Processing math: 28%

Your data matches 5 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)
Matching statistic: St000064
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00045: Integer partitions reading tableauStandard tableaux
Mp00081: Standard tableaux reading word permutationPermutations
St000064: Permutations ⟶ ℤResult quality: 100% values known / values provided: 100%distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[2,-1] => [2]
=> [[1,2]]
=> [1,2] => 2
[-2,1] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,-2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,-2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,-2,-3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[1,3,-2] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,-3,2] => [2]
=> [[1,2]]
=> [1,2] => 2
[-1,3,-2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-1,-3,2] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[2,-1,3] => [2]
=> [[1,2]]
=> [1,2] => 2
[2,-1,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-2,1,3] => [2]
=> [[1,2]]
=> [1,2] => 2
[-2,1,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[-2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[-2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[-3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[-3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[3,2,-1] => [2]
=> [[1,2]]
=> [1,2] => 2
[3,-2,-1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-3,2,1] => [2]
=> [[1,2]]
=> [1,2] => 2
[-3,-2,1] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,-2,-3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,2,-3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[-1,-2,3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,-2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[-1,-2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4
[1,2,4,-3] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,2,-4,3] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,-2,4,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[1,-2,-4,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-1,2,4,-3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-1,2,-4,3] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[-1,-2,4,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[-1,-2,4,-3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[-1,-2,-4,3] => [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[-1,-2,-4,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2
[1,3,-2,4] => [2]
=> [[1,2]]
=> [1,2] => 2
[1,3,-2,-4] => [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
Description
The number of one-box pattern of a permutation. This is the number of i for which there is a j such that (i,σi) and (j,σj) have distance 2 in the taxi metric on the Z2 grid.
Matching statistic: St001880
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00179: Integer partitions to skew partitionSkew partitions
Mp00185: Skew partitions cell posetPosets
St001880: Posets ⟶ ℤResult quality: 46% values known / values provided: 46%distinct values known / distinct values provided: 67%
Values
[-1,-2] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[2,-1] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[-2,1] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,-2,-3] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,2,-3] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,-2,3] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,-2,-3] => [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,-2] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,-3,2] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,3,-2] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-1,-3,2] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[2,-1,3] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[2,-1,-3] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-2,1,3] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[-2,1,-3] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[2,3,-1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[2,-3,1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-2,3,1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-2,-3,-1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[3,1,-2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[3,-1,2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-3,1,2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-3,-1,-2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[3,2,-1] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[3,-2,-1] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-3,2,1] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[-3,-2,1] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[1,2,-3,-4] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,-2,3,-4] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,-2,-3,4] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,-2,-3,-4] => [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-1,2,3,-4] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,2,-3,4] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,2,-3,-4] => [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-1,-2,3,4] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,-2,3,-4] => [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-1,-2,-3,4] => [1,1,1]
=> [[1,1,1],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-1,-2,-3,-4] => [1,1,1,1]
=> [[1,1,1,1],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[1,2,4,-3] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,2,-4,3] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,-2,4,-3] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[1,-2,-4,3] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-1,2,4,-3] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-1,2,-4,3] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-1,-2,4,3] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,-2,4,-3] => [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 3
[-1,-2,-4,3] => [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 3
[-1,-2,-4,-3] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,3,-2,4] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,3,-2,-4] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[1,-3,2,4] => [2]
=> [[2],[]]
=> ([(0,1)],2)
=> ? = 2
[1,-3,2,-4] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-1,3,2,-4] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[-1,3,-2,4] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-1,3,-2,-4] => [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 3
[-1,-3,2,4] => [2,1]
=> [[2,1],[]]
=> ([(0,1),(0,2)],3)
=> ? = 2
[-1,-3,2,-4] => [2,1,1]
=> [[2,1,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 3
[-1,-3,-2,-4] => [1,1]
=> [[1,1],[]]
=> ([(0,1)],2)
=> ? = 2
[1,3,4,-2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,3,-4,2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,-3,4,2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,-3,-4,-2] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-1,3,4,-2] => [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4
