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Your data matches 64 different statistics following compositions of up to 3 maps.
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Matching statistic: St001200
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00122: Dyck paths —Elizalde-Deutsch bijection⟶ Dyck paths
St001200: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> 3
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[-1,-2,-4,-3] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,3,2,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[-1,-3,2,4] => [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 3
[-1,-3,-2,-4] => [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 2
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> [1,1,0,0,1,0]
=> 2
Description
The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St000782
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00146: Dyck paths —to tunnel matching⟶ Perfect matchings
St000782: Perfect matchings ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [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]
=> [1,0,1,1,0,0]
=> [(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)]
=> ? = 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]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(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)]
=> ? = 2 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 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]
=> [1,0,1,0,1,0]
=> [(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)]
=> ? = 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)]
=> ? = 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)]
=> ? = 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)]
=> ? = 2 - 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)]
=> ? = 3 - 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]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(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)]
=> ? = 2 - 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 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] => [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]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(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]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(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] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(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)]
=> ? = 3 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [(1,4),(2,3),(5,8),(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)]
=> ? = 3 - 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [(1,6),(2,5),(3,4),(7,8)]
=> ? = 2 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [(1,6),(2,3),(4,5),(7,8)]
=> ? = 2 - 1
[3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[3,-2,1,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[-3,-2,-1,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 1
[4,-2,-3,1] => [1,1]
=> [1,0,1,1,0,0]
=> [(1,2),(3,6),(4,5)]
=> 1 = 2 - 1
[-4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [(1,2),(3,4),(5,6)]
=> 1 = 2 - 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 type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00201: Dyck paths —Ringel⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [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]
=> [1,0,1,1,0,0]
=> [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] => ? = 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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] => ? = 2 + 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 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]
=> [1,0,1,0,1,0]
=> [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] => ? = 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] => ? = 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] => ? = 2 + 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [3,1,4,5,2] => ? = 2 + 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] => ? = 3 + 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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] => ? = 2 + 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 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] => [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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 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] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3 + 1
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3 + 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3 + 1
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,4,1,5,3] => ? = 3 + 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [2,3,5,1,4] => ? = 2 + 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [5,3,1,2,4] => ? = 2 + 1
[3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[3,-2,1,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[-3,-2,-1,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 1
[4,-2,-3,1] => [1,1]
=> [1,0,1,1,0,0]
=> [3,1,4,2] => 3 = 2 + 1
[-4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [4,1,2,3] => 3 = 2 + 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 type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001722: Binary words ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [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]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 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]
=> [1,0,1,0,1,0]
=> 101010 => 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 => ? = 2 - 1
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 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 => ? = 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 => ? = 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 => ? = 2 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 10111000 => ? = 2 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => ? = 3 - 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]
=> [1,0,1,0,1,0]
=> 101010 => 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 => ? = 2 - 1
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 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] => [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]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 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]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 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] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> 11001100 => ? = 3 - 1
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> 11100010 => ? = 2 - 1
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> 11010010 => ? = 2 - 1
[3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[3,-2,1,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[-3,-2,-1,-4] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 1
[4,-2,-3,1] => [1,1]
=> [1,0,1,1,0,0]
=> 101100 => 1 = 2 - 1
[-4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> 101010 => 1 = 2 - 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
$$ 0110 < 1010 < 1101 < 1110 < 1111. $$
Matching statistic: St001816
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00033: Dyck paths —to two-row standard tableau⟶ Standard tableaux
St001816: Standard tableaux ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,-2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,2,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,-2,3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,-2,-3] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 2 - 2
[-1,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,-1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-2,1,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[2,-3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[-2,3,1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[-2,-3,-1] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[3,1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[3,-1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[-3,1,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[-3,-1,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[3,-2,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-3,-2,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,2,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,-2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,-2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,-2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 2 - 2
[-1,2,3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,2,-3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,2,-3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 2 - 2
[-1,-2,3,4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,-2,3,-4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 2 - 2
[-1,-2,-3,4] => [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [[1,3,4,5],[2,6,7,8]]
=> ? = 2 - 2
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[1,3,4,5,6],[2,7,8,9,10]]
=> ? = 3 - 2
