6],[5]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
[[1,6],[6]]
=> [2,1]
=> 1010 => 1001 => 2 = 1 + 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001631
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001631: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001631: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Values
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,1,3],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,1,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,2,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 - 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[[1,5],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,5],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,5],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[4,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,1,4],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,1,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,1,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,2,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,2,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,2,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[2,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[3,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 0 = 1 - 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 - 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 - 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 - 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 - 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 - 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 - 1
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 - 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 - 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 - 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 - 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 4 - 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 - 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 - 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 4 - 1
[[1,6],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,6],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,6],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,6],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
[[1,6],[6]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0 = 1 - 1
Description
The number of simple modules $S$ with $dim Ext^1(S,A)=1$ in the incidence algebra $A$ of the poset.
Matching statistic: St001632
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Mp00087: Permutations —inverse first fundamental transformation⟶ Permutations
Mp00209: Permutations —pattern poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 10% ●values known / values provided: 16%●distinct values known / distinct values provided: 10%
Values
[[1,2],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[3]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,4],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,4],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,4],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[3],[4]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,3],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,2,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[2,2,3],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1],[2,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[2,2],[3,3]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[1,3],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,5],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,5],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,5],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,5],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[4,5],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[2],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[3],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[2],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[3],[4],[5]]
=> [3,2,1] => [2,3,1] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,1,4],[2]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,2,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[2,2,4],[3]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[2,2,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[2,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[3,3,4],[4]]
=> [3,1,2,4] => [3,2,1,4] => ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> 2 = 1 + 1
[[1,1],[2,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[2,2],[3,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[2,2],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[2,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[3,3],[4,4]]
=> [3,4,1,2] => [3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(1,6),(1,7),(2,6),(2,7),(3,6),(3,7),(4,6),(4,7),(6,5),(7,5)],8)
=> ? = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[1,4],[2],[3]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1,4],[2],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => [4,2,3,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 1 + 1
[[2,4],[3],[4]]
=> [3,2,1,4] => [2,3,1,4] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1],[2],[3],[4]]
=> [4,3,2,1] => [3,2,4,1] => ([(0,1),(0,2),(0,3),(1,6),(2,4),(2,6),(3,4),(3,6),(4,5),(6,5)],7)
=> ? = 3 + 1
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => [4,3,2,1,5] => ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,2,3],[2,3]]
=> [2,4,1,3,5] => [4,3,1,2,5] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 1 + 1
[[1,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[2,2,2],[3,3]]
=> [4,5,1,2,3] => [5,3,1,4,2] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[2,2,3],[3,3]]
=> [3,4,1,2,5] => [3,1,4,2,5] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 2 + 1
[[1,1,1],[2],[3]]