[-1,3,-4,2] => [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4
[-1,-3,4,2] => [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4
[-1,-3,-4,-2] => [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4
[1,4,2,-3] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,4,-2,3] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,-4,2,3] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[1,-4,-2,-3] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-1,4,2,-3] => [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4
[-1,4,-2,3] => [3,1]
=> [[3,1],[]]
=> ([(0,2),(0,3),(3,1)],4)
=> ? = 4
[2,-1,4,-3] => [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,-1,-4,3] => [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[-2,1,4,-3] => [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[-2,1,-4,3] => [2,2]
=> [[2,2],[]]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 4
[2,3,-1,4] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[2,-3,1,4] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-2,3,1,4] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-2,-3,-1,4] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[2,3,4,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,3,-4,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,-3,4,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,-3,-4,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[-2,3,4,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[-2,3,-4,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[-2,-3,4,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[-2,-3,-4,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,1,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,-1,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,-4,1,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,-4,-1,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[-2,4,1,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[-2,4,-1,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[-2,-4,1,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[-2,-4,-1,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 4
[2,4,3,-1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[2,-4,3,1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-2,4,3,1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
[-2,-4,3,-1] => [3]
=> [[3],[]]
=> ([(0,2),(2,1)],3)
=> 3
Description
The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice.
Matching statistic: St000782
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00146: Dyck paths to tunnel matchingPerfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [(1,2),(3,8),(4,7),(5,6)]
=> ? = 3 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [(1,2),(3,10),(4,9),(5,8),(6,7)]
=> ? = 4 - 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,3,2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-3,-2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,4,-3,2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-1,-4,-3,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [(1,4),(2,3),(5,6)]
=> 1 = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [(1,2),(3,8),(4,5),(6,7)]
=> ? = 3 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 4 - 1
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 4 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 4 - 1
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(6,7)]
=> ? = 4 - 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 3 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 4 - 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: St001583
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00201: Dyck paths RingelPermutations
St001583: Permutations ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 3 + 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [3,1,4,5,6,2] => ? = 4 + 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-3,-2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,4,-3,2] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-1,-4,-3,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[2,1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> [2,4,1,3] => 3 = 2 + 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [5,1,4,2,3] => ? = 3 + 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 4 + 1
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 4 + 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 4 + 1
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 4 + 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 3 + 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 4 + 1
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001722
Mp00169: Signed permutations odd cycle typeInteger partitions
Mp00043: Integer partitions to Dyck pathDyck paths
Mp00093: Dyck paths to binary wordBinary words
St001722: Binary words ⟶ ℤResult quality: 17% values known / values provided: 26%distinct values known / distinct values provided: 17%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[1,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,-1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,1,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[3,2,-1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,2,1] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 3 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 4 - 1
[1,2,4,-3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,2,-4,3] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,3,-2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-3,2,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,3,2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-3,-2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[1,4,3,-2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[1,-4,3,2] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,4,-3,2] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-1,-4,-3,-2] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[2,1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[2,-1,3,4] => [2]
=> [1,1,0,0,1,0]
=> 110010 => 1 = 2 - 1
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 10110100 => ? = 3 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 4 - 1
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 4 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 4 - 1
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 4 - 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 3 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 4 - 1
Description
The number of minimal chains with small intervals between a binary word and the top element. A valley in a binary word is a subsequence 01, or a trailing 0. A peak is a subsequence 10 or a trailing 1. Let P be the lattice on binary words of length n, where the covering elements of a word are obtained by replacing a valley with a peak. An interval [w_1, w_2] in P is small if w_2 is obtained from w_1 by replacing some valleys with peaks. This statistic counts the number of chains w = w_1 < \dots < w_d = 1\dots 1 to the top element of minimal length. For example, there are two such chains for the word 0110: 0110 < 1011 < 1101 < 1110 < 1111 and 0110 < 1010 < 1101 < 1110 < 1111.