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Your data matches 175 different statistics following compositions of up to 3 maps.
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Matching statistic: St001811
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St001811: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 0
[1,2,3,4] => 0
[1,2,4,3] => 2
[1,3,2,4] => 1
[1,3,4,2] => 1
[1,4,2,3] => 1
[1,4,3,2] => 2
[2,1,3,4] => 0
[2,1,4,3] => 2
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 1
[2,4,3,1] => 1
[3,1,2,4] => 0
[3,1,4,2] => 1
[3,2,1,4] => 0
[3,2,4,1] => 0
[3,4,1,2] => 0
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 1
[4,2,1,3] => 0
[4,2,3,1] => 0
[4,3,1,2] => 0
[4,3,2,1] => 0
[1,2,3,4,5] => 0
[1,2,3,5,4] => 3
[1,2,4,3,5] => 2
[1,2,4,5,3] => 2
[1,2,5,3,4] => 2
[1,2,5,4,3] => 4
[1,3,2,4,5] => 1
[1,3,2,5,4] => 4
[1,3,4,2,5] => 1
[1,3,4,5,2] => 1
[1,3,5,2,4] => 3
[1,3,5,4,2] => 3
[1,4,2,3,5] => 1
[1,4,2,5,3] => 3
[1,4,3,2,5] => 2
[1,4,3,5,2] => 2
[1,4,5,2,3] => 2
[1,4,5,3,2] => 2
Description
The Castelnuovo-Mumford regularity of a permutation.
The ''Castelnuovo-Mumford regularity'' of a permutation $\sigma$ is the ''Castelnuovo-Mumford regularity'' of the ''matrix Schubert variety'' $X_\sigma$.
Equivalently, it is the difference between the degrees of the ''Grothendieck polynomial'' and the ''Schubert polynomial'' for $\sigma$. It can be computed by subtracting the ''Coxeter length'' [[St000018]] from the ''Rajchgot index'' [[St001759]].
Matching statistic: St000142
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St000142: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,0,1,0]
=> [1]
=> [1]
=> 0
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,1]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,2,1,1,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 1
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,1,1,1,1,1]
=> 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,2,2]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,2,2,1,1]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2,2,1,1]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2,1,1,1,1]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
Description
The number of even parts of a partition.
Matching statistic: St000319
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000319: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,-1] => []
=> ? = 0
[2,1] => [2,1] => [1,-2] => [1]
=> 0
[1,2,3] => [1,2,3] => [2,3,-1] => []
=> ? ∊ {0,0}
[1,3,2] => [1,3,2] => [3,2,-1] => [1]
=> 0
[2,1,3] => [2,1,3] => [1,3,-2] => [1]
=> 0
[2,3,1] => [2,3,1] => [1,2,-3] => [1,1]
=> 0
[3,1,2] => [3,1,2] => [3,1,-2] => []
=> ? ∊ {0,0}
[3,2,1] => [3,2,1] => [2,1,-3] => [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => []
=> ? ∊ {0,0,0,0,1,2}
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1]
=> 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [1]
=> 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [1,1]
=> 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => []
=> ? ∊ {0,0,0,0,1,2}
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1]
=> 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,1]
=> 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,1,1]
=> 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1]
=> 0
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [2,1]
=> 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => []
=> ? ∊ {0,0,0,0,1,2}
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [1]
=> 0
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2]
=> 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => []
=> ? ∊ {0,0,0,0,1,2}
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3]
=> 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2]
=> 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => []
=> ? ∊ {0,0,0,0,1,2}
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => []
=> ? ∊ {0,0,0,0,1,2}
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3]
=> 2
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [1]
=> 0
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [2,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1]
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [1]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [1,1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [1,1,1]
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [1]
=> 0
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [2,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [1]
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [2]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [2,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [3]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [2]
=> 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [3]
=> 2
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [2,1]
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1]
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,1]
=> 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,1,1]
=> 0
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [2,1]
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,1]
=> 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,1,1]
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,1,1]
=> 0
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,5,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,5,3,1,2] => [4,5,3,1,2] => [5,3,1,2,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,1,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,3,4,2] => [5,1,3,4,2] => [5,3,4,1,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,1,4,3] => [5,2,1,4,3] => [2,5,4,1,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,3,1,4] => [5,2,3,1,4] => [2,3,5,1,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,1,3,2] => [5,4,1,3,2] => [5,4,2,1,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,2,1,3] => [5,4,2,1,3] => [3,5,2,1,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
Description
The spin of an integer partition.