=> [5,4,1,2,3] => [4,2,5,3,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,12),(2,8),(2,10),(2,12),(3,7),(3,10),(3,12),(4,6),(4,10),(4,12),(5,6),(5,7),(5,8),(5,12),(6,11),(6,13),(7,11),(7,13),(8,11),(8,13),(10,13),(11,9),(12,11),(12,13),(13,9)],14)
=> ? = 1 + 1
[[1,1,3],[2],[3]]
=> [4,3,1,2,5] => [4,2,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 4 + 1
[[1,2,3],[2],[3]]
=> [4,2,1,3,5] => [2,4,3,1,5] => ([(0,1),(0,2),(0,3),(0,4),(1,5),(1,10),(2,8),(2,9),(2,10),(3,7),(3,9),(3,10),(4,5),(4,7),(4,8),(5,11),(7,11),(7,12),(8,11),(8,12),(9,12),(10,11),(10,12),(11,6),(12,6)],13)
=> ? = 2 + 1
[[1,1],[2,2],[3]]
=> [5,3,4,1,2] => [5,2,3,4,1] => ([(0,2),(0,3),(0,4),(1,9),(2,5),(2,7),(3,5),(3,6),(4,1),(4,6),(4,7),(5,10),(6,9),(6,10),(7,9),(7,10),(9,8),(10,8)],11)
=> ? = 2 + 1
[[1,1],[2,3],[3]]
=> [4,3,5,1,2] => [4,1,5,2,3] => ([(0,2),(0,3),(0,4),(0,5),(1,11),(1,12),(2,7),(2,10),(3,6),(3,10),(4,6),(4,8),(4,10),(5,1),(5,7),(5,8),(5,10),(6,12),(7,11),(7,12),(8,11),(8,12),(10,11),(10,12),(11,9),(12,9)],13)
=> ? = 4 + 1
[[1,6],[2]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,6],[3]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,6],[4]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,6],[5]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[[1,6],[6]]
=> [2,1,3] => [2,1,3] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001621
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00075: Semistandard tableaux —reading word permutation⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 12%●distinct values known / distinct values provided: 10%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00195: Posets —order ideals⟶ Lattices
St001621: Lattices ⟶ ℤResult quality: 10% ●values known / values provided: 12%●distinct values known / distinct values provided: 10%
Values
[[1,2],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,3],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,3],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1],[2],[3]]
=> [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)
=> ? = 1 + 1
[[1,1,2],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,1],[2,2]]
=> [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)
=> ? = 1 + 1
[[1,4],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,4],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[3,4],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1],[2],[4]]
=> [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)
=> ? = 1 + 1
[[1],[3],[4]]
=> [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)
=> ? = 1 + 1
[[2],[3],[4]]
=> [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)
=> ? = 1 + 1
[[1,1,3],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,1,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[2,2,3],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,1],[2,3]]
=> [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)
=> ? = 1 + 1
[[1,1],[3,3]]
=> [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)
=> ? = 1 + 1
[[1,2],[3,3]]
=> [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)
=> ? = 1 + 1
[[2,2],[3,3]]
=> [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)
=> ? = 1 + 1
[[1,1],[2],[3]]
=> [4,3,1,2] => ([(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)
=> ? = 1 + 1
[[1,3],[2],[3]]
=> [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)
=> ? = 3 + 1
[[1,1,1,2],[2]]
=> [4,1,2,3,5] => ([(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)
=> ? = 1 + 1
[[1,1,1],[2,2]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[[1,1,2],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[[1,5],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,5],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,5],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,5],[5]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,5],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,5],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,5],[5]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[3,5],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[3,5],[5]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[4,5],[5]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1],[2],[5]]
=> [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)
=> ? = 1 + 1
[[1],[3],[5]]
=> [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)
=> ? = 1 + 1
[[1],[4],[5]]
=> [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)
=> ? = 1 + 1
[[2],[3],[5]]
=> [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)
=> ? = 1 + 1
[[2],[4],[5]]
=> [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)
=> ? = 1 + 1
[[3],[4],[5]]
=> [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)
=> ? = 1 + 1
[[1,1,4],[2]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,1,4],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,1,4],[4]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,2,4],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,2,4],[4]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,3,4],[4]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[2,2,4],[3]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[2,2,4],[4]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[2,3,4],[4]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[3,3,4],[4]]
=> [3,1,2,4] => ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> 2 = 1 + 1
[[1,1],[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)
=> ? = 1 + 1
[[1,1],[3,4]]
=> [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)
=> ? = 1 + 1
[[1,1],[4,4]]
=> [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)
=> ? = 1 + 1
[[1,2],[3,4]]
=> [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)
=> ? = 1 + 1
[[1,2],[4,4]]
=> [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)
=> ? = 1 + 1
[[1,3],[4,4]]
=> [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)
=> ? = 1 + 1
[[2,2],[3,4]]
=> [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)
=> ? = 1 + 1
