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Your data matches 157 different statistics following compositions of up to 3 maps.
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Matching statistic: St001006
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001006: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 1
[1,0,1,0]
=> 1
[1,1,0,0]
=> 2
[1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> 3
[1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> 2
[1,0,1,1,1,0,0,1,0,0]
=> 2
[1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> 1
Description
Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St000993
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00024: Dyck paths —to 321-avoiding permutation⟶ Permutations
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00086: Permutations —first fundamental transformation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1] => [1] => [1]
=> ? = 1
[1,0,1,0]
=> [2,1] => [2,1] => [2]
=> 1
[1,1,0,0]
=> [1,2] => [1,2] => [1,1]
=> 2
[1,0,1,0,1,0]
=> [2,1,3] => [2,1,3] => [2,1]
=> 1
[1,0,1,1,0,0]
=> [2,3,1] => [3,2,1] => [3]
=> 1
[1,1,0,0,1,0]
=> [3,1,2] => [2,3,1] => [2,1]
=> 1
[1,1,0,1,0,0]
=> [1,3,2] => [1,3,2] => [2,1]
=> 1
[1,1,1,0,0,0]
=> [1,2,3] => [1,2,3] => [1,1,1]
=> 3
[1,0,1,0,1,0,1,0]
=> [2,1,4,3] => [2,1,4,3] => [2,2]
=> 2
[1,0,1,0,1,1,0,0]
=> [2,4,1,3] => [3,2,4,1] => [3,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [2,1,3,4] => [2,1,3,4] => [2,1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,3,1,4] => [3,2,1,4] => [3,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [4,2,3,1] => [3,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,1,4,2] => [3,4,1,2] => [2,1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,4,3,1] => [3,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1,2,4] => [2,3,1,4] => [2,1,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,3,2,4] => [1,3,2,4] => [2,1,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,3,4,2] => [1,4,3,2] => [3,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,3,4,1] => [2,1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,4,2,3] => [1,3,4,2] => [2,1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,2,4,3] => [1,2,4,3] => [2,1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,2,3,4] => [1,1,1,1]
=> 4
[1,0,1,0,1,0,1,0,1,0]
=> [2,1,4,3,5] => [2,1,4,3,5] => [2,2,1]
=> 2
[1,0,1,0,1,0,1,1,0,0]
=> [2,4,1,3,5] => [3,2,4,1,5] => [3,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [3,2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [2,4,1,5,3] => [4,2,5,1,3] => [3,1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,4,5,1,3] => [3,2,5,4,1] => [3,2]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [2,1,5,3,4] => [2,1,4,5,3] => [2,2,1]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [2,5,1,3,4] => [3,2,4,5,1] => [3,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [2,1,3,5,4] => [2,1,3,5,4] => [2,2,1]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [2,3,1,5,4] => [3,2,1,5,4] => [3,2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [4,2,3,5,1] => [3,1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,4,5] => [2,1,3,4,5] => [2,1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [2,3,1,4,5] => [3,2,1,4,5] => [3,1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,3,4,1,5] => [4,2,3,1,5] => [3,1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [5,2,3,4,1] => [3,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [3,1,4,2,5] => [3,4,1,2,5] => [2,1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,4,1,2,5] => [2,4,3,1,5] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [3,1,4,5,2] => [3,5,1,4,2] => [3,1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,4,1,5,2] => [4,5,3,1,2] => [3,1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,5,3,4,1] => [3,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [3,1,5,2,4] => [3,4,1,5,2] => [2,2,1]
=> 2
[1,1,0,1,0,0,1,1,0,0]
=> [3,5,1,2,4] => [2,4,3,5,1] => [3,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [3,1,2,5,4] => [2,3,1,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,3,2,5,4] => [1,3,2,5,4] => [2,2,1]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [1,4,3,5,2] => [3,1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,4,5] => [2,3,1,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,3,2,4,5] => [1,3,2,4,5] => [2,1,1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,3,4,5,2] => [1,5,3,4,2] => [3,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,1,5,2,3] => [4,3,5,1,2] => [3,1,1]
=> 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001199
(load all 82 compositions to match this statistic)
(load all 82 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 92%●distinct values known / distinct values provided: 33%
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 92%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> [1,0]
=> [1,1,0,0]
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> [1,1,1,0,0,0]
=> ? = 2
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,3}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,3}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,4}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,4}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,2,4}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [1,1,1,1,0,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 2
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {2,2,2,5}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,3,6}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,3,6}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,2,3,6}
Description
The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA.