The Ferrers shape of an integer partition $\lambda$ can be decomposed into border strips. The spin is then defined to be the total number of crossings of border strips of $\lambda$ with the vertical lines in the Ferrers shape.
The following example is taken from Appendix B in [1]: Let $\lambda = (5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1), (4,3,3,1), (2,2), (1), ().$$
The first strip $(5,5,4,4,2,1) \setminus (4,3,3,1)$ crosses $4$ times, the second strip $(4,3,3,1) \setminus (2,2)$ crosses $3$ times, the strip $(2,2) \setminus (1)$ crosses $1$ time, and the remaining strip $(1) \setminus ()$ does not cross.
This yields the spin of $(5,5,4,4,2,1)$ to be $4+3+1 = 8$.
Matching statistic: St000320
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St000320: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,-1] => []
=> ? = 0
[2,1] => [2,1] => [1,-2] => [1]
=> 0
[1,2,3] => [1,2,3] => [2,3,-1] => []
=> ? ∊ {0,0}
[1,3,2] => [1,3,2] => [3,2,-1] => [1]
=> 0
[2,1,3] => [2,1,3] => [1,3,-2] => [1]
=> 0
[2,3,1] => [2,3,1] => [1,2,-3] => [1,1]
=> 0
[3,1,2] => [3,1,2] => [3,1,-2] => []
=> ? ∊ {0,0}
[3,2,1] => [3,2,1] => [2,1,-3] => [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => []
=> ? ∊ {0,0,0,0,1,2}
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1]
=> 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [1]
=> 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [1,1]
=> 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => []
=> ? ∊ {0,0,0,0,1,2}
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1]
=> 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,1]
=> 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,1,1]
=> 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1]
=> 0
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [2,1]
=> 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => []
=> ? ∊ {0,0,0,0,1,2}
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [1]
=> 0
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2]
=> 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => []
=> ? ∊ {0,0,0,0,1,2}
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3]
=> 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2]
=> 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => []
=> ? ∊ {0,0,0,0,1,2}
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => []
=> ? ∊ {0,0,0,0,1,2}
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3]
=> 2
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [1]
=> 0
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [2,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1]
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [1]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [1,1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [1,1,1]
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [1]
=> 0
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [2,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [1]
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [2]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [2,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [3]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [2]
=> 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [3]
=> 2
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [2,1]
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1]
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,1]
=> 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,1,1]
=> 0
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [2,1]
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,1]
=> 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,1,1]
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,1,1]
=> 0
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,5,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,5,3,1,2] => [4,5,3,1,2] => [5,3,1,2,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,1,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,3,4,2] => [5,1,3,4,2] => [5,3,4,1,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,1,4,3] => [5,2,1,4,3] => [2,5,4,1,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,3,1,4] => [5,2,3,1,4] => [2,3,5,1,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,1,3,2] => [5,4,1,3,2] => [5,4,2,1,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,2,1,3] => [5,4,2,1,3] => [3,5,2,1,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
Description
The dinv adjustment of an integer partition.
The Ferrers shape of an integer partition $\lambda = (\lambda_1,\ldots,\lambda_k)$ can be decomposed into border strips. For $0 \leq j < \lambda_1$ let $n_j$ be the length of the border strip starting at $(\lambda_1-j,0)$.
The dinv adjustment is then defined by
$$\sum_{j:n_j > 0}(\lambda_1-1-j).$$
The following example is taken from Appendix B in [2]: Let $\lambda=(5,5,4,4,2,1)$. Removing the border strips successively yields the sequence of partitions
$$(5,5,4,4,2,1),(4,3,3,1),(2,2),(1),(),$$
and we obtain $(n_0,\ldots,n_4) = (10,7,0,3,1)$.