[[2,2],[4,4]]
=> [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)
=> ? = 1 + 1
[[2,3],[4,4]]
=> [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)
=> ? = 1 + 1
[[3,3],[4,4]]
=> [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)
=> ? = 1 + 1
[[1,1],[2],[4]]
=> [4,3,1,2] => ([(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)
=> ? = 1 + 1
[[1,1],[3],[4]]
=> [4,3,1,2] => ([(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)
=> ? = 1 + 1
[[1,2],[3],[4]]
=> [4,3,1,2] => ([(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)
=> ? = 1 + 1
[[1,4],[2],[3]]
=> [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)
=> ? = 3 + 1
[[1,4],[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)
=> ? = 3 + 1
[[1,4],[3],[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)
=> ? = 3 + 1
[[2,2],[3],[4]]
=> [4,3,1,2] => ([(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)
=> ? = 1 + 1
[[2,4],[3],[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)
=> ? = 3 + 1
[[1],[2],[3],[4]]
=> [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)
=> ? = 3 + 1
[[1,1,1,3],[2]]
=> [4,1,2,3,5] => ([(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)
=> ? = 1 + 1
[[1,1,1,3],[3]]
=> [4,1,2,3,5] => ([(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)
=> ? = 1 + 1
[[1,1,2,3],[3]]
=> [4,1,2,3,5] => ([(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)
=> ? = 1 + 1
[[1,2,2,3],[3]]
=> [4,1,2,3,5] => ([(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)
=> ? = 1 + 1
[[2,2,2,3],[3]]
=> [4,1,2,3,5] => ([(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)
=> ? = 1 + 1
[[1,1,1],[2,3]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[[1,1,1],[3,3]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[[1,1,3],[2,2]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[[1,1,2],[3,3]]
=> [4,5,1,2,3] => ([(0,3),(1,4),(4,2)],5)
=> ([(0,4),(0,5),(1,9),(2,3),(2,11),(3,8),(4,1),(4,10),(5,2),(5,10),(7,6),(8,6),(9,7),(10,9),(10,11),(11,7),(11,8)],12)
=> ? = 1 + 1
[[1,1,3],[2,3]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[[1,1,3],[3,3]]
=> [3,4,1,2,5] => ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(2,8),(3,7),(4,3),(4,6),(5,2),(5,6),(6,7),(6,8),(7,9),(8,9),(9,1)],10)
=> ? = 2 + 1
[[1,6],[2]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,6],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,6],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,6],[5]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[1,6],[6]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,6],[3]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,6],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,6],[5]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[2,6],[6]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[3,6],[4]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[3,6],[5]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[3,6],[6]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[4,6],[5]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[4,6],[6]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
[[5,6],[6]]
=> [2,1,3] => ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2 = 1 + 1
Description
The number of atoms of a lattice.
An element of a lattice is an '''atom''' if it covers the least element.
The following 48 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001623The number of doubly irreducible elements of a lattice. St001624The breadth of a lattice. St001625The Möbius invariant of a lattice. 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. St000189The number of elements in the poset. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001926Sparre Andersen's position of the maximum of a signed permutation. St000181The number of connected components of the Hasse diagram for the poset. St001722The number of minimal chains with small intervals between a binary word and the top element. St001805The maximal overlap of a cylindrical tableau associated with a semistandard tableau. St001890The maximum magnitude of the Möbius function of a poset. St000084The number of subtrees. St000168The number of internal nodes of an ordered tree. St000328The maximum number of child nodes in a tree. St000679The pruning number of an ordered tree. St001637The number of (upper) dissectors of a poset. St001668The number of points of the poset minus the width of the poset. St001857The number of edges in the reduced word graph of a signed permutation. St001964The interval resolution global dimension of a poset. St000075The orbit size of a standard tableau under promotion. St000718The largest Laplacian 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. St001626The number of maximal proper sublattices of a lattice. St001754The number of tolerances of a finite lattice. St000116The major index of a semistandard tableau obtained by standardizing. St000327The number of cover relations in a poset. St000635The number of strictly order preserving maps of a poset into itself. St000973The length of the boundary of an ordered tree. St000975The length of the boundary minus the length of the trunk of an ordered tree. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St000415The size of the automorphism group of the rooted tree underlying the ordered tree. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000080The rank of the 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 $K[x]/(x^n)$. 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 $K[x]/(x^n)$. St000100The number of linear extensions of a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of $K[x]/(x^n)$. St000180The number of chains of a poset. St001909The number of interval-closed sets of a poset. St000529The number of permutations whose descent word is the given binary word. St000634The number of endomorphisms of a poset. St001645The pebbling number of a connected graph.
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