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Your data matches 73 different statistics following compositions of up to 3 maps.
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Matching statistic: St001466
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001466: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 1
[1,2,3] => 0
[1,3,2] => 1
[2,1,3] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,2,1] => 1
[1,2,3,4] => 0
[1,2,4,3] => 1
[1,3,2,4] => 1
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 1
[2,1,4,3] => 2
[2,3,1,4] => 0
[2,3,4,1] => 0
[2,4,1,3] => 0
[2,4,3,1] => 0
[3,1,2,4] => 0
[3,1,4,2] => 0
[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] => 0
[4,2,1,3] => 0
[4,2,3,1] => 1
[4,3,1,2] => 0
[4,3,2,1] => 2
[1,2,3,4,5] => 0
[1,2,3,5,4] => 1
[1,2,4,3,5] => 1
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 1
[1,3,2,5,4] => 2
[1,3,4,2,5] => 0
[1,3,4,5,2] => 0
[1,3,5,2,4] => 0
[1,3,5,4,2] => 0
[1,4,2,3,5] => 0
[1,4,2,5,3] => 0
[1,4,3,2,5] => 0
[1,4,3,5,2] => 0
[1,4,5,2,3] => 0
Description
The number of transpositions swapping cyclically adjacent numbers in a permutation.
Put differently, this is the number of adjacent two-cycles in the chord diagram of a permutation.
Matching statistic: St000934
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 93%●distinct values known / distinct values provided: 75%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 93%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [3]
=> []
=> ? ∊ {0,0,1,1,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> 0
[1,3,4,2] => [2,1,1]
=> [1,1]
=> 0
[1,4,2,3] => [2,1,1]
=> [1,1]
=> 0
[1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> 1
[2,3,1,4] => [2,1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,1,3] => [2,1,1]
=> [1,1]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[3,1,2,4] => [2,1,1]
=> [1,1]
=> 0
[3,1,4,2] => [2,2]
=> [2]
=> 1
[3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[3,4,1,2] => [2,1,1]
=> [1,1]
=> 0
[3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,1,2,3] => [2,1,1]
=> [1,1]
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 0
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 0
[1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 0
[2,1,4,5,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,3,4] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,4,3] => [3,2]
=> [2]
=> 1
[2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> 0
[2,5,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,4,2,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,2,1,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,2,5,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,5,2,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,3,2,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,4,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,2,1,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,2,4,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,2,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,2,1,3] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,2,3,1] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,3,1,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,3,2,1] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,6,5,4,3,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[2,6,5,4,3,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[3,6,5,4,2,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[4,6,5,3,2,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[5,4,3,2,1,6] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[5,4,3,2,6,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[5,4,3,6,2,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[5,4,6,3,2,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[5,6,4,3,2,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[6,1,5,4,3,2] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[6,2,5,4,3,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[6,3,5,4,2,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[6,4,3,2,1,5] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[6,4,3,2,5,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
[6,4,3,5,2,1] => [5,1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3}
Description
The 2-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 2 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
Matching statistic: St000621
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
St000621: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1]
=> ? ∊ {0,1}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,1}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [2,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,3,4,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,4,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,4,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [2]
=> 1
[2,3,1,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[2,3,4,1] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[2,4,1,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[2,4,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[3,1,2,4] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[3,1,4,2] => [2,2]
=> [2]
=> [2]
=> 1
[3,2,1,4] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[3,2,4,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[3,4,1,2] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[3,4,2,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,1,2,3] => [2,1,1]
=> [1,1]
=> [1,1]
=> 0
[4,1,3,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,2,1,3] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,2,3,1] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,3,1,2] => [3,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[4,3,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,2,2}
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[1,3,4,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[1,5,4,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,1,4,5,3] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,1,5,3,4] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,1,5,4,3] => [3,2]
=> [2]
=> [2]
=> 1
[2,3,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [2,1]
=> [3]
=> 0
[2,3,4,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1,1]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [1,1]
=> [1,1]
=> 0
[2,5,4,3,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,4,2,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,2,1,5] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,2,5,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,5,2,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,3,2,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,4,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,4,3,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,2,1,4] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,2,4,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,2,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,3,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,2,1,3] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,2,3,1] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,3,1,2] => [4,1]
=> [1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,3,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,6,5,4,3,2] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[2,6,5,4,3,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[3,6,5,4,2,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[4,6,5,3,2,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,4,3,2,1,6] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,4,3,2,6,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,4,3,6,2,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,4,6,3,2,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[5,6,4,3,2,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,5,4,3,2] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,2,5,4,3,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,3,5,4,2,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,4,3,2,1,5] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,4,3,2,5,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,4,3,5,2,1] => [5,1]
=> [1]
=> [1]
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even.
