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Your data matches 718 different statistics following compositions of up to 3 maps.
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Matching statistic: St001604
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,3,4,-1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,3,-4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,-3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,-3,-4,-1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-2,3,-4,-1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-2,-3,4,-1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-2,-3,-4,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,-3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,4,-1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,-4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,-4,-1,-3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-2,4,-1,-3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-2,-4,1,-3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-2,-4,-1,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,4,-2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,-4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,-1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,-1,-4,-2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-3,1,-4,-2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-3,-1,4,-2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-3,-1,-4,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,4,2,-1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,4,-2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,-4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,-4,-2,-1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-3,4,-2,-1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-3,-4,2,-1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-3,-4,-2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,2,-3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,-2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,-1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,-1,-2,-3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-4,1,-2,-3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-4,-1,2,-3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-4,-1,-2,3] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,1,-2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,-1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,-3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[4,-3,-1,-2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-4,3,-1,-2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-4,-3,1,-2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-4,-3,-1,2] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [2,2,1]
=> [2,1]
=> 0
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St001217
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001217: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[2,3,4,-1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,3,-4,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,-3,4,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,-3,-4,-1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-2,3,4,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-2,3,-4,-1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-2,-3,4,-1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-2,-3,-4,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,4,1,-3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,4,-1,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,-4,1,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,-4,-1,-3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-2,4,1,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-2,4,-1,-3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-2,-4,1,-3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-2,-4,-1,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,1,4,-2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,1,-4,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,-1,4,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,-1,-4,-2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-3,1,4,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-3,1,-4,-2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-3,-1,4,-2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-3,-1,-4,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,4,2,-1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,4,-2,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,-4,2,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[3,-4,-2,-1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-3,4,2,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-3,4,-2,-1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-3,-4,2,-1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-3,-4,-2,1] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,1,2,-3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,1,-2,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,-1,2,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,-1,-2,-3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-4,1,2,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-4,1,-2,-3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-4,-1,2,-3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-4,-1,-2,3] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,3,1,-2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,3,-1,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,-3,1,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[4,-3,-1,-2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-4,3,1,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-4,3,-1,-2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-4,-3,1,-2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-4,-3,-1,2] => [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 0
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[-8,-6,1,-5,-4,2,3,7] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-7,4,-6,5,-8,-3,1,2] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,-6,-5,1,3,4] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-3,-6,1,2,4,5] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[6,-5,-4,1,2,3] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-6,-4,1,2,3,5] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-6,1,-5,2,3,4] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-2,1,-6,3,4,5] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-5,4,2,-6,1,3] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-5,4,-6,3,1,2] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-5,-3,2,-6,-4,1] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-2,-3,-6,5,-4,1] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,-3,2,5,-6,1] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,-4,5,6,-3,1] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-2,4,5,6,-3,1] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,-6,-5,3,1,2] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-4,-5,-6,-2,1,3] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[2,-4,-6,5,1,3] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[6,-4,-5,2,1,3] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-5,4,-6,1,2,3] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,1,-6,-5,2,3] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[1,7,-6,-5,2,3,4] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-7,2,-5,1,3,4,6] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[-4,2,-7,1,3,5,6] => [4,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 1
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 0
[3,1,5,2,8,4,6,-7] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,5,1,2,8,3,6,-7] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[7,4,1,8,2,3,5,-6] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,1,5,2,7,4,6,-3] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,5,1,2,7,3,6,-4] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,4,1,6,2,3,5,-7] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,6,7,1,2,3,4,-5] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,6,1,7,2,3,5,-4] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,1,6,7,2,4,5,-3] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,6,8,1,2,3,4,-7] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[6,7,8,1,2,3,4,-5] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[6,1,7,8,2,4,5,-3] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,1,7,2,8,4,6,-3] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,7,1,2,8,3,6,-4] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,6,1,8,2,3,5,-7] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[3,1,6,8,2,4,5,-7] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,1,2,7,8,5,6,-3] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[3,1,8,2,7,4,6,-5] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[3,1,8,7,2,4,5,-6] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,8,1,7,2,3,5,-6] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[4,8,1,2,7,3,6,-5] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[5,8,7,1,2,3,4,-6] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[6,8,5,1,2,3,4,-7] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[7,8,1,6,2,3,5,-4] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
[8,7,5,1,2,3,4,-6] => [8]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 0
Description
The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1.
Matching statistic: St001517
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001517: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001517: Permutations ⟶ ℤResult quality: 86% ●values known / values provided: 86%●distinct values known / distinct values provided: 100%
Values
[2,3,4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[2,3,-4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[2,-3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[2,-3,-4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-2,3,-4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-2,-3,4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-2,-3,-4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[2,4,1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[2,4,-1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[2,-4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[2,-4,-1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-2,4,-1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-2,-4,1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-2,-4,-1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[3,1,4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[3,1,-4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[3,-1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[3,-1,-4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-3,1,-4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-3,-1,4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-3,-1,-4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[3,4,2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[3,4,-2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[3,-4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[3,-4,-2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-3,4,-2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-3,-4,2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-3,-4,-2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[4,1,2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[4,1,-2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[4,-1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[4,-1,-2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-4,1,-2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-4,-1,2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-4,-1,-2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[4,3,1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[4,3,-1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[4,-3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[4,-3,-1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-4,3,-1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-4,-3,1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-4,-3,-1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 2 = 0 + 2
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 2 = 0 + 2
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 2 = 0 + 2
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 2
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 2
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1 + 2
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1 + 2
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1 + 2
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 2
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 2
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 2
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 2
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 2
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 1 + 2
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 2
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 2
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 2
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 2
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 2
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 2
[-5,1,7,-6,2,3,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 0 + 2
[-7,4,-3,1,2,5,6] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 2
[-6,5,7,-3,1,2,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 0 + 2
[4,1,-7,-6,-5,2,3] => [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [4,2,5,1,3,6,7] => ? = 0 + 2
[3,7,-5,-2,1,4,6] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 0 + 2
[3,-7,-6,-5,-2,1,4] => [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [4,5,6,1,2,3,7] => ? = 0 + 2
[3,1,5,2,8,4,6,-7] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[4,5,1,2,8,3,6,-7] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[7,4,1,8,2,3,5,-6] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[8,1,5,2,7,4,6,-3] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[8,5,1,2,7,3,6,-4] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[8,4,1,6,2,3,5,-7] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[8,6,7,1,2,3,4,-5] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[8,6,1,7,2,3,5,-4] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[8,1,6,7,2,4,5,-3] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[5,6,8,1,2,3,4,-7] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[6,7,8,1,2,3,4,-5] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[6,1,7,8,2,4,5,-3] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[5,1,7,2,8,4,6,-3] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[5,7,1,2,8,3,6,-4] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[4,6,1,8,2,3,5,-7] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[3,1,6,8,2,4,5,-7] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[4,1,2,7,8,5,6,-3] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[3,1,8,2,7,4,6,-5] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[3,1,8,7,2,4,5,-6] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[4,8,1,7,2,3,5,-6] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[4,8,1,2,7,3,6,-5] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[5,8,7,1,2,3,4,-6] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[6,8,5,1,2,3,4,-7] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[7,8,1,6,2,3,5,-4] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[8,7,5,1,2,3,4,-6] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[4,5,2,3,8,1,6,-7] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
[7,5,1,3,8,2,6,-4] => [8]
=> [[1,2,3,4,5,6,7,8]]
=> [1,2,3,4,5,6,7,8] => ? = 0 + 2
Description
The length of a longest pair of twins in a permutation.
A pair of twins in a permutation is a pair of two disjoint subsequences which are order isomorphic.