The dinv adjustment is thus $4+3+1+0 = 8$.
Matching statistic: St000749
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Mp00123: Dyck paths —Barnabei-Castronuovo involution⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000749: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> [1]
=> 0
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> []
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,1,0,1,0,0]
=> [1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [2,2]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,1,1,1,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 0
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 0
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> 2
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 0
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,4,4,4}
Description
The 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.
For example, restricting $S_{(6,3)}$ to $\mathfrak S_8$ yields $$S_{(5,3)}\oplus S_{(6,2)}$$ of degrees (number of standard Young tableaux) 28 and 20, none of which are odd. Restricting to $\mathfrak S_7$ yields $$S_{(4,3)}\oplus 2S_{(5,2)}\oplus S_{(6,1)}$$ of degrees 14, 14 and 6. However, restricting to $\mathfrak S_6$ yields
$$S_{(3,3)}\oplus 3S_{(4,2)}\oplus 3S_{(5,1)}\oplus S_6$$ of degrees 5,9,5 and 1. Therefore, the statistic on the partition $(6,3)$ gives 3.
This is related to $2$-saturations of Welter's game, see [1, Corollary 1.2].
Matching statistic: St001252
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
St001252: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1]
=> [1]
=> 0
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,1]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,2,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,2,1,1,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,1,1,1,1,1]
=> 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 0
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> 0
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,2,2]
=> 3
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,2,2,1,1]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 3
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> 3
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2,2,1,1]
=> 2
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> 3
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2,1,1,1,1]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,3,4,4}
Description
Half the sum of the even parts of a partition.
Matching statistic: St001382
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001382: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1,0,1,0]
=> [1]
=> []
=> 0
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> []
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> []
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> 2
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> 1
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> []
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> []
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> []
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> 2
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> 4
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> 4
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> 3
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> 3
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 3
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> 3
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> 2
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> 2
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> 2
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> 0
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,4}
Description
The number of boxes in the diagram of a partition that do not lie in its Durfee square.
Matching statistic: St001587
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00313: Integer partitions —Glaisher-Franklin inverse⟶ Integer partitions
St001587: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,0,1,0]
=> [1]
=> [1]
=> 0
[2,1] => [1,1,0,0]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [2]
=> 1
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [4,1]
=> 2
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> 1
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> 1
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> 1
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> 0
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [4]
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> 0
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> 1
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,0,0,0,1,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,1,1,1,1,1,1,1]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> 3
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [4,1,1,1,1,1]
=> 2
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [4,3,1]
=> 2
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> 2
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [4,1,1,1]
=> 2
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,1,1,1,1]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> 3
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1,1,1,1,1]
=> 1
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 2
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [4,2]
=> 2
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,1,1,1,1,1]
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,1,1,1,1,1]
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [2,2]
=> 1
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [2,2]
=> 1
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [2,2]
=> 1
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [2,2]
=> 1
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [2,2]
=> 1
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [2,2]
=> 1
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,1,1,1,1,1,1]
=> 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> 3
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [4,1,1,1,1]
=> 2
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
Description
Half of the largest even part of an integer partition.
The largest even part is recorded by [[St000995]].