To be precise, this is given for a partition $\lambda \vdash n$ by the number of standard tableaux $T$ of shape $\lambda$ such that $\min\big( \operatorname{Des}(T) \cup \{n\} \big)$ is even.
This notion was used in [1, Proposition 2.3], see also [2, Theorem 1.1].
The case of an odd minimum is [[St000620]].
Matching statistic: St000205
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,3,1,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,3,4,1,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,4,5,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,5,1,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,4,1,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,5,1,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,1,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,4,1,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,1,2,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,1,4,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,4,5,2] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,2,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,4,1,2,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,5,1,2] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,1,2,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,5,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,1,2,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,2,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,4,5,6] => [6] => [[6],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000206
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000206: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 86%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,3,1,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,3,4,1,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,4,5,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,5,1,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,4,1,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,5,1,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,1,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,4,1,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,1,2,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,1,4,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,4,5,2] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,2,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,4,1,2,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,5,1,2] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,1,2,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,5,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,1,2,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,2,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,4,5,6] => [6] => [[6],[]]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight.
Given $\lambda$ count how many ''integer compositions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
See also [[St000205]].
Each value in this statistic is greater than or equal to corresponding value in [[St000205]].
Matching statistic: St001175
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001175: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[2,3,1,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,3,4,1,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,4,5,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,5,1,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,4,1,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,5,1,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,1,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,4,1,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,1,2,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,1,4,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,4,5,2] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 0
[3,2,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,4,1,2,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,5,1,2] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,1,2,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,5,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,1,2,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,2,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,4,5,6] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3}
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St001586
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00248: Permutations —DEX composition⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[1] => [1] => [[1],[]]
=> []
=> ? = 0
[1,2] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[2,1] => [2] => [[2],[]]
=> []
=> ? ∊ {0,1}
[1,2,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[1,3,2] => [1,2] => [[2,1],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,1,3] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[2,3,1] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,1,2] => [3] => [[3],[]]
=> []
=> ? ∊ {0,0,1,1,1}
[3,2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 0
[1,2,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,2,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[1,3,2,4] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,3,4,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,2,3] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[1,4,3,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[2,1,3,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,1,4,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[2,3,1,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,3,4,1] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,1,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[2,4,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[3,1,2,4] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,1,4,2] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,1,4] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,2,4,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[3,4,1,2] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[3,4,2,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,1,2,3] => [4] => [[4],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,1,3,2] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,2,1,3] => [2,2] => [[3,2],[1]]
=> [1]
=> 0
[4,2,3,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 0
[4,3,1,2] => [1,3] => [[3,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,2,2}
[4,3,2,1] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 0
[1,2,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[1,2,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[1,2,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[1,3,2,4,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,2,5,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,3,4,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,4,5,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,5,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,3,5,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,4,2,3,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,2,5,3] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,2,5] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,3,5,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,4,5,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,4,5,3,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,2,3,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,5,2,4,3] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,3,2,4] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 0
[1,5,3,4,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 0
[1,5,4,2,3] => [1,1,3] => [[3,1,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,5,4,3,2] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 0