[1,-2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,4,-3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,2,-4,3] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-2,4,3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,-2,4,-3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 3 - 2
[-1,-2,-4,3] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 3 - 2
[-1,-2,-4,-3] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,3,-2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-3,2,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,3,2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,3,-2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,3,-2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 3 - 2
[-1,-3,2,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-3,2,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 3 - 2
[-1,-3,-2,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[1,3,4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[1,3,-4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[1,-3,4,2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[1,-3,-4,-2] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[-1,3,4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[-1,3,-4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[-1,-3,4,2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[-1,-3,-4,-2] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[1,4,2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[1,4,-2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[1,-4,2,3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[1,-4,-2,-3] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[-1,4,2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[-1,4,-2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[-1,-4,2,3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[-1,-4,-2,-3] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[1,4,-3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[1,-4,-3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,4,3,-2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,4,-3,2] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-1,4,-3,-2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 3 - 2
[-1,-4,3,2] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-1,-4,-3,2] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 3 - 2
[-1,-4,-3,-2] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[2,1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[2,-1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,-1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[2,-1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 3 - 2
[-2,1,3,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-2,1,-3,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-2,1,-3,-4] => [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [[1,3,4,6],[2,5,7,8]]
=> ? = 3 - 2
[-2,-1,-3,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[2,-1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 3 - 2
[2,-1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 3 - 2
[-2,1,4,-3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 3 - 2
[-2,1,-4,3] => [2,2]
=> [1,1,0,0,1,1,0,0]
=> [[1,2,5,6],[3,4,7,8]]
=> ? = 3 - 2
[2,3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[2,3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[2,-3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[2,-3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[-2,3,1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[-2,3,1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[-2,-3,-1,4] => [3]
=> [1,1,1,0,0,0,1,0]
=> [[1,2,3,7],[4,5,6,8]]
=> ? = 2 - 2
[-2,-3,-1,-4] => [3,1]
=> [1,1,0,1,0,0,1,0]
=> [[1,2,4,7],[3,5,6,8]]
=> ? = 2 - 2
[3,2,-1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[3,-2,1,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[3,-2,-1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-3,2,1,-4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-3,-2,1,4] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[-3,-2,-1,-4] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[4,2,-3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[4,-2,3,-1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
[4,-2,-3,1] => [1,1]
=> [1,0,1,1,0,0]
=> [[1,3,4],[2,5,6]]
=> 0 = 2 - 2
[-4,2,-3,1] => [2,1]
=> [1,0,1,0,1,0]
=> [[1,3,5],[2,4,6]]
=> 0 = 2 - 2
Description
Eigenvalues of the top-to-random operator acting on a simple module.
These eigenvalues are given in [1] and [3].
The simple module of the symmetric group indexed by a partition $\lambda$ has dimension equal to the number of standard tableaux of shape $\lambda$. Hence, the eigenvalues of any linear operator defined on this module can be indexed by standard tableaux of shape $\lambda$; this statistic gives all the eigenvalues of the operator acting on the module.
This statistic bears different names, such as the type in [2] or eig in [3].
Similarly, the eigenvalues of the random-to-random operator acting on a simple module is [[St000508]].
Matching statistic: St001491
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00093: Dyck paths —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 6% ●values known / values provided: 6%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,-3] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 2 - 1
[-1,3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[2,-1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-2,1,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[2,3,-1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[2,-3,1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-2,3,1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-2,-3,-1] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[3,1,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[3,-1,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-3,1,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-3,-1,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[3,-2,-1] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-3,-2,1] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 2 - 1
[-1,2,3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 2 - 1
[-1,-2,3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,3,-4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 2 - 1
[-1,-2,-3,4] => [1,1,1]
=> [1,1,0,1,0,0]
=> 110100 => ? = 2 - 1
[-1,-2,-3,-4] => [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 11010100 => ? = 3 - 1
[1,-2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,-2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,2,4,-3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,2,-4,3] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,-2,4,3] => [1,1]
=> [1,1,0,0]
=> 1100 => 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]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,3,-2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,-3,2,-4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,3,2,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,3,-2,4] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 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]
=> [1,0,1,1,0,0]
=> 101100 => ? = 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]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,3,4,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,3,-4,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,-3,4,2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,-3,-4,-2] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-1,3,4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 2 - 1
[-1,3,-4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 2 - 1
[-1,-3,4,2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 2 - 1
[-1,-3,-4,-2] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 2 - 1
[1,4,2,-3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,4,-2,3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,-4,2,3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[1,-4,-2,-3] => [3]
=> [1,0,1,0,1,0]
=> 101010 => ? = 2 - 1
[-1,4,2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 2 - 1
[-1,4,-2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 2 - 1
[-1,-4,2,3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 2 - 1
[-1,-4,-2,-3] => [3,1]
=> [1,0,1,0,1,1,0,0]
=> 10101100 => ? = 2 - 1
[1,4,-3,-2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[1,-4,-3,2] => [2,1]
=> [1,0,1,1,0,0]
=> 101100 => ? = 2 - 1