Matching statistic: St001011
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 89%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001011: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 89%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,3}
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,3}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,2,5}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,2,5}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,6}
Description
Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path.
Matching statistic: St001493
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 89%●distinct values known / distinct values provided: 33%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001493: Dyck paths ⟶ ℤResult quality: 33% ●values known / values provided: 89%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,3}
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,3}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,3}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 2
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,2,5}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,2,5}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {2,2,2,2,3,6}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {2,2,2,2,3,6}
Description
The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra.
Matching statistic: St000667
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000667: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000698
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000698: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [2,1] => [2]
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,2] => [1,1]
=> [1]
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [3,2,1] => [3]
=> []
=> ? ∊ {1,1,1,3}
[1,0,1,1,0,0]
=> [2,3,1] => [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
[1,1,0,0,1,0]
=> [3,1,2] => [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
[1,1,0,1,0,0]
=> [2,1,3] => [2,1]
=> [1]
=> ? ∊ {1,1,1,3}
[1,1,1,0,0,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,0,1,0]
=> [4,3,2,1] => [4]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,0,1,1,0,0]
=> [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,1,0,0,1,0]
=> [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,1,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,1,1,0,0,0]
=> [2,3,4,1] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,0,0,1,1,0,0]
=> [3,4,1,2] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,0,1,0,1,0,0]
=> [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,0,1,1,0,0,0]
=> [2,3,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [4,1,2,3] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1] => [5]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [4,5,3,2,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [5,3,4,2,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,1,0,0,0]
=> [3,4,5,2,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,3,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,1,0,0]
=> [4,5,2,3,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,1,1,0,0,0]
=> [3,4,2,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [4,2,3,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [3,2,4,5,1] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1,2] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,1,0,0]
=> [4,5,3,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [5,3,4,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [4,3,5,1,2] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [3,4,5,1,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1,3] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,0,1,1,0,0]
=> [4,5,2,1,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1,4] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,1,0,1,1,0,0,0]
=> [3,4,2,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [5,2,3,1,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [5,4,1,2,3] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [4,5,1,2,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [5,3,1,2,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,4,1,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [5,2,1,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [5,1,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [3,1,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 2
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [6,4,5,3,2,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,5,6,3,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [6,5,3,4,2,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [5,6,3,4,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [6,4,3,5,2,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [4,5,3,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [6,3,4,5,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,4,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,5,6,2,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [3,4,5,6,2,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,3,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [5,6,4,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [6,4,5,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [5,4,6,2,3,1] => [4,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,1,0,0,0]
=> [4,5,6,2,3,1] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,4,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,5,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1,2] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1,3] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2,1,4] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,0,1,0,1,0,1,0,0,1,0]
=> [6,4,3,2,1,5] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => [5,1]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,6}
Description
The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core.
For any positive integer k, one associates a k-core to a partition by repeatedly removing all rim hooks of size k.
This statistic counts the 2-rim hooks that are removed in this process to obtain a 2-core.
Matching statistic: St001571
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Mp00233: Dyck paths —skew partition⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001571: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1,0]
=> [[1],[]]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [[1,1],[]]
=> []
=> ? ∊ {1,2}
[1,1,0,0]
=> [1,1,0,0]
=> [[2],[]]
=> []
=> ? ∊ {1,2}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [[1,1,1],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [[2,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [[2,1],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [[2,2],[]]
=> []
=> ? ∊ {1,1,1,3}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [[1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [[2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [[2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [[2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [[2,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [[3,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [[2,2,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [[3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1],[1]]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [[2,2,2,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [[3,3,3],[2]]
=> [2]
=> 2
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [[3,2,2],[1]]
=> [1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [[3,3,2],[1,1]]
=> [1,1]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3],[]]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [[1,1,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [[2,1,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [[2,2,1,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [[2,2,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [[3,2,1,1],[1]]
=> [1]
=> 1
[1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [[2,2,1,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [[3,3,2,1],[2,1]]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [[3,2,2,1],[1,1]]
=> [1,1]
=> 1
[1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [[3,3,1,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [[3,3,3,1],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [[4,4,4],[]]
=> []
=> ? ∊ {1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
Description
The Cartan determinant of the integer partition.