Matching statistic: St001918
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00170: Permutations —to signed permutation⟶ Signed permutations
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Mp00167: Signed permutations —inverse Kreweras complement⟶ Signed permutations
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
St001918: Integer partitions ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,2] => [2,-1] => []
=> ? = 0
[2,1] => [2,1] => [1,-2] => [1]
=> 0
[1,2,3] => [1,2,3] => [2,3,-1] => []
=> ? ∊ {0,0}
[1,3,2] => [1,3,2] => [3,2,-1] => [1]
=> 0
[2,1,3] => [2,1,3] => [1,3,-2] => [1]
=> 0
[2,3,1] => [2,3,1] => [1,2,-3] => [1,1]
=> 0
[3,1,2] => [3,1,2] => [3,1,-2] => []
=> ? ∊ {0,0}
[3,2,1] => [3,2,1] => [2,1,-3] => [2]
=> 1
[1,2,3,4] => [1,2,3,4] => [2,3,4,-1] => []
=> ? ∊ {0,0,0,0,1,2}
[1,2,4,3] => [1,2,4,3] => [2,4,3,-1] => [1]
=> 0
[1,3,2,4] => [1,3,2,4] => [3,2,4,-1] => [1]
=> 0
[1,3,4,2] => [1,3,4,2] => [4,2,3,-1] => [1,1]
=> 0
[1,4,2,3] => [1,4,2,3] => [3,4,2,-1] => []
=> ? ∊ {0,0,0,0,1,2}
[1,4,3,2] => [1,4,3,2] => [4,3,2,-1] => [2]
=> 1
[2,1,3,4] => [2,1,3,4] => [1,3,4,-2] => [1]
=> 0
[2,1,4,3] => [2,1,4,3] => [1,4,3,-2] => [1,1]
=> 0
[2,3,1,4] => [2,3,1,4] => [1,2,4,-3] => [1,1]
=> 0
[2,3,4,1] => [2,3,4,1] => [1,2,3,-4] => [1,1,1]
=> 0
[2,4,1,3] => [2,4,1,3] => [1,4,2,-3] => [1]
=> 0
[2,4,3,1] => [2,4,3,1] => [1,3,2,-4] => [2,1]
=> 1
[3,1,2,4] => [3,1,2,4] => [3,1,4,-2] => []
=> ? ∊ {0,0,0,0,1,2}
[3,1,4,2] => [3,1,4,2] => [4,1,3,-2] => [1]
=> 0
[3,2,1,4] => [3,2,1,4] => [2,1,4,-3] => [2]
=> 1
[3,2,4,1] => [3,2,4,1] => [2,1,3,-4] => [2,1]
=> 1
[3,4,1,2] => [3,4,1,2] => [4,1,2,-3] => []
=> ? ∊ {0,0,0,0,1,2}
[3,4,2,1] => [3,4,2,1] => [3,1,2,-4] => [3]
=> 2
[4,1,2,3] => [4,1,2,3] => [3,4,1,-2] => [2]
=> 1
[4,1,3,2] => [4,1,3,2] => [4,3,1,-2] => []
=> ? ∊ {0,0,0,0,1,2}
[4,2,1,3] => [4,2,1,3] => [2,4,1,-3] => []
=> ? ∊ {0,0,0,0,1,2}
[4,2,3,1] => [4,2,3,1] => [2,3,1,-4] => [3]
=> 2
[4,3,1,2] => [4,3,1,2] => [4,2,1,-3] => [1]
=> 0
[4,3,2,1] => [4,3,2,1] => [3,2,1,-4] => [2,1]
=> 1
[1,2,3,4,5] => [1,2,3,4,5] => [2,3,4,5,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,2,3,5,4] => [1,2,3,5,4] => [2,3,5,4,-1] => [1]
=> 0
[1,2,4,3,5] => [1,2,4,3,5] => [2,4,3,5,-1] => [1]
=> 0
[1,2,4,5,3] => [1,2,4,5,3] => [2,5,3,4,-1] => [1,1]
=> 0
[1,2,5,3,4] => [1,2,5,3,4] => [2,4,5,3,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,2,5,4,3] => [1,2,5,4,3] => [2,5,4,3,-1] => [2]
=> 1
[1,3,2,4,5] => [1,3,2,4,5] => [3,2,4,5,-1] => [1]
=> 0
[1,3,2,5,4] => [1,3,2,5,4] => [3,2,5,4,-1] => [1,1]
=> 0
[1,3,4,2,5] => [1,3,4,2,5] => [4,2,3,5,-1] => [1,1]
=> 0
[1,3,4,5,2] => [1,3,4,5,2] => [5,2,3,4,-1] => [1,1,1]
=> 0
[1,3,5,2,4] => [1,3,5,2,4] => [4,2,5,3,-1] => [1]
=> 0
[1,3,5,4,2] => [1,3,5,4,2] => [5,2,4,3,-1] => [2,1]
=> 1
[1,4,2,3,5] => [1,4,2,3,5] => [3,4,2,5,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,4,2,5,3] => [1,4,2,5,3] => [3,5,2,4,-1] => [1]
=> 0
[1,4,3,2,5] => [1,4,3,2,5] => [4,3,2,5,-1] => [2]
=> 1
[1,4,3,5,2] => [1,4,3,5,2] => [5,3,2,4,-1] => [2,1]
=> 1