[-1,4,-3,2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-4,-3,-2] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[2,1,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-2,-1,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[3,-2,1,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-3,-2,-1,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[4,-2,-3,1] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-4,-2,-3,-1] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,2,3,-4,-5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,2,-3,4,-5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,2,-3,-4,5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,3,4,-5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,3,-4,5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-3,4,5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,3,4,-5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,3,-4,5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3,4,5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,3,4,5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-3,5,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-3,-5,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3,5,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-3,-5,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,3,5,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,3,-5,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,4,3,-5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[1,-2,-4,-3,-5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,4,3,-5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,2,-4,-3,-5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,4,3,5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,-4,-3,5] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,4,5,3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,4,-5,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,-4,5,-3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,-4,-5,3] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,5,3,4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 1
[-1,-2,5,-3,-4] => [1,1]
=> [1,1,0,0]
=> 1100 => 1 = 2 - 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: St000207
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000207: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2 - 1
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[2,3,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[2,-3,1] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,3,1] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[3,1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[3,-1,2] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,1,2] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 2 - 1
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,2,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,3,2,-4] => [2]
=> []
=> ?
=> ? = 2 - 1
[-1,3,-2,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,3,-2,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-3,2,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-3,2,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-3,-2,-4] => [2]
=> []
=> ?
=> ? = 2 - 1
[1,3,4,-2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,3,-4,2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,4,2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,-4,-2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,6,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,5,6,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,5,4,6,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,6,3,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,4,6,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,3,4,6,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,6,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,2,4,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,3,6,4,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,2,3,4,6,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,2,3,6,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,2,6,4,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,1,3,4,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,6,3,4,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-3,-2,1,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-3,1,2,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-4,2,3,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,-3,1,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,1,4,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,3,2,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,3,1,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,6,4,3,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,3,4,2,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,3,4,1,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,7,2,1,5,6,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,7,2,4,1,6,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,7,2,4,5,1,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,7,3,2,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,7,3,5,2,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[5,2,7,3,1,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,7,3,2,1,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,7,3,5,1,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,7,3,2,5,1,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has all vertices in integer lattice points.
Matching statistic: St000208
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000208: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2 - 1
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[2,3,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[2,-3,1] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,3,1] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[3,1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[3,-1,2] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,1,2] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 2 - 1
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,2,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,3,2,-4] => [2]
=> []
=> ?
=> ? = 2 - 1
[-1,3,-2,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,3,-2,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-3,2,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-3,2,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-3,-2,-4] => [2]
=> []
=> ?
=> ? = 2 - 1
[1,3,4,-2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,3,-4,2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,4,2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,-4,-2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,6,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,5,6,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,5,4,6,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,6,3,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,4,6,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,3,4,6,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,6,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,2,4,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,3,6,4,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,2,3,4,6,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,2,3,6,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,2,6,4,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,1,3,4,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,6,3,4,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-3,-2,1,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-3,1,2,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-4,2,3,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,-3,1,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,1,4,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,3,2,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,3,1,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,6,4,3,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,3,4,2,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,3,4,1,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,7,2,1,5,6,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,7,2,4,1,6,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,7,2,4,5,1,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,7,3,2,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,7,3,5,2,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[5,2,7,3,1,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,7,3,2,1,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,7,3,5,1,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,7,3,2,5,1,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has only integer lattice points as vertices.
See also [[St000205]], [[St000206]] and [[St000207]].
Matching statistic: St000618
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000618: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2 - 1
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[2,3,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[2,-3,1] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,3,1] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[3,1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[3,-1,2] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,1,2] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 2 - 1
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,2,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,3,2,-4] => [2]
=> []
=> ?
=> ? = 2 - 1
[-1,3,-2,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,3,-2,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-3,2,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-3,2,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-3,-2,-4] => [2]
=> []
=> ?