Matching statistic: St001435
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[2,3,4,-1] => [4]
=> [[4],[]]
=> 0
[2,3,-4,1] => [4]
=> [[4],[]]
=> 0
[2,-3,4,1] => [4]
=> [[4],[]]
=> 0
[2,-3,-4,-1] => [4]
=> [[4],[]]
=> 0
[-2,3,4,1] => [4]
=> [[4],[]]
=> 0
[-2,3,-4,-1] => [4]
=> [[4],[]]
=> 0
[-2,-3,4,-1] => [4]
=> [[4],[]]
=> 0
[-2,-3,-4,1] => [4]
=> [[4],[]]
=> 0
[2,4,1,-3] => [4]
=> [[4],[]]
=> 0
[2,4,-1,3] => [4]
=> [[4],[]]
=> 0
[2,-4,1,3] => [4]
=> [[4],[]]
=> 0
[2,-4,-1,-3] => [4]
=> [[4],[]]
=> 0
[-2,4,1,3] => [4]
=> [[4],[]]
=> 0
[-2,4,-1,-3] => [4]
=> [[4],[]]
=> 0
[-2,-4,1,-3] => [4]
=> [[4],[]]
=> 0
[-2,-4,-1,3] => [4]
=> [[4],[]]
=> 0
[3,1,4,-2] => [4]
=> [[4],[]]
=> 0
[3,1,-4,2] => [4]
=> [[4],[]]
=> 0
[3,-1,4,2] => [4]
=> [[4],[]]
=> 0
[3,-1,-4,-2] => [4]
=> [[4],[]]
=> 0
[-3,1,4,2] => [4]
=> [[4],[]]
=> 0
[-3,1,-4,-2] => [4]
=> [[4],[]]
=> 0
[-3,-1,4,-2] => [4]
=> [[4],[]]
=> 0
[-3,-1,-4,2] => [4]
=> [[4],[]]
=> 0
[3,4,2,-1] => [4]
=> [[4],[]]
=> 0
[3,4,-2,1] => [4]
=> [[4],[]]
=> 0
[3,-4,2,1] => [4]
=> [[4],[]]
=> 0
[3,-4,-2,-1] => [4]
=> [[4],[]]
=> 0
[-3,4,2,1] => [4]
=> [[4],[]]
=> 0
[-3,4,-2,-1] => [4]
=> [[4],[]]
=> 0
[-3,-4,2,-1] => [4]
=> [[4],[]]
=> 0
[-3,-4,-2,1] => [4]
=> [[4],[]]
=> 0
[4,1,2,-3] => [4]
=> [[4],[]]
=> 0
[4,1,-2,3] => [4]
=> [[4],[]]
=> 0
[4,-1,2,3] => [4]
=> [[4],[]]
=> 0
[4,-1,-2,-3] => [4]
=> [[4],[]]
=> 0
[-4,1,2,3] => [4]
=> [[4],[]]
=> 0
[-4,1,-2,-3] => [4]
=> [[4],[]]
=> 0
[-4,-1,2,-3] => [4]
=> [[4],[]]
=> 0
[-4,-1,-2,3] => [4]
=> [[4],[]]
=> 0
[4,3,1,-2] => [4]
=> [[4],[]]
=> 0
[4,3,-1,2] => [4]
=> [[4],[]]
=> 0
[4,-3,1,2] => [4]
=> [[4],[]]
=> 0
[4,-3,-1,-2] => [4]
=> [[4],[]]
=> 0
[-4,3,1,2] => [4]
=> [[4],[]]
=> 0
[-4,3,-1,-2] => [4]
=> [[4],[]]
=> 0
[-4,-3,1,-2] => [4]
=> [[4],[]]
=> 0
[-4,-3,-1,2] => [4]
=> [[4],[]]
=> 0
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[-6,-4,-2,1,3,5] => [6]
=> [[6],[]]
=> ? = 1
[4,5,2,3,-6,1] => [6]
=> [[6],[]]
=> ? = 1
[2,-6,-4,-5,1,3] => [6]
=> [[6],[]]
=> ? = 1
[-6,-5,1,-4,2,3] => [3,2,1]
=> [[3,2,1],[]]
=> ? = 0
[-5,3,6,1,2,4] => [6]
=> [[6],[]]
=> ? = 1
[4,-5,-6,-3,1,2] => [6]
=> [[6],[]]
=> ? = 1
[-4,3,-5,2,-6,1] => [6]
=> [[6],[]]
=> ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[6,1],[]]
=> ? = 0
[-8,-6,1,-5,-4,2,3,7] => [4,2]
=> [[4,2],[]]
=> ? = 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[5,3],[]]
=> ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[5,3],[]]
=> ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[5,3],[]]
=> ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-7,4,-6,5,-8,-3,1,2] => [4,2]
=> [[4,2],[]]
=> ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[6,1],[]]
=> ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[6,1],[]]
=> ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[6,1],[]]
=> ? = 0
[6,1,2,3,4,-5] => [6]
=> [[6],[]]
=> ? = 1
[5,1,2,3,6,-4] => [6]
=> [[6],[]]
=> ? = 1
[5,1,2,6,4,-3] => [6]
=> [[6],[]]
=> ? = 1
[4,1,2,5,6,-3] => [6]
=> [[6],[]]
=> ? = 1
[5,1,6,3,4,-2] => [6]
=> [[6],[]]
=> ? = 1
[4,1,5,3,6,-2] => [6]
=> [[6],[]]
=> ? = 1
[5,1,6,2,3,-4] => [6]
=> [[6],[]]
=> ? = 1
[3,1,5,6,4,-2] => [6]
=> [[6],[]]
=> ? = 1
[4,1,5,6,2,-3] => [6]
=> [[6],[]]
=> ? = 1
[3,1,4,5,6,-2] => [6]
=> [[6],[]]
=> ? = 1
[5,6,2,3,4,-1] => [6]
=> [[6],[]]
=> ? = 1
[4,5,2,3,6,-1] => [6]
=> [[6],[]]
=> ? = 1
[3,5,2,6,4,-1] => [6]
=> [[6],[]]
=> ? = 1
[3,4,2,5,6,-1] => [6]
=> [[6],[]]
=> ? = 1
[5,6,1,2,4,-3] => [6]
=> [[6],[]]
=> ? = 1
[4,5,1,2,6,-3] => [6]
=> [[6],[]]
=> ? = 1
[4,5,1,6,3,-2] => [6]
=> [[6],[]]
=> ? = 1
[2,5,6,3,4,-1] => [6]
=> [[6],[]]
=> ? = 1
[2,4,5,3,6,-1] => [6]
=> [[6],[]]
=> ? = 1
[4,5,6,2,3,-1] => [6]
=> [[6],[]]
=> ? = 1
[3,5,6,1,4,-2] => [6]
=> [[6],[]]
=> ? = 1
[3,4,5,1,6,-2] => [6]
=> [[6],[]]
=> ? = 1
[2,3,5,6,4,-1] => [6]
=> [[6],[]]
=> ? = 1
[3,4,5,6,2,-1] => [6]
=> [[6],[]]
=> ? = 1
Description
The number of missing boxes in the first row.