Let p=[p1,...,pr] be a given integer partition with highest part t. Let A=K[x]/(xt) be the finite dimensional algebra over the field K and M the direct sum of the indecomposable A-modules of vector space dimension pi for each i. Then the Cartan determinant of p is the Cartan determinant of the endomorphism algebra of M over A.
Explicitly, this is the determinant of the matrix (min, where \bar p is the set of distinct parts of the partition.
Matching statistic: St001934
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00119: Dyck paths —to 321-avoiding permutation (Krattenthaler)⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 33% ●values known / values provided: 79%●distinct values known / distinct values provided: 33%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> ? = 2
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? ∊ {1,3}
[1,1,1,0,0,0]
=> [3,1,2] => [3]
=> []
=> ? ∊ {1,3}
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,2,3] => [3,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,1,2,4}
[1,1,0,1,1,0,0,0]
=> [2,4,1,3] => [4]
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,0,0,0,1,0]
=> [3,1,2,4] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,1,4,2] => [4]
=> []
=> ? ∊ {1,1,2,4}
[1,1,1,0,1,0,0,0]
=> [3,4,1,2] => [2,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0]
=> [4,1,2,3] => [4]
=> []
=> ? ∊ {1,1,2,4}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,2,4] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,2,5,3] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,2,3,4] => [4,1]
=> [1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,3,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,1,4] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,1,3,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,5,3] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,1,3] => [3,2]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,1,3,4] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,1,0,0,0,1,0,1,0]
=> [3,1,2,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,1,2,5,4] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,1,4,2,5] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,1,4,5,2] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,1,0,0,1,1,0,0,0]
=> [3,1,5,2,4] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,1,2,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,1,5,2] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,1,2] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,1,2,4] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4,1,2,3,5] => [4,1]
=> [1]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [4,1,2,5,3] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,1,1,0,0,1,0,0,0]
=> [4,1,5,2,3] => [3,2]
=> [2]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [4,5,1,2,3] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,1,1,1,1,0,0,0,0,0]
=> [5,1,2,3,4] => [5]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,2,5}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> [2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,0,1,1,0,0,1,0,0]
=> [2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,0,1,1,1,0,0,0,0]
=> [2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,1,0,0,1,0,1,0,0]
=> [2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,1,0,0,1,1,0,0,0]
=> [2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> [2,4,5,6,1,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> [2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,5,6,1,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,1,4,5,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> [3,1,4,6,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,0,0,1,1,0,0,1,0,0]
=> [3,1,5,2,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,0,0,1,1,1,0,0,0,0]
=> [3,1,6,2,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,0,1,0,1,0,0,1,0,0]
=> [3,4,5,1,6,2] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [3,4,6,1,2,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,1,2,5,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,0,0,0,1,1,0,0,0]
=> [4,1,2,6,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [4,1,5,6,2,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,5,1,2,6,3] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,1,2,3,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,1,2,3,6,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,1,6,2,3,4] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,1,2,3,4,5] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,3,6}
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation \pi\in\mathfrak S_n with \ell cycles, including fixed points, is a tuple of r = n - \ell transpositions
(a_1, b_1),\dots,(a_r, b_r)
with b_1 \leq \dots \leq b_r and a_i < b_i for all i, whose product, in this order, is \pi.
For example, the cycle (2,3,1) has the two factorizations (2,3)(1,3) and (1,2)(2,3).