=> ? = 2 - 1
[1,3,4,-2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,3,-4,2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,4,2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,-4,-2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,6,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,5,6,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,5,4,6,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,6,3,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,4,6,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,3,4,6,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,6,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,2,4,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,3,6,4,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,2,3,4,6,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,2,3,6,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,2,6,4,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,1,3,4,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,6,3,4,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-3,-2,1,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-3,1,2,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-4,2,3,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,-3,1,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,1,4,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,3,2,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,3,1,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,6,4,3,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,3,4,2,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,3,4,1,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,7,2,1,5,6,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,7,2,4,1,6,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,7,2,4,5,1,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,7,3,2,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,7,3,5,2,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[5,2,7,3,1,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,7,3,2,1,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,7,3,5,1,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,7,3,2,5,1,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
The number of self-evacuating tableaux of given shape.
This is the same as the number of standard domino tableaux of the given shape.
Matching statistic: St000667
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 2% ●values known / values provided: 2%●distinct values known / distinct values provided: 33%
Values
[-1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[1,-2,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[-1,3,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[-1,-3,2] => []
=> ?
=> ?
=> ? = 2 - 1
[2,-1,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,1,-3] => []
=> ?
=> ?
=> ? = 2 - 1
[2,3,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[2,-3,1] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,3,1] => []
=> ?
=> ?
=> ? = 2 - 1
[-2,-3,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[3,1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[3,-1,2] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,1,2] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,-1,-2] => []
=> ?
=> ?
=> ? = 2 - 1
[3,-2,-1] => []
=> ?
=> ?
=> ? = 2 - 1
[-3,-2,1] => []
=> ?
=> ?
=> ? = 2 - 1
[1,2,-3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[1,-2,-3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,3,-4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,2,-3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,2,-3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,3,4] => [1,1]
=> [1]
=> []
=> ? = 2 - 1
[-1,-2,3,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,-3,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[1,-2,4,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-2,-4,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,4,-3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,2,-4,3] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,4,3] => [2]
=> []
=> ?
=> ? = 2 - 1
[-1,-2,4,-3] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-2,-4,3] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-2,-4,-3] => [2]
=> []
=> ?
=> ? = 2 - 1
[1,3,-2,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,2,-4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,3,2,-4] => [2]
=> []
=> ?
=> ? = 2 - 1
[-1,3,-2,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,3,-2,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-3,2,4] => [1]
=> []
=> ?
=> ? = 2 - 1
[-1,-3,2,-4] => []
=> ?
=> ?
=> ? = 3 - 1
[-1,-3,-2,-4] => [2]
=> []
=> ?
=> ? = 2 - 1
[1,3,4,-2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,3,-4,2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,4,2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,-3,-4,-2] => [1]
=> []
=> ?
=> ? = 2 - 1
[1,2,3,-4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,-3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,3,-4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,2,-3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-1,-2,3,4,5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,6,4,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,3,5,6,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,5,4,6,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,6,3,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,4,6,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,5,3,4,6,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,4,3,6,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,2,4,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,3,6,4,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,2,3,4,6,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[4,2,3,6,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3,2,6,4,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,1,3,4,5,-2] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,6,3,4,5,-1] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-3,-2,1,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[-3,1,2,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,-4,2,3,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[2,-3,1,4,5,6] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,1,4,5,-3] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,3,2,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,3,1,5,-4] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,2,6,4,3,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,3,4,2,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,3,4,1,-5] => [1,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,1,3,4,5,6,-2] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,7,2,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[4,7,2,1,5,6,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,7,2,4,1,6,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,7,2,4,5,1,-3] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,2,1,4,5,6,-3] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,7,3,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,5,7,3,2,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,6,7,3,5,2,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[1,7,3,2,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[5,2,7,3,1,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[5,7,3,2,1,6,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,2,7,3,5,1,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[6,7,3,2,5,1,-4] => [2,1,1]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[7,2,3,1,5,6,-4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
[1,2,3,7,4,6,-5] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1 = 2 - 1
Description
The greatest common divisor of the parts of the partition.
The following 54 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the partition. St001527The cyclic permutation representation number of an integer partition. St001571The Cartan determinant of the integer partition. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001602The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on endofunctions. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001627The number of coloured connected graphs such that the multiplicities of colours are given by a partition. St001763The Hurwitz number of an integer partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001175The size of a partition minus the hook length of the base cell. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001248Sum of the even parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001541The Gini index of an integer partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St001128The exponens consonantiae of a partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
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