Matching statistic: St001438
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[2,3,4,-1] => [4]
=> [[4],[]]
=> 0
[2,3,-4,1] => [4]
=> [[4],[]]
=> 0
[2,-3,4,1] => [4]
=> [[4],[]]
=> 0
[2,-3,-4,-1] => [4]
=> [[4],[]]
=> 0
[-2,3,4,1] => [4]
=> [[4],[]]
=> 0
[-2,3,-4,-1] => [4]
=> [[4],[]]
=> 0
[-2,-3,4,-1] => [4]
=> [[4],[]]
=> 0
[-2,-3,-4,1] => [4]
=> [[4],[]]
=> 0
[2,4,1,-3] => [4]
=> [[4],[]]
=> 0
[2,4,-1,3] => [4]
=> [[4],[]]
=> 0
[2,-4,1,3] => [4]
=> [[4],[]]
=> 0
[2,-4,-1,-3] => [4]
=> [[4],[]]
=> 0
[-2,4,1,3] => [4]
=> [[4],[]]
=> 0
[-2,4,-1,-3] => [4]
=> [[4],[]]
=> 0
[-2,-4,1,-3] => [4]
=> [[4],[]]
=> 0
[-2,-4,-1,3] => [4]
=> [[4],[]]
=> 0
[3,1,4,-2] => [4]
=> [[4],[]]
=> 0
[3,1,-4,2] => [4]
=> [[4],[]]
=> 0
[3,-1,4,2] => [4]
=> [[4],[]]
=> 0
[3,-1,-4,-2] => [4]
=> [[4],[]]
=> 0
[-3,1,4,2] => [4]
=> [[4],[]]
=> 0
[-3,1,-4,-2] => [4]
=> [[4],[]]
=> 0
[-3,-1,4,-2] => [4]
=> [[4],[]]
=> 0
[-3,-1,-4,2] => [4]
=> [[4],[]]
=> 0
[3,4,2,-1] => [4]
=> [[4],[]]
=> 0
[3,4,-2,1] => [4]
=> [[4],[]]
=> 0
[3,-4,2,1] => [4]
=> [[4],[]]
=> 0
[3,-4,-2,-1] => [4]
=> [[4],[]]
=> 0
[-3,4,2,1] => [4]
=> [[4],[]]
=> 0
[-3,4,-2,-1] => [4]
=> [[4],[]]
=> 0
[-3,-4,2,-1] => [4]
=> [[4],[]]
=> 0
[-3,-4,-2,1] => [4]
=> [[4],[]]
=> 0
[4,1,2,-3] => [4]
=> [[4],[]]
=> 0
[4,1,-2,3] => [4]
=> [[4],[]]
=> 0
[4,-1,2,3] => [4]
=> [[4],[]]
=> 0
[4,-1,-2,-3] => [4]
=> [[4],[]]
=> 0
[-4,1,2,3] => [4]
=> [[4],[]]
=> 0
[-4,1,-2,-3] => [4]
=> [[4],[]]
=> 0
[-4,-1,2,-3] => [4]
=> [[4],[]]
=> 0
[-4,-1,-2,3] => [4]
=> [[4],[]]
=> 0
[4,3,1,-2] => [4]
=> [[4],[]]
=> 0
[4,3,-1,2] => [4]
=> [[4],[]]
=> 0
[4,-3,1,2] => [4]
=> [[4],[]]
=> 0
[4,-3,-1,-2] => [4]
=> [[4],[]]
=> 0
[-4,3,1,2] => [4]
=> [[4],[]]
=> 0
[-4,3,-1,-2] => [4]
=> [[4],[]]
=> 0
[-4,-3,1,-2] => [4]
=> [[4],[]]
=> 0
[-4,-3,-1,2] => [4]
=> [[4],[]]
=> 0
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[-6,-4,-2,1,3,5] => [6]
=> [[6],[]]
=> ? = 1
[4,5,2,3,-6,1] => [6]
=> [[6],[]]
=> ? = 1
[2,-6,-4,-5,1,3] => [6]
=> [[6],[]]
=> ? = 1
[-6,-5,1,-4,2,3] => [3,2,1]
=> [[3,2,1],[]]
=> ? = 0
[-5,3,6,1,2,4] => [6]
=> [[6],[]]
=> ? = 1
[4,-5,-6,-3,1,2] => [6]
=> [[6],[]]
=> ? = 1
[-4,3,-5,2,-6,1] => [6]
=> [[6],[]]
=> ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[6,1],[]]
=> ? = 0
[-8,-6,1,-5,-4,2,3,7] => [4,2]
=> [[4,2],[]]
=> ? = 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[5,3],[]]
=> ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[5,3],[]]
=> ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[5,3],[]]
=> ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-7,4,-6,5,-8,-3,1,2] => [4,2]
=> [[4,2],[]]
=> ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[6,1],[]]
=> ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[6,1],[]]
=> ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[6,1],[]]
=> ? = 0
[6,1,2,3,4,-5] => [6]
=> [[6],[]]
=> ? = 1
[5,1,2,3,6,-4] => [6]
=> [[6],[]]
=> ? = 1
[5,1,2,6,4,-3] => [6]
=> [[6],[]]
=> ? = 1
[4,1,2,5,6,-3] => [6]
=> [[6],[]]
=> ? = 1
[5,1,6,3,4,-2] => [6]
=> [[6],[]]
=> ? = 1
[4,1,5,3,6,-2] => [6]
=> [[6],[]]
=> ? = 1
[5,1,6,2,3,-4] => [6]
=> [[6],[]]
=> ? = 1
[3,1,5,6,4,-2] => [6]
=> [[6],[]]
=> ? = 1
[4,1,5,6,2,-3] => [6]
=> [[6],[]]
=> ? = 1
[3,1,4,5,6,-2] => [6]
=> [[6],[]]
=> ? = 1
[5,6,2,3,4,-1] => [6]
=> [[6],[]]
=> ? = 1
[4,5,2,3,6,-1] => [6]
=> [[6],[]]
=> ? = 1
[3,5,2,6,4,-1] => [6]
=> [[6],[]]
=> ? = 1
[3,4,2,5,6,-1] => [6]
=> [[6],[]]
=> ? = 1
[5,6,1,2,4,-3] => [6]
=> [[6],[]]
=> ? = 1
[4,5,1,2,6,-3] => [6]
=> [[6],[]]
=> ? = 1
[4,5,1,6,3,-2] => [6]
=> [[6],[]]
=> ? = 1
[2,5,6,3,4,-1] => [6]
=> [[6],[]]
=> ? = 1
[2,4,5,3,6,-1] => [6]
=> [[6],[]]
=> ? = 1
[4,5,6,2,3,-1] => [6]
=> [[6],[]]
=> ? = 1
[3,5,6,1,4,-2] => [6]
=> [[6],[]]
=> ? = 1
[3,4,5,1,6,-2] => [6]
=> [[6],[]]
=> ? = 1
[2,3,5,6,4,-1] => [6]
=> [[6],[]]
=> ? = 1
[3,4,5,6,2,-1] => [6]
=> [[6],[]]
=> ? = 1
Description
The number of missing boxes of a skew partition.