[1,4,5,2,3] => [1,4,5,2,3] => [4,5,2,3,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,4,5,3,2] => [1,4,5,3,2] => [5,4,2,3,-1] => [3]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [3,4,5,2,-1] => [2]
=> 1
[1,5,2,4,3] => [1,5,2,4,3] => [3,5,4,2,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => [1,5,3,2,4] => [4,3,5,2,-1] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[1,5,3,4,2] => [1,5,3,4,2] => [5,3,4,2,-1] => [3]
=> 2
[1,5,4,2,3] => [1,5,4,2,3] => [4,5,3,2,-1] => [1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [5,4,3,2,-1] => [2,1]
=> 1
[2,1,3,4,5] => [2,1,3,4,5] => [1,3,4,5,-2] => [1]
=> 0
[2,1,3,5,4] => [2,1,3,5,4] => [1,3,5,4,-2] => [1,1]
=> 0
[2,1,4,3,5] => [2,1,4,3,5] => [1,4,3,5,-2] => [1,1]
=> 0
[2,1,4,5,3] => [2,1,4,5,3] => [1,5,3,4,-2] => [1,1,1]
=> 0
[2,1,5,3,4] => [2,1,5,3,4] => [1,4,5,3,-2] => [1]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [1,5,4,3,-2] => [2,1]
=> 1
[2,3,1,4,5] => [2,3,1,4,5] => [1,2,4,5,-3] => [1,1]
=> 0
[2,3,1,5,4] => [2,3,1,5,4] => [1,2,5,4,-3] => [1,1,1]
=> 0
[2,3,4,1,5] => [2,3,4,1,5] => [1,2,3,5,-4] => [1,1,1]
=> 0
[3,1,2,4,5] => [3,1,2,4,5] => [3,1,4,5,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,1,5,2,4] => [3,1,5,2,4] => [4,1,5,3,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,4,1,2,5] => [3,4,1,2,5] => [4,1,2,5,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,4,5,1,2] => [3,4,5,1,2] => [5,1,2,3,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,5,1,4,2] => [3,5,1,4,2] => [5,1,4,2,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[3,5,2,1,4] => [3,5,2,1,4] => [3,1,5,2,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,1,3,2,5] => [4,1,3,2,5] => [4,3,1,5,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,1,5,3,2] => [4,1,5,3,2] => [5,4,1,3,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,2,1,3,5] => [4,2,1,3,5] => [2,4,1,5,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,2,5,1,3] => [4,2,5,1,3] => [2,5,1,3,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,5,1,2,3] => [4,5,1,2,3] => [4,5,1,2,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[4,5,3,1,2] => [4,5,3,1,2] => [5,3,1,2,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,2,3,4] => [5,1,2,3,4] => [3,4,5,1,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,1,3,4,2] => [5,1,3,4,2] => [5,3,4,1,-2] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,1,4,3] => [5,2,1,4,3] => [2,5,4,1,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,2,3,1,4] => [5,2,3,1,4] => [2,3,5,1,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,1,3,2] => [5,4,1,3,2] => [5,4,2,1,-3] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
[5,4,2,1,3] => [5,4,2,1,3] => [3,5,2,1,-4] => []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,4,4,4}
Description
The degree of the cyclic sieving polynomial corresponding to an integer partition.
Let $\lambda$ be an integer partition of $n$ and let $N$ be the least common multiple of the parts of $\lambda$. Fix an arbitrary permutation $\pi$ of cycle type $\lambda$. Then $\pi$ induces a cyclic action of order $N$ on $\{1,\dots,n\}$.