Matching statistic: St000627
(load all 16 compositions to match this statistic)
(load all 16 compositions to match this statistic)
Mp00093: Dyck paths —to binary word⟶ Binary words
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Mp00135: Binary words —rotate front-to-back⟶ Binary words
Mp00096: Binary words —Foata bijection⟶ Binary words
St000627: Binary words ⟶ ℤResult quality: 77% ●values known / values provided: 77%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> 10 => 01 => 01 => 1
[1,0,1,0]
=> 1010 => 0101 => 1001 => 1
[1,1,0,0]
=> 1100 => 1001 => 0101 => 2
[1,0,1,0,1,0]
=> 101010 => 010101 => 110001 => 1
[1,0,1,1,0,0]
=> 101100 => 011001 => 010101 => 3
[1,1,0,0,1,0]
=> 110010 => 100101 => 101001 => 1
[1,1,0,1,0,0]
=> 110100 => 101001 => 011001 => 1
[1,1,1,0,0,0]
=> 111000 => 110001 => 001101 => 1
[1,0,1,0,1,0,1,0]
=> 10101010 => 01010101 => 11100001 => 1
[1,0,1,0,1,1,0,0]
=> 10101100 => 01011001 => 01100101 => 1
[1,0,1,1,0,0,1,0]
=> 10110010 => 01100101 => 10101001 => 1
[1,0,1,1,0,1,0,0]
=> 10110100 => 01101001 => 01101001 => 1
[1,0,1,1,1,0,0,0]
=> 10111000 => 01110001 => 00101101 => 1
[1,1,0,0,1,0,1,0]
=> 11001010 => 10010101 => 11010001 => 1
[1,1,0,0,1,1,0,0]
=> 11001100 => 10011001 => 01010101 => 4
[1,1,0,1,0,0,1,0]
=> 11010010 => 10100101 => 10110001 => 1
[1,1,0,1,0,1,0,0]
=> 11010100 => 10101001 => 01110001 => 1
[1,1,0,1,1,0,0,0]
=> 11011000 => 10110001 => 00110101 => 1
[1,1,1,0,0,0,1,0]
=> 11100010 => 11000101 => 10011001 => 2
[1,1,1,0,0,1,0,0]
=> 11100100 => 11001001 => 01011001 => 1
[1,1,1,0,1,0,0,0]
=> 11101000 => 11010001 => 00111001 => 1
[1,1,1,1,0,0,0,0]
=> 11110000 => 11100001 => 00011101 => 1
[1,0,1,0,1,0,1,0,1,0]
=> 1010101010 => 0101010101 => 1111000001 => 1
[1,0,1,0,1,0,1,1,0,0]
=> 1010101100 => 0101011001 => 0111000101 => 1
[1,0,1,0,1,1,0,0,1,0]
=> 1010110010 => 0101100101 => 1011001001 => 1
[1,0,1,0,1,1,0,1,0,0]
=> 1010110100 => 0101101001 => 0111001001 => 1
[1,0,1,0,1,1,1,0,0,0]
=> 1010111000 => 0101110001 => 0011001101 => 1
[1,0,1,1,0,0,1,0,1,0]
=> 1011001010 => 0110010101 => 1101010001 => 1
[1,0,1,1,0,0,1,1,0,0]
=> 1011001100 => 0110011001 => 0101010101 => 5
[1,0,1,1,0,1,0,0,1,0]
=> 1011010010 => 0110100101 => 1011010001 => 1
[1,0,1,1,0,1,0,1,0,0]
=> 1011010100 => 0110101001 => 0111010001 => 1
[1,0,1,1,0,1,1,0,0,0]
=> 1011011000 => 0110110001 => 0011010101 => 1
[1,0,1,1,1,0,0,0,1,0]
=> 1011100010 => 0111000101 => 1001011001 => 1
[1,0,1,1,1,0,0,1,0,0]
=> 1011100100 => 0111001001 => 0101011001 => 1
[1,0,1,1,1,0,1,0,0,0]
=> 1011101000 => 0111010001 => 0011011001 => ? ∊ {1,2,2,2,2,2,2}
[1,0,1,1,1,1,0,0,0,0]
=> 1011110000 => 0111100001 => 0001011101 => 1
[1,1,0,0,1,0,1,0,1,0]
=> 1100101010 => 1001010101 => 1110100001 => 1
[1,1,0,0,1,0,1,1,0,0]
=> 1100101100 => 1001011001 => 0110100101 => 1
[1,1,0,0,1,1,0,0,1,0]
=> 1100110010 => 1001100101 => 1010101001 => 1
[1,1,0,0,1,1,0,1,0,0]
=> 1100110100 => 1001101001 => 0110101001 => 1
[1,1,0,0,1,1,1,0,0,0]
=> 1100111000 => 1001110001 => 0010101101 => 1
[1,1,0,1,0,0,1,0,1,0]
=> 1101001010 => 1010010101 => 1101100001 => 1
[1,1,0,1,0,0,1,1,0,0]
=> 1101001100 => 1010011001 => 0101100101 => 1
[1,1,0,1,0,1,0,0,1,0]
=> 1101010010 => 1010100101 => 1011100001 => 1
[1,1,0,1,0,1,0,1,0,0]
=> 1101010100 => 1010101001 => 0111100001 => 1
[1,1,0,1,0,1,1,0,0,0]
=> 1101011000 => 1010110001 => 0011100101 => ? ∊ {1,2,2,2,2,2,2}
[1,1,0,1,1,0,0,0,1,0]
=> 1101100010 => 1011000101 => 1001101001 => 1
[1,1,0,1,1,0,0,1,0,0]
=> 1101100100 => 1011001001 => 0101101001 => 1
[1,1,0,1,1,0,1,0,0,0]
=> 1101101000 => 1011010001 => 0011101001 => ? ∊ {1,2,2,2,2,2,2}
[1,1,0,1,1,1,0,0,0,0]
=> 1101110000 => 1011100001 => 0001101101 => 1
[1,1,1,0,0,0,1,0,1,0]
=> 1110001010 => 1100010101 => 1100110001 => 1
[1,1,1,0,0,0,1,1,0,0]
=> 1110001100 => 1100011001 => 0100110101 => 1
[1,1,1,0,0,1,0,0,1,0]
=> 1110010010 => 1100100101 => 1010110001 => 1
[1,1,1,0,1,0,1,0,0,0]
=> 1110101000 => 1101010001 => 0011110001 => ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,0,0,0,0,1,0]
=> 1111000010 => 1110000101 => 1000111001 => ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,0,0,0,1,0,0]
=> 1111000100 => 1110001001 => 0100111001 => ? ∊ {1,2,2,2,2,2,2}
[1,1,1,1,0,0,1,0,0,0]
=> 1111001000 => 1110010001 => 0010111001 => ? ∊ {1,2,2,2,2,2,2}