Matching statistic: St001487
(load all 21 compositions to match this statistic)
(load all 21 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[2,3,4,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,3,-4,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,-3,4,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,-3,-4,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,3,4,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,3,-4,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,-3,4,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,-3,-4,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,4,1,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,4,-1,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,-4,1,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,-4,-1,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,4,1,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,4,-1,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,-4,1,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,-4,-1,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,1,4,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,1,-4,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,-1,4,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,-1,-4,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,1,4,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,1,-4,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,-1,4,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,-1,-4,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,4,2,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,4,-2,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,-4,2,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,-4,-2,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,4,2,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,4,-2,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,-4,2,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,-4,-2,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,1,2,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,1,-2,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,-1,2,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,-1,-2,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,1,2,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,1,-2,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,-1,2,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,-1,-2,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,3,1,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,3,-1,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,-3,1,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,-3,-1,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,3,1,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,3,-1,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,-3,1,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,-3,-1,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[-6,-4,-2,1,3,5] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,2,3,-6,1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[2,-6,-4,-5,1,3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[-6,-5,1,-4,2,3] => [3,2,1]
=> [[3,2,1],[]]
=> ? = 0 + 1
[-5,3,6,1,2,4] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,-5,-6,-3,1,2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[-4,3,-5,2,-6,1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[6,1],[]]
=> ? = 0 + 1
[-8,-6,1,-5,-4,2,3,7] => [4,2]
=> [[4,2],[]]
=> ? = 1 + 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[5,3],[]]
=> ? = 1 + 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[5,3],[]]
=> ? = 1 + 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[5,3],[]]
=> ? = 1 + 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-7,4,-6,5,-8,-3,1,2] => [4,2]
=> [[4,2],[]]
=> ? = 1 + 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1 + 1
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 0 + 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[6,1],[]]
=> ? = 0 + 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[6,1],[]]
=> ? = 0 + 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[6,1],[]]
=> ? = 0 + 1
[6,1,2,3,4,-5] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,1,2,3,6,-4] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,1,2,6,4,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,1,2,5,6,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,1,6,3,4,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,1,5,3,6,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,1,6,2,3,-4] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,1,5,6,4,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,1,5,6,2,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,1,4,5,6,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,6,2,3,4,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,2,3,6,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,5,2,6,4,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,4,2,5,6,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,6,1,2,4,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,1,2,6,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,1,6,3,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[2,5,6,3,4,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[2,4,5,3,6,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,6,2,3,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,5,6,1,4,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,4,5,1,6,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[2,3,5,6,4,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,4,5,6,2,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
(load all 28 compositions to match this statistic)
(load all 28 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[2,3,4,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,3,-4,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,-3,4,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,-3,-4,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,3,4,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,3,-4,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,-3,4,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,-3,-4,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,4,1,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,4,-1,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,-4,1,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[2,-4,-1,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,4,1,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,4,-1,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,-4,1,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-2,-4,-1,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,1,4,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,1,-4,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,-1,4,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,-1,-4,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,1,4,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,1,-4,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,-1,4,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,-1,-4,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,4,2,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,4,-2,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,-4,2,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[3,-4,-2,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,4,2,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,4,-2,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,-4,2,-1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-3,-4,-2,1] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,1,2,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,1,-2,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,-1,2,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,-1,-2,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,1,2,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,1,-2,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,-1,2,-3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,-1,-2,3] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,3,1,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,3,-1,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,-3,1,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[4,-3,-1,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,3,1,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,3,-1,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,-3,1,-2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-4,-3,-1,2] => [4]
=> [[4],[]]
=> 1 = 0 + 1
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[-6,-4,-2,1,3,5] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,2,3,-6,1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[2,-6,-4,-5,1,3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[-6,-5,1,-4,2,3] => [3,2,1]
=> [[3,2,1],[]]
=> ? = 0 + 1
[-5,3,6,1,2,4] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,-5,-6,-3,1,2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[-4,3,-5,2,-6,1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[6,1],[]]
=> ? = 0 + 1
[-8,-6,1,-5,-4,2,3,7] => [4,2]
=> [[4,2],[]]
=> ? = 1 + 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[5,3],[]]
=> ? = 1 + 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[5,3],[]]
=> ? = 1 + 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[5,3],[]]
=> ? = 1 + 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-7,4,-6,5,-8,-3,1,2] => [4,2]
=> [[4,2],[]]
=> ? = 1 + 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 1 + 1
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[3,2,1,1],[]]
=> ? = 0 + 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[6,1],[]]
=> ? = 0 + 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ? = 0 + 1
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[6,1],[]]
=> ? = 0 + 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[6,1],[]]
=> ? = 0 + 1
[6,1,2,3,4,-5] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,1,2,3,6,-4] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,1,2,6,4,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,1,2,5,6,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,1,6,3,4,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,1,5,3,6,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,1,6,2,3,-4] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,1,5,6,4,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,1,5,6,2,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,1,4,5,6,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,6,2,3,4,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,2,3,6,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,5,2,6,4,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,4,2,5,6,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[5,6,1,2,4,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,1,2,6,-3] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,1,6,3,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[2,5,6,3,4,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[2,4,5,3,6,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[4,5,6,2,3,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,5,6,1,4,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,4,5,1,6,-2] => [6]
=> [[6],[]]
=> ? = 1 + 1
[2,3,5,6,4,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
[3,4,5,6,2,-1] => [6]
=> [[6],[]]
=> ? = 1 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St001811
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001811: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[2,3,4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,3,-4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,-3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,-3,-4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-2,3,-4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-2,-3,4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-2,-3,-4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,4,1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,4,-1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,-4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[2,-4,-1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-2,4,-1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-2,-4,1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-2,-4,-1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,1,4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,1,-4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,-1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,-1,-4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-3,1,-4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-3,-1,4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-3,-1,-4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,4,2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,4,-2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,-4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[3,-4,-2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-3,4,-2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-3,-4,2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-3,-4,-2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,1,2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,1,-2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,-1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,-1,-2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-4,1,-2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-4,-1,2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-4,-1,-2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,-1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,-3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,-3,-1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-4,3,-1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-4,-3,1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-4,-3,-1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 0
[-6,-4,-2,1,3,5] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,5,2,3,-6,1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[2,-6,-4,-5,1,3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[-6,-5,1,-4,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ? = 0
[-5,3,6,1,2,4] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,-5,-6,-3,1,2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[-4,3,-5,2,-6,1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[-8,-6,1,-5,-4,2,3,7] => [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ? = 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-7,4,-6,5,-8,-3,1,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ? = 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 1
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0
[6,1,2,3,4,-5] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[5,1,2,3,6,-4] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[5,1,2,6,4,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,1,2,5,6,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[5,1,6,3,4,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,1,5,3,6,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[5,1,6,2,3,-4] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[3,1,5,6,4,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,1,5,6,2,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[3,1,4,5,6,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[5,6,2,3,4,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,5,2,3,6,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[3,5,2,6,4,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[3,4,2,5,6,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[5,6,1,2,4,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,5,1,2,6,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,5,1,6,3,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[2,5,6,3,4,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[2,4,5,3,6,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[4,5,6,2,3,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[3,5,6,1,4,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[3,4,5,1,6,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[2,3,5,6,4,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
[3,4,5,6,2,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1
Description
The Castelnuovo-Mumford regularity of a permutation.
The ''Castelnuovo-Mumford regularity'' of a permutation σ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' Xσ.