The corresponding character can be identified with the cyclic sieving polynomial $C_\lambda(q)$ of this action, modulo $q^N-1$. Explicitly, it is
$$
\sum_{p\in\lambda} [p]_{q^{N/p}},
$$
where $[p]_q = 1+\dots+q^{p-1}$ is the $q$-integer.
This statistic records the degree of $C_\lambda(q)$. Equivalently, it equals
$$
\left(1 - \frac{1}{\lambda_1}\right) N,
$$
where $\lambda_1$ is the largest part of $\lambda$.
The statistic is undefined for the empty partition.
Matching statistic: St001964
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Mp00121: Dyck paths —Cori-Le Borgne involution⟶ Dyck paths
Mp00242: Dyck paths —Hessenberg poset⟶ Posets
St001964: Posets ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[1,2] => [1,0,1,0]
=> [1,0,1,0]
=> ([(0,1)],2)
=> 0
[2,1] => [1,1,0,0]
=> [1,1,0,0]
=> ([],2)
=> 0
[1,2,3] => [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> ([(0,2),(2,1)],3)
=> 0
[1,3,2] => [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> ([(1,2)],3)
=> ? = 1
[2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> ([(0,1),(0,2)],3)
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> ([(0,2),(1,2)],3)
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 0
[3,2,1] => [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> ([],3)
=> 0
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> ([(0,3),(2,1),(3,2)],4)
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3)],4)
=> 0
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(3,1)],4)
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> ([(0,3),(1,2),(2,3)],4)
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {0,1,2,2}
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> ([(2,3)],4)
=> ? ∊ {0,1,2,2}
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ([(0,3),(3,1),(3,2)],4)
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> ([(0,3),(1,3),(3,2)],4)
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,0,0]
=> ([(1,3),(2,3)],4)
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {0,1,2,2}
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> ([(1,2),(1,3)],4)
=> ? ∊ {0,1,2,2}
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> ([(0,1),(0,2),(0,3)],4)
=> 1
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> ([],4)
=> 0
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(1,3),(2,4)],5)
=> 2
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> ([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 1
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> ([(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ([(0,4),(3,2),(4,1),(4,3)],5)
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> ([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> 2
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> ([(0,2),(0,3),(1,4),(2,4),(3,1)],5)
=> 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> 1
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 1
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(2,4)],5)
=> 1
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> 1
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> ([(0,4),(1,2),(1,3),(1,4)],5)
=> 1
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> ([(0,2),(0,3),(0,4),(4,1)],5)
=> 1
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> ([(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ([(0,3),(3,4),(4,1),(4,2)],5)
=> 1
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> ([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[2,1,4,5,3] => [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> ([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
[2,1,5,3,4] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[2,3,1,4,5] => [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> 1
[2,3,1,5,4] => [1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
[2,3,4,1,5] => [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> ([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 1
[3,5,1,2,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[3,5,1,4,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[3,5,2,1,4] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[3,5,2,4,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[3,5,4,1,2] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[3,5,4,2,1] => [1,1,1,0,1,1,0,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,1,2,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,1,2,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,1,3,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,1,3,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,2,1,3,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,2,1,5,3] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,2,3,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,2,3,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,3,1,2,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,3,1,5,2] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,3,2,1,5] => [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> ([(2,3),(2,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
[4,3,2,5,1] => [1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> ([(1,2),(1,3),(1,4)],5)
=> ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,3,3,4,4,4}
Description
The interval resolution global dimension of a poset.
This is the cardinality of the longest chain of right minimal approximations by interval modules of an indecomposable module over the incidence algebra.