[1,0,1,0,1,0,1,1,1,0,0,0]
=> 101010111000 => 010101110001 => 001110001101 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,0,1,1,0,1,1,0,0,0]
=> 101011011000 => 010110110001 => 001110010101 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,0,1,1,1,0,1,0,0,0]
=> 101011101000 => 010111010001 => 001110011001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,0,1,0,1,1,0,0,0]
=> 101101011000 => 011010110001 => 001110100101 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,0,1,1,0,1,0,0,0]
=> 101101101000 => 011011010001 => 001110101001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,1,0,1,0,1,0,0,0]
=> 101110101000 => 011101010001 => 001110110001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,1,1,0,0,0,0,1,0]
=> 101111000010 => 011110000101 => 100010111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,1,1,0,0,0,1,0,0]
=> 101111000100 => 011110001001 => 010010111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,1,1,0,0,1,0,0,0]
=> 101111001000 => 011110010001 => 001010111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,0,1,1,1,1,0,1,0,0,0,0]
=> 101111010000 => 011110100001 => 000110111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,0,1,1,1,0,1,0,0,0]
=> 110011101000 => 100111010001 => 001101011001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,0,1,0,1,1,0,0,0]
=> 110101011000 => 101010110001 => 001111000101 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> 110101101000 => 101011010001 => 001111001001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,1,0,1,0,1,0,0,0]
=> 110110101000 => 101101010001 => 001111010001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,1,1,0,0,0,0,1,0]
=> 110111000010 => 101110000101 => 100011011001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,1,1,0,0,0,1,0,0]
=> 110111000100 => 101110001001 => 010011011001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> 110111001000 => 101110010001 => 001011011001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,0,1,1,1,0,1,0,0,0,0]
=> 110111010000 => 101110100001 => 000111011001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,0,1,0,1,1,0,0,0]
=> 111001011000 => 110010110001 => 001101100101 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,0,1,1,0,1,0,0,0]
=> 111001101000 => 110011010001 => 001101101001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,0,0,0,1,1,0,0]
=> 111010001100 => 110100011001 => 010011100101 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,0,0,1,1,0,0,0]
=> 111010011000 => 110100110001 => 001011100101 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> 111010101000 => 110101010001 => 001111100001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> 111010110000 => 110101100001 => 000111100101 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,1,0,0,0,0,1,0]
=> 111011000010 => 110110000101 => 100011101001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> 111011000100 => 110110001001 => 010011101001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,1,0,0,1,0,0,0]
=> 111011001000 => 110110010001 => 001011101001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> 111011010000 => 110110100001 => 000111101001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> 111100101000 => 111001010001 => 001101110001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,1,0,0,0,0,1,0]
=> 111101000010 => 111010000101 => 100011110001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> 111101000100 => 111010001001 => 010011110001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> 111101001000 => 111010010001 => 001011110001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> 111101010000 => 111010100001 => 000111110001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> 111110000010 => 111100000101 => 100001111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> 111110000100 => 111100001001 => 010001111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> 111110001000 => 111100010001 => 001001111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> 111110010000 => 111100100001 => 000101111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> 111110100000 => 111101000001 => 000011111001 => ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
Description
The exponent of a binary word.