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for σ. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St000181
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
Mp00185: Skew partitions —cell poset⟶ Posets
St000181: Posets ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[2,3,4,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,3,-4,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,-3,4,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,-3,-4,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-2,3,4,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-2,3,-4,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-2,-3,4,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-2,-3,-4,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,1,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,4,-1,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,-4,1,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[2,-4,-1,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-2,4,1,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-2,4,-1,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-2,-4,1,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-2,-4,-1,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,4,-2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,1,-4,2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,-1,4,2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,-1,-4,-2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-3,1,4,2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-3,1,-4,-2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-3,-1,4,-2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-3,-1,-4,2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,2,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,4,-2,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,-4,2,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[3,-4,-2,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-3,4,2,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-3,4,-2,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-3,-4,2,-1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-3,-4,-2,1] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,2,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,1,-2,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,-1,2,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,-1,-2,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-4,1,2,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-4,1,-2,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-4,-1,2,-3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-4,-1,-2,3] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,1,-2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,3,-1,2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,-3,1,2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[4,-3,-1,-2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-4,3,1,2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-4,3,-1,-2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-4,-3,1,-2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-4,-3,-1,2] => [4]
=> [[4],[]]
=> ([(0,3),(2,1),(3,2)],4)
=> 1 = 0 + 1
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1 = 0 + 1
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[2,1,1,1],[]]
=> ([(0,2),(0,4),(3,1),(4,3)],5)
=> 1 = 0 + 1
[-6,-4,-2,1,3,5] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,5,2,3,-6,1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[2,-6,-4,-5,1,3] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[-6,-5,1,-4,2,3] => [3,2,1]
=> [[3,2,1],[]]
=> ([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> ? = 0 + 1
[-5,3,6,1,2,4] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,-5,-6,-3,1,2] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[-4,3,-5,2,-6,1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 0 + 1
[-8,-6,1,-5,-4,2,3,7] => [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1 + 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? = 0 + 1
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ? = 1 + 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ? = 1 + 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[5,3],[]]
=> ([(0,2),(0,5),(2,6),(3,4),(3,7),(4,1),(5,3),(5,6),(6,7)],8)
=> ? = 1 + 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? = 0 + 1
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? = 0 + 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? = 0 + 1
[-7,4,-6,5,-8,-3,1,2] => [4,2]
=> [[4,2],[]]
=> ([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> ? = 1 + 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? = 0 + 1
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? = 0 + 1
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[2,2,2,2],[]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ? = 1 + 1
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[3,2,1,1],[]]
=> ([(0,4),(0,5),(3,2),(4,3),(4,6),(5,1),(5,6)],7)
=> ? = 0 + 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? = 0 + 1
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 0 + 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[6,2],[]]
=> ([(0,2),(0,6),(2,7),(3,5),(4,3),(5,1),(6,4),(6,7)],8)
=> ? = 0 + 1
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 0 + 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[6,1],[]]
=> ([(0,2),(0,6),(3,5),(4,3),(5,1),(6,4)],7)
=> ? = 0 + 1
[6,1,2,3,4,-5] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[5,1,2,3,6,-4] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[5,1,2,6,4,-3] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,1,2,5,6,-3] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[5,1,6,3,4,-2] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,1,5,3,6,-2] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[5,1,6,2,3,-4] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[3,1,5,6,4,-2] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,1,5,6,2,-3] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[3,1,4,5,6,-2] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[5,6,2,3,4,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,5,2,3,6,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[3,5,2,6,4,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[3,4,2,5,6,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[5,6,1,2,4,-3] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,5,1,2,6,-3] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,5,1,6,3,-2] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[2,5,6,3,4,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[2,4,5,3,6,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[4,5,6,2,3,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[3,5,6,1,4,-2] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[3,4,5,1,6,-2] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[2,3,5,6,4,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
[3,4,5,6,2,-1] => [6]
=> [[6],[]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],6)
=> ? = 1 + 1
Description
The number of connected components of the Hasse diagram for the poset.
Matching statistic: St001208
(load all 251 compositions to match this statistic)
(load all 251 compositions to match this statistic)
Mp00169: Signed permutations —odd cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001208: Permutations ⟶ ℤResult quality: 50% ●values known / values provided: 62%●distinct values known / distinct values provided: 50%
Values
[2,3,4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[2,3,-4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[2,-3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[2,-3,-4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-2,3,4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-2,3,-4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-2,-3,4,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-2,-3,-4,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[2,4,1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[2,4,-1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[2,-4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[2,-4,-1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-2,4,1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-2,4,-1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-2,-4,1,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-2,-4,-1,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,1,4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,1,-4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,-1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,-1,-4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-3,1,4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-3,1,-4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-3,-1,4,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-3,-1,-4,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,4,2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,4,-2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,-4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[3,-4,-2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-3,4,2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-3,4,-2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-3,-4,2,-1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-3,-4,-2,1] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,1,2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,1,-2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,-1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,-1,-2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-4,1,2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-4,1,-2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-4,-1,2,-3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-4,-1,-2,3] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,3,1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,3,-1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,-3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[4,-3,-1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-4,3,1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-4,3,-1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-4,-3,1,-2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-4,-3,-1,2] => [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 1 = 0 + 1
[-1,-2,-3,-4,-5] => [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1 = 0 + 1
[-1,-2,-3,5,-4] => [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => 1 = 0 + 1
[-6,-4,-2,1,3,5] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,5,2,3,-6,1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[2,-6,-4,-5,1,3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[-6,-5,1,-4,2,3] => [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ? = 0 + 1
[-5,3,6,1,2,4] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,-5,-6,-3,1,2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[-4,3,-5,2,-6,1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[3,2,8,-6,-5,1,4,7] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 1
[-8,-6,1,-5,-4,2,3,7] => [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ? = 1 + 1
[2,8,-6,-5,1,3,4,7] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 1
[-5,4,-8,-6,-2,1,3,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1 + 1
[4,-5,-8,-6,-3,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1 + 1
[3,-8,-5,-6,-4,1,2,7] => [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [4,5,6,1,2,3,7,8] => ? = 1 + 1
[5,8,-4,3,-7,1,2,6] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 1
[-7,-5,4,6,-8,-2,1,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 1
[-7,-6,5,3,-8,-4,1,2] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 1
[-7,4,-6,5,-8,-3,1,2] => [4,2]
=> [[1,2,5,6],[3,4]]
=> [3,4,1,2,5,6] => ? = 1 + 1
[3,-7,-5,-6,2,4,-8,1] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 1
[-8,-4,-6,2,3,-7,1,5] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 1
[-7,-3,2,6,-8,-4,1,5] => [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? = 1 + 1
[-7,-8,-3,2,5,-6,1,4] => [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [5,3,2,6,1,4,7] => ? = 0 + 1
[-8,6,7,5,-4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 1
[-8,2,5,7,4,-6,1,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 1
[-8,-7,5,6,4,1,2,3] => [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [3,4,1,2,5,6,7,8] => ? = 0 + 1
[8,6,-7,-4,5,1,2,3] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 1
[5,7,-8,4,3,-6,1,2] => [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [2,1,3,4,5,6,7] => ? = 0 + 1
[6,1,2,3,4,-5] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[5,1,2,3,6,-4] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[5,1,2,6,4,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,1,2,5,6,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[5,1,6,3,4,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,1,5,3,6,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[5,1,6,2,3,-4] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[3,1,5,6,4,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,1,5,6,2,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[3,1,4,5,6,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[5,6,2,3,4,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,5,2,3,6,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[3,5,2,6,4,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[3,4,2,5,6,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[5,6,1,2,4,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,5,1,2,6,-3] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,5,1,6,3,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[2,5,6,3,4,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[2,4,5,3,6,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[4,5,6,2,3,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[3,5,6,1,4,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[3,4,5,1,6,-2] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[2,3,5,6,4,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
[3,4,5,6,2,-1] => [6]
=> [[1,2,3,4,5,6]]
=> [1,2,3,4,5,6] => ? = 1 + 1
Description
The number of connected components of the quiver of A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra A of K[x]/(xn).