The following 165 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001435The number of missing boxes in the first row. St000954Number of times the corresponding LNakayama algebra has $Ext^i(D(A),A)=0$ for $i>0$. St001164Number of indecomposable injective modules whose socle has projective dimension at most g-1 (g the global dimension) minus the number of indecomposable projective-injective modules. St001233The number of indecomposable 2-dimensional modules with projective dimension one. St000456The monochromatic index of a connected graph. St001541The Gini index of an integer partition. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000346The number of coarsenings of a partition. St000533The minimum of the number of parts and the size of the first part of an integer partition. St000783The side length of the largest staircase partition fitting into a partition. St001432The order dimension of the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001438The number of missing boxes of a skew partition. St001651The Frankl number of a lattice. St000741The Colin de Verdière graph invariant. St000567The sum of the products of all pairs of parts. 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. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001176The size of a partition minus its first part. 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. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000938The number of zeros of the symmetric group character corresponding to the partition. 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. St000929The constant term of the character polynomial of an integer partition. St000419The number of Dyck paths that are weakly above the Dyck path, except for the path itself. St000455The second largest eigenvalue of a graph if it is integral. St000478Another weight of a partition according to Alladi. St000674The number of hills of a Dyck path. St000683The number of points below the Dyck path such that the diagonal to the north-east hits the path between two down steps, and the diagonal to the north-west hits the path between two up steps. St000693The modular (standard) major index of a standard tableau. St000934The 2-degree of an integer partition. St000946The sum of the skew hook positions in a Dyck path. St000984The number of boxes below precisely one peak. St001032The number of horizontal steps in the bicoloured Motzkin path associated with the Dyck path. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. 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. St001139The number of occurrences of hills of size 2 in a Dyck path. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001204Call 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. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001480The number of simple summands of the module J^2/J^3. St000225Difference between largest and smallest parts in a partition. St001175The size of a partition minus the hook length of the base cell. St001912The length of the preperiod in Bulgarian solitaire corresponding to an integer partition. St000454The largest eigenvalue of a graph if it is integral. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000137The Grundy value of an integer partition. St000460The hook length of the last cell along the main diagonal of an integer partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St000870The product of the hook lengths of the diagonal cells in an integer partition. St001122The multiplicity of the sign representation in the Kronecker square corresponding to a partition. St001247The number of parts of a partition that are not congruent 2 modulo 3. St001248Sum of the even parts of a partition. St001249Sum of the odd parts of a partition. St001279The sum of the parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001525The number of symmetric hooks on the diagonal of a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001606The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001657The number of twos in an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St001914The size of the orbit of an integer partition in Bulgarian solitaire. 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. St001961The sum of the greatest common divisors of all pairs of parts. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001866The nesting alignments of a signed permutation. St000668The least common multiple of the parts of the partition. 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. St000933The number of multipartitions of sizes given by an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001250The number of parts of a partition that are not congruent 0 modulo 3. St001360The number of covering relations in Young's lattice below a partition. St001380The number of monomer-dimer tilings of a Ferrers diagram. St001593This is the number of standard Young tableaux of the given shifted shape. St001600The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. 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. St001628The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple connected graphs. St001933The largest multiplicity of a part in an integer partition. 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. St001845The number of join irreducibles minus the rank of a lattice. St000909The number of maximal chains of maximal size in a poset. St000939The number of characters of the symmetric group whose value on the partition is positive. St000993The multiplicity of the largest part of an integer partition. St001846The number of elements which do not have a complement in the lattice. St001570The minimal number of edges to add to make a graph Hamiltonian. St000527The width of the poset. St001868The number of alignments of type NE of a signed permutation. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001864The number of excedances of a signed permutation. St001556The number of inversions of the third entry of a permutation. St000422The energy of a graph, if it is integral. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000632The jump number of the poset. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite 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. St000181The number of connected components of the Hasse diagram for the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St000908The length of the shortest maximal antichain in a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001472The permanent of the Coxeter matrix of the poset. St001510The number of self-evacuating linear extensions of a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001645The pebbling number of a connected graph. St001779The order of promotion on the set of linear extensions of a poset. St001330The hat guessing number of a graph. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000936The number of even values of the symmetric group character corresponding to the partition. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001128The exponens consonantiae of a partition. St001875The number of simple modules with projective dimension at most 1. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000706The product of the factorials of the multiplicities of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. 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. 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. St001568The smallest positive integer that does not appear twice in the partition. St001487The number of inner corners of a skew partition. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000907The number of maximal antichains of minimal length in a poset. St001545The second Elser number of a connected graph. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000906The length of the shortest maximal chain in a poset.
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