This is the largest number e such that w is the concatenation of e identical factors. This statistic is also called '''frequency'''.
The following 147 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000781The number of proper colouring schemes of a Ferrers diagram. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000617The number of global maxima of a Dyck path. St000284The Plancherel distribution on integer partitions. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000934The 2-degree of an integer partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001901The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition. St000003The number of standard Young tableaux of the partition. St000053The number of valleys of the Dyck path. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000291The number of descents of a binary word. St000306The bounce count of a Dyck path. St000326The position of the first one in a binary word after appending a 1 at the end. St000346The number of coarsenings of a partition. St000390The number of runs of ones in 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. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000913The number of ways to refine the partition into singletons. St000955Number of times one has Ext^i(D(A),A)>0 for i>0 for the corresponding LNakayama algebra. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in 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. 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. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the 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 \{ i \geq 0 | Ext^i(S,A) \neq 0 \} at least two in the Nakayama algebra A corresponding to the Dyck path. St001192The maximal dimension of Ext_A^2(S,A) for a simple module S over the corresponding Nakayama algebra A. 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. 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. 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. St001256Number of simple reflexive modules that are 2-stable reflexive. 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. 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. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. 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. 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. 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. St001597The Frobenius rank of a skew partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. 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. St000260The radius of a connected graph. St001432The order dimension of the partition. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. 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. St001933The largest multiplicity of a part in an integer partition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001344The neighbouring number of a permutation. St001162The minimum jump of a permutation. St000570The Edelman-Greene number 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. St000100The number of linear extensions of a poset. St000633The size of the automorphism group of a poset. St000640The rank of the largest boolean interval in a poset. St000910The number of maximal chains of minimal length in a poset. St000914The sum of the values of the Möbius function of a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001632The number of indecomposable injective modules I with dim Ext^1(I,A)=1 for the incidence algebra A of a poset. St001890The maximum magnitude of the Möbius function of a poset. St001130The number of two successive successions in a permutation. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001208The 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]/(x^n). St001330The hat guessing number of a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000456The monochromatic index of a connected graph. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000755The number of real roots of the characteristic polynomial of a linear recurrence associated with an integer partition. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001780The order of promotion on the set of standard tableaux of given shape. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St000181The number of connected components of the Hasse diagram for the poset. St000145The Dyson rank of a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000478Another weight of a partition according to Alladi. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000618The number of self-evacuating tableaux of given shape. St000681The Grundy value of Chomp on Ferrers diagrams. St000939The number of characters of the symmetric group whose value on the partition is positive. St001280The number of parts of an integer partition that are at least two. St001384The number of boxes in the diagram of a partition that do not lie in the largest triangle it contains. 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. 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. St001593This is the number of standard Young tableaux of the given shifted shape. St001599The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on rooted trees. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001608The number of coloured rooted 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001938The number of transitive monotone factorizations of genus zero of a permutation of given cycle type. St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. St001820The size of the image of the pop stack sorting operator. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001846The number of elements which do not have a complement in the lattice. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000741The Colin de Verdière graph invariant. St001877Number of indecomposable injective modules with projective dimension 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000668The least common multiple of the parts of the partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001964The interval resolution global dimension of a poset. St000657The smallest part of an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St001267The length of the Lyndon factorization of the binary word. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001437The flex of a binary word. 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. St001060The distinguishing index of a graph.
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