The following 708 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001890The maximum magnitude of the Möbius function of a poset. St000787The number of flips required to make a perfect matching noncrossing. St000788The number of nesting-similar perfect matchings of a perfect matching. St000623The number of occurrences of the pattern 52341 in a permutation. St001741The largest integer such that all patterns of this size are contained in the permutation. St001260The permanent of an alternating sign matrix. St000629The defect of a binary word. St000002The number of occurrences of the pattern 123 in a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000142The number of even parts of a partition. St000143The largest repeated part of a partition. St000149The number of cells of the partition whose leg is zero and arm is odd. St000150The floored half-sum of the multiplicities of a partition. St000210Minimum over maximum difference of elements in cycles. St000256The number of parts from which one can substract 2 and still get an integer partition. St000257The number of distinct parts of a partition that occur at least twice. St000317The cycle descent number of a permutation. St000352The Elizalde-Pak rank of a permutation. St000357The number of occurrences of the pattern 12-3. St000360The number of occurrences of the pattern 32-1. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000367The number of simsun double descents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000405The number of occurrences of the pattern 1324 in a permutation. St000406The number of occurrences of the pattern 3241 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000480The number of lower covers of a partition in dominance order. St000486The number of cycles of length at least 3 of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000546The number of global descents of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000731The number of double exceedences of a permutation. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000962The 3-shifted major index of a permutation. St000995The largest even part of an integer partition. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001082The number of boxed occurrences of 123 in a permutation. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001091The number of parts in an integer partition whose next smaller part has the same size. St001092The number of distinct even parts of a partition. St001130The number of two successive successions in a permutation. St001214The aft of an integer partition. St001251The number of parts of a partition that are not congruent 1 modulo 3. St001252Half the sum of the even parts of a partition. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001381The fertility of a permutation. St001429The number of negative entries in a signed permutation. St001434The number of negative sum pairs of a signed permutation. St001513The number of nested exceedences of a permutation. St001550The number of inversions between exceedances where the greater exceedance is linked. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001588The number of distinct odd parts smaller than the largest even part in an integer partition. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001715The number of non-records in a permutation. St001728The number of invisible descents of a permutation. St001793The difference between the clique number and the chromatic number of a graph. St001947The number of ties in a parking function. St000003The number of standard Young tableaux of the partition. St000049The number of set partitions whose sorted block sizes correspond to the partition. St000054The first entry of the permutation. St000088The row sums of the character table of the symmetric group. St000182The number of permutations whose cycle type is the given integer partition. St000212The number of standard Young tableaux for an integer partition such that no two consecutive entries appear in the same row. St000275Number of permutations whose sorted list of non zero multiplicities of the Lehmer code is the given partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000321The number of integer partitions of n that are dominated by an integer partition. St000345The number of refinements of a partition. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000517The Kreweras number of an integer partition. St000542The number of left-to-right-minima of a permutation. St000705The number of semistandard tableaux on a given integer partition of n with maximal entry n. St000781The number of proper colouring schemes of a Ferrers diagram. St000913The number of ways to refine the partition into singletons. St000935The number of ordered refinements of an integer partition. St000990The first ascent of a permutation. St001121The multiplicity of the irreducible representation indexed by the partition in the Kronecker square corresponding to the partition. St001129The product of the squares of the parts of a partition. St001344The neighbouring number of a permutation. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001468The smallest fixpoint of a permutation. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001711The number of permutations such that conjugation with a permutation of given cycle type yields the squared permutation. St001667The maximal size of a pair of weak twins for a permutation. St000009The charge of a standard tableau. St000039The number of crossings of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000057The Shynar inversion number of a standard tableau. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000133The "bounce" of a permutation. St000185The weighted size of a partition. St000217The number of occurrences of the pattern 312 in a permutation. St000219The number of occurrences of the pattern 231 in a permutation. St000221The number of strong fixed points of a permutation. St000223The number of nestings in the permutation. St000234The number of global ascents of a permutation. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000283The size of the preimage of the map 'to graph' from Binary trees to Graphs. St000315The number of isolated vertices of a graph. St000358The number of occurrences of the pattern 31-2. St000359The number of occurrences of the pattern 23-1. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St000386The number of factors DDU in a Dyck path. St000407The number of occurrences of the pattern 2143 in a permutation. St000432The number of occurrences of the pattern 231 or of the pattern 312 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000461The rix statistic of a permutation. St000481The number of upper covers of a partition in dominance order. St000513The number of invariant subsets of size 2 when acting with a permutation of given cycle type. St000547The number of even non-empty partial sums of an integer partition. St000552The number of cut vertices of a graph. St000561The number of occurrences of the pattern {{1,2,3}} in a set partition. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000647The number of big descents of a permutation. St000649The number of 3-excedences of a permutation. St000660The number of rises of length at least 3 of a Dyck path. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000768The number of peaks in an integer composition. St000779The tier of a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St000807The sum of the heights of the valleys of the associated bargraph. St000872The number of very big descents of a permutation. St000873The aix statistic of a permutation. St000879The number of long braid edges in the graph of braid moves of a permutation. St000961The shifted major index of a permutation. St000963The 2-shifted major index of a permutation. St000989The number of final rises of a permutation. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001174The Gorenstein dimension of the algebra A/I when I is the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001305The number of induced cycles on four vertices in a graph. St001306The number of induced paths on four vertices in a graph. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001327The minimal number of occurrences of the split-pattern in a linear ordering of the vertices of the graph. St001353The number of prime nodes in the modular decomposition of a graph. St001356The number of vertices in prime modules of a graph. St001371The length of the longest Yamanouchi prefix of a binary word. St001394The genus of a permutation. St001430The number of positive entries in a signed permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001520The number of strict 3-descents. St001537The number of cyclic crossings of a permutation. St001549The number of restricted non-inversions between exceedances. St001552The number of inversions between excedances and fixed points of a permutation. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001577The minimal number of edges to add or remove to make a graph a cograph. St001593This is the number of standard Young tableaux of the given shifted shape. St001596The number of two-by-two squares inside a skew partition. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001696The natural major index of a standard Young tableau. St001705The number of occurrences of the pattern 2413 in a permutation. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001847The number of occurrences of the pattern 1432 in a permutation. St001850The number of Hecke atoms of a permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001948The number of augmented double ascents of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000048The multinomial of the parts of a partition. St000056The decomposition (or block) number of a permutation. St000069The number of maximal elements of a poset. St000193The row of the unique '1' in the first column of the alternating sign matrix. St000287The number of connected components of a graph. St000314The number of left-to-right-maxima of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000346The number of coarsenings of a partition. St000374The number of exclusive right-to-left minima of a permutation. St000570The Edelman-Greene number of a permutation. St000654The first descent of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000805The number of peaks of the associated bargraph. St000810The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to monomial symmetric functions. St000811The sum of the entries in the column specified by the partition of the change of basis matrix from powersum symmetric functions to Schur symmetric functions. St000814The sum of the entries in the column specified by the partition of the change of basis matrix from elementary symmetric functions to Schur symmetric functions. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000886The number of permutations with the same antidiagonal sums. St000991The number of right-to-left minima of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001103The number of words with multiplicities of the letters given by the partition, avoiding the consecutive pattern 123. St001162The minimum jump of a permutation. St001192The maximal dimension of Ext2A(S,A) for a simple module S over the corresponding Nakayama algebra A. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001256Number of simple reflexive modules that are 2-stable reflexive. St001272The number of graphs with the same degree sequence. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001387Number of standard Young tableaux of the skew shape tracing the border of the given partition. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001461The number of topologically connected components of the chord diagram of a permutation. St001518The number of graphs with the same ordinary spectrum as the given graph. St001590The crossing number of a perfect matching. St001591The number of graphs with the given composition of multiplicities of Laplacian eigenvalues. St001597The Frobenius rank of a skew partition. St001710The number of permutations such that conjugation with a permutation of given cycle type yields the inverse permutation. St001737The number of descents of type 2 in a permutation. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000058The order of a permutation. St000485The length of the longest cycle of a permutation. St000842The breadth of a permutation. St001198The number of simple modules in the algebra eAe with projective dimension at most 1 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001206The maximal dimension of an indecomposable projective eAe-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001555The order of a signed permutation. St001569The maximal modular displacement of a permutation. St001738The minimal order of a graph which is not an induced subgraph of the given graph. 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. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000929The constant term of the character polynomial of an integer partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001175The size of a partition minus the hook length of the base cell. 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. St001525The number of symmetric hooks on the diagonal of a partition. St001561The value of the elementary symmetric function evaluated at 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000706The product of the factorials of the multiplicities of an integer partition. St000993The multiplicity of the largest part of an integer partition. St001280The number of parts of an integer partition that are at least two. St001364The number of permutations whose cube equals a fixed permutation of given cycle type. St001568The smallest positive integer that does not appear twice in the partition. St001933The largest multiplicity of a part in an integer partition. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St000478Another weight of a partition according to Alladi. St000934The 2-degree of an integer partition. St000944The 3-degree of an integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001939The number of parts that are equal to their multiplicity in the integer partition. St001940The number of distinct parts that are equal to their multiplicity in the integer partition. St000667The greatest common divisor of the parts of the partition. St000928The sum of the coefficients of the character polynomial of an integer partition. St001571The Cartan determinant of the integer partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. 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. 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. 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. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000207Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St000208Number of integral Gelfand-Tsetlin polytopes with prescribed top row and integer partition weight. St000618The number of self-evacuating tableaux of given shape. St001389The number of partitions of the same length below the given integer partition. St001432The order dimension of the 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. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St000016The number of attacking pairs of a standard tableau. St000017The number of inversions of a standard tableau. St000117The number of centered tunnels of a Dyck path. St000292The number of ascents of a binary word. St000295The length of the border of a binary word. St000296The length of the symmetric border of a binary word. St000348The non-inversion sum of a binary word. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000682The Grundy value of Welter's game on a binary word. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St000790The number of pairs of centered tunnels, one strictly containing the other, of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000877The depth of the binary word interpreted as a path. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001125The number of simple modules that satisfy the 2-regular condition in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001172The number of 1-rises at odd height of a Dyck path. St001176The size of a partition minus its first part. St001177Twice the mean value of the major index among all standard Young tableaux of a partition. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001193The dimension of Ext1A(A/AeA,A) in the corresponding Nakayama algebra A such that eA is a minimal faithful projective-injective module. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001584The area statistic between a Dyck path and its bounce path. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. St001910The height of the middle non-run of a Dyck path. St001961The sum of the greatest common divisors of all pairs of parts. St000005The bounce statistic of a Dyck path. St000006The dinv of a Dyck path. St000026The position of the first return of a Dyck path. St000053The number of valleys of the Dyck path. St000075The orbit size of a standard tableau under promotion. St000120The number of left tunnels of a Dyck path. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000331The number of upper interactions of a Dyck path. St000390The number of runs of ones in a binary word. St000476The sum of the semi-lengths of tunnels before a valley of a Dyck path. St000627The exponent of a binary word. St000628The balance of a binary word. St000655The length of the minimal rise of a Dyck path. St000675The number of centered multitunnels of a Dyck path. St000691The number of changes of a binary word. St000734The last entry in the first row of a standard tableau. St000847The number of standard Young tableaux whose descent set is the binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000932The number of occurrences of the pattern UDU in a Dyck path. St000947The major index east count of a Dyck path. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000965The sum of the dimension of Ext^i(D(A),A) for i=1,. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001067The number of simple modules of dominant dimension at least two in the corresponding Nakayama algebra. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001161The major index north count of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001188The number of simple modules S with grade inf at least two in the Nakayama algebra A corresponding to the Dyck path. St001191Number of simple modules S with Ext_A^i(S,A)=0 for all i=0,1,...,g-1 in the corresponding Nakayama algebra A with global dimension g. St001196The global dimension of A minus the global dimension of eAe for the corresponding Nakayama algebra with minimal faithful projective-injective module eA. St001197The global dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001216The number of indecomposable injective modules in the corresponding Nakayama algebra that have non-vanishing second Ext-group with the regular module. St001223Number of indecomposable projective non-injective modules P such that the modules X and Y in a an Auslander-Reiten sequence ending at P are torsionless. St001225The vector space dimension of the first extension group between J and itself when J is the Jacobson radical of the corresponding Nakayama algebra. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001274The number of indecomposable injective modules with projective dimension equal to two. St001276The number of 2-regular indecomposable modules in the corresponding Nakayama algebra. St001278The number of indecomposable modules that are fixed by \tau \Omega^1 composed with its inverse in the corresponding Nakayama algebra. St001294The maximal torsionfree index of a simple non-projective module in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001313The number of Dyck paths above the lattice path given by a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001462The number of factors of a standard tableaux under concatenation. St001481The minimal height of a peak of a Dyck path. St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001498The normalised height of a Nakayama algebra with magnitude 1. St001499The number of indecomposable projective-injective modules of a magnitude 1 Nakayama algebra. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001506Half the projective dimension of the unique simple module with even projective dimension in a magnitude 1 Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001595The number of standard Young tableaux of the skew partition. St001721The degree of a binary word. St001722The number of minimal chains with small intervals between a binary word and the top element. St001732The number of peaks visible from the left. St001786The number of total orderings of the north steps of a Dyck path such that steps after the k-th east step are not among the first k positions in the order. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001884The number of borders of a binary word. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001929The number of meanders with top half given by the noncrossing matching corresponding to the Dyck path. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St000015The number of peaks of a Dyck path. St000439The position of the first down step of a Dyck path. St000630The length of the shortest palindromic decomposition of a binary word. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000983The length of the longest alternating subword. St001024Maximum of dominant dimensions of the simple modules in the Nakayama algebra corresponding to the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c_0,c_1,...,c_{n−1}] such that n=c_0 < c_i for all i > 0 a special CNakayama algebra. St001203We associate to a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c_0,c_1,...,c_{n-1}] such that n=c_0 < c_i for all i > 0 a Dyck path as follows:
St001275The projective dimension of the second term in a minimal injective coresolution of the regular module. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001299The product of all non-zero projective dimensions of simple modules of the corresponding Nakayama algebra. St001365The number of lattice paths of the same length weakly above the path given by a binary word. St001471The magnitude of a Dyck path. St001500The global dimension of magnitude 1 Nakayama algebras. St001530The depth of a Dyck path. St001733The number of weak left to right maxima of a Dyck path. St000969We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c_0,c_1,...,c_{n-1}] by adding c_0 to c_{n-1}. St001028Number of simple modules with injective dimension equal to the dominant dimension in the Nakayama algebra corresponding to the Dyck path. St001166Number of indecomposable projective non-injective modules with dominant dimension equal to the global dimension plus the number of indecomposable projective injective modules in the corresponding Nakayama algebra. St001182Number of indecomposable injective modules with codominant dimension at least two in the corresponding Nakayama algebra. St001255The vector space dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001200The number of simple modules in eAe with projective dimension at most 2 in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000693The modular (standard) major index of a standard tableau. St000753The Grundy value for the game of Kayles on a binary word. St001171The vector space dimension of Ext_A^1(I_o,A) when I_o is the tilting module corresponding to the permutation o in the Auslander algebra A of K[x]/(x^n). St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St001413Half the length of the longest even length palindromic prefix of a binary word. St001485The modular major index of a binary word. St001524The degree of symmetry of a binary word. St001556The number of inversions of the third entry of a permutation. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001583The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001686The order of promotion on a Gelfand-Tsetlin pattern. St001885The number of binary words with the same proper border set. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001423The number of distinct cubes in a binary word. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001414Half the length of the longest odd length palindromic prefix of a binary word. St001424The number of distinct squares in a binary word. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000186The sum of the first row in a Gelfand-Tsetlin pattern. St000735The last entry on the main diagonal of a standard tableau. St000744The length of the path to the largest entry in a standard Young tableau. St001002Number of indecomposable modules with projective and injective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001406The number of nonzero entries in a Gelfand Tsetlin pattern. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001684The reduced word complexity of a permutation. St000044The number of vertices of the unicellular map given by a perfect matching. St000393The number of strictly increasing runs in a binary word. St000950Number of tilting modules of the corresponding LNakayama algebra, where a tilting module is a generalised tilting module of projective dimension 1. St001267The length of the Lyndon factorization of the binary word. St001404The number of distinct entries in a Gelfand Tsetlin pattern. St000519The largest length of a factor maximising the subword complexity. St000922The minimal number such that all substrings of this length are unique. St001415The length of the longest palindromic prefix of a binary word. St001416The length of a longest palindromic factor of a binary word. St001417The length of a longest palindromic subword of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001003The number of indecomposable modules with projective dimension at most 1 in the Nakayama algebra corresponding to the Dyck path. St001838The number of nonempty primitive factors of a binary word. St000826The stopping time of the decimal representation of the binary word for the 3x+1 problem. St000294The number of distinct factors of a binary word. St000391The sum of the positions of the ones in a binary word. St001697The shifted natural comajor index of a standard Young tableau. St001930The weak major index of a binary word. St000545The number of parabolic double cosets with minimal element being the given permutation. St001168The vector space dimension of the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(x^n). St000631The number of distinct palindromic decompositions of a binary word. St000949Gives the number of generalised tilting modules of the corresponding LNakayama algebra. St000033The number of permutations greater than or equal to the given permutation in (strong) Bruhat order. St000958The number of Bruhat factorizations of a permutation. St000347The inversion sum of a binary word. St000518The number of distinct subsequences in a binary word. St001916The number of transient elements in the orbit of Bulgarian solitaire corresponding to a necklace. St001915The size of the component corresponding to a necklace in Bulgarian solitaire. St001560The product of the cardinalities of the lower order ideal and upper order ideal generated by a permutation in weak order. St001243The sum of coefficients in the Schur basis of certain LLT polynomials associated with a Dyck path. St000289The decimal representation of a binary word. St000827The decimal representation of a binary word with a leading 1. St001242The toal dimension of certain Sn modules determined by LLT polynomials associated with a Dyck path. St000921The number of internal inversions of a binary word. St000264The girth of a graph, which is not a tree. St001141The number of occurrences of hills of size 3 in a Dyck path. St000697The number of 3-rim hooks removed from an integer partition to obtain its associated 3-core. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the 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. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000369The dinv deficit of a Dyck path. St000567The sum of the products of all pairs of parts. St000661The number of rises of length 3 of a Dyck path. St000674The number of hills of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St000442The maximal area to the right of an up step of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000874The position of the last double rise in a Dyck path. St000946The sum of the skew hook positions in a Dyck path. St000976The sum of the positions of double up-steps of a Dyck path. St001031The height of the bicoloured Motzkin path associated with the Dyck path. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St000444The length of the maximal rise of a Dyck path. St000260The radius of a connected graph. St000046The largest eigenvalue of the random to random operator acting on the simple module corresponding to the given partition. St000137The Grundy value of an integer partition. St001383The BG-rank of an integer partition. St000068The number of minimal elements in a poset. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001845The number of join irreducibles minus the rank of a lattice. St001772The number of occurrences of the signed pattern 12 in a signed permutation. St001863The number of weak excedances of a signed permutation. St001864The number of excedances of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001868The number of alignments of type NE of a signed permutation. St001889The size of the connectivity set of a signed permutation. St001861The number of Bruhat lower covers of a permutation. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St001527The cyclic permutation representation number of an integer partition. 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. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000045The number of linear extensions of a binary tree. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001472The permanent of the Coxeter matrix of the poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St000680The Grundy value for Hackendot on posets. St000912The number of maximal antichains in a poset. St001858The number of covering elements of a signed permutation in absolute order. St000188The area of the Dyck path corresponding to a parking function and the total displacement of a parking function. St000195The number of secondary dinversion pairs of the dyck path corresponding to a parking function. St000943The number of spots the most unlucky car had to go further in a parking function. St001903The number of fixed points of a parking function. St001624The breadth of a lattice. St000135The number of lucky cars of the parking function. St001927Sparre Andersen's number of positives of a signed permutation. St000540The sum of the entries of a parking function minus its length. St000165The sum of the entries of a parking function. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St001095The number of non-isomorphic posets with precisely one further covering relation. St001510The number of self-evacuating linear extensions of a finite poset. St001534The alternating sum of the coefficients of the Poincare polynomial of the poset cone. St001631The number of simple modules S with dim Ext^1(S,A)=1 in the incidence algebra A of the poset. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St000640The rank of the largest boolean interval in a poset. St000911The number of maximal antichains of maximal size in a poset. St000942The number of critical left to right maxima of the parking functions. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001636The number of indecomposable injective modules with projective dimension at most one in the incidence algebra of the poset. St001718The number of non-empty open intervals in a poset. St001768The number of reduced words of a signed permutation. St001860The number of factors of the Stanley symmetric function associated with a signed permutation. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001904The length of the initial strictly increasing segment of a parking function. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000642The size of the smallest orbit of antichains under Panyushev complementation. St000907The number of maximal antichains of minimal length in a poset. St000524The number of posets with the same order polynomial. St000526The number of posets with combinatorially isomorphic order polytopes. St000528The height of a poset. St000717The number of ordinal summands of a poset. St000906The length of the shortest maximal chain in a poset. St001343The dimension of the reduced incidence algebra of a poset. St001926Sparre Andersen's position of the maximum of a signed permutation. St000643The size of the largest orbit of antichains under Panyushev complementation. St001625The Möbius invariant of a lattice. St001782The order of rowmotion on the set of order ideals of a poset. St001621The number of atoms of a lattice. St001623The number of doubly irreducible elements of a lattice. St001626The number of maximal proper sublattices of a lattice. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000550The number of modular elements of a lattice. St000551The number of left modular elements of a lattice. St001854The size of the left Kazhdan-Lusztig cell, St000782The indicator function of whether a given perfect matching is an L & P matching. St000850The number of 1/2-balanced pairs in a poset. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St000180The number of chains of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St001397Number of pairs of incomparable elements in a finite poset. St001398Number of subsets of size 3 of elements in a poset that form a "v". St001902The number of potential covers of a poset. St001964The interval resolution global dimension of a poset. St000525The number of posets with the same zeta polynomial. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001268The size of the largest ordinal summand in the poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001635The trace of the square of the Coxeter matrix of the incidence algebra of a poset. St001779The order of promotion on the set of linear extensions of a poset. St000080The rank of the poset. St001300The rank of the boundary operator in degree 1 of the chain complex of the order complex of the poset. St000189The number of elements in the poset. St000656The number of cuts of a poset. St001717The largest size of an interval in a poset. St000104The number of facets in the order polytope of this poset. St000151The number of facets in the chain polytope of the poset. St001664The number of non-isomorphic subposets of a poset. St000641The number of non-empty boolean intervals in a poset. St000639The number of relations in a poset. St001909The number of interval-closed sets of a poset. St001709The number of homomorphisms to the three element chain of a poset. St001815The number of order preserving surjections from a poset to a total order. St000634The number of endomorphisms of a poset. St001813The product of the sizes of the principal order filters in a poset. St000716The dimension of the irreducible representation of Sp(6) labelled by an integer partition. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000477The weight of a partition according to Alladi. St000997The even-odd crank of an integer partition. St000515The number of invariant set partitions when acting with a permutation of given cycle type.
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