[2,1,3,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,1,3,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,1,4,3,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,4,5,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,3,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,1,5,4,3] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[2,3,1,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,1,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,3,4,1,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,4,5,1] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,5,1,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,3,5,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,4,1,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,1,5,3] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,1,5] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,3,5,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,4,5,1,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,4,5,3,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,1,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[2,5,1,4,3] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,3,1,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[2,5,3,4,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 0
[2,5,4,1,3] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[2,5,4,3,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,1,2,4,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,1,2,5,4] => [3,2] => [[4,3],[2]]
=> [2]
=> 0
[3,1,4,2,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,4,5,2] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,2,4] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,1,5,4,2] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1
[3,2,1,4,5] => [2,3] => [[4,2],[1]]
=> [1]
=> 0
[3,4,1,2,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,4,5,1,2] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[3,5,1,2,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,1,2,3,5] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,1,2,5] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,3,5,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[4,5,1,2,3] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,2,3,4] => [5] => [[5],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [1,4] => [[4,1],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2}
[1,2,3,4,5,6] => [6] => [[6],[]]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,2,2,3,3}
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St001122
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001122: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1,0]
=> []
=> ? = 0
[1,2] => [1,2] => [1,0,1,0]
=> [1]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> []
=> ? = 0
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1}
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1}
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,3,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,4,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,4,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,5,3,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,5,4,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,2,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,2,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,4,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,4,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,5,2,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,5,4,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,2,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,2,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,3,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,3,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
The multiplicity of the sign representation in the Kronecker square corresponding to a partition.
The Kronecker coefficient is the multiplicity $g_{\mu,\nu}^\lambda$ of the Specht module $S^\lambda$ in $S^\mu\otimes S^\nu$:
$$ S^\mu\otimes S^\nu = \bigoplus_\lambda g_{\mu,\nu}^\lambda S^\lambda $$
This statistic records the Kronecker coefficient $g_{\lambda,\lambda}^{1^n}$, for $\lambda\vdash n$. It equals $1$ if and only if $\lambda$ is self-conjugate.
Matching statistic: St001137
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001137: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 82%●distinct values known / distinct values provided: 75%
Values
[1] => [1,0]
=> []
=> []
=> ? = 0
[1,2] => [1,0,1,0]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [1,1,0,0]
=> []
=> []
=> ? = 1
[1,2,3] => [1,0,1,0,1,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,3] => [1,1,0,0,1,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,3,1] => [1,1,0,1,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1}
[3,2,1] => [1,1,1,0,0,0]
=> []
=> []
=> ? ∊ {1,1}
[1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 0
[1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 0
[2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 0
[3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,2,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,2,2}
[4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,2,2}
[4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,2,2}
[4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,2,2}
[4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,2,2}
[4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,1,2,2}
[1,2,3,4,5] => [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,2,3,5,4] => [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,2,4,3,5] => [1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> 1
[1,2,4,5,3] => [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 0
[1,2,5,3,4] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,2,5,4,3] => [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,3,2,4,5] => [1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> 1
[1,3,2,5,4] => [1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> 0
[1,3,4,2,5] => [1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> 0
[1,3,4,5,2] => [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[1,3,5,2,4] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[1,3,5,4,2] => [1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 0
[1,4,2,3,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,2,5,3] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,4,3,2,5] => [1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 0
[1,4,3,5,2] => [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 0
[1,4,5,2,3] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[1,4,5,3,2] => [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 0
[1,5,2,3,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,2,4,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,3,2,4] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,3,4,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,4,2,3] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[2,1,3,4,5] => [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> 0
[2,1,3,5,4] => [1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> 1
[2,1,4,3,5] => [1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> 0
[5,1,2,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,2,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,3,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,3,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,4,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,1,3,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,3,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,3,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2}
[6,1,2,3,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,3,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,4,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,4,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,5,3,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,5,4,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,2,4,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,2,5,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,4,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,4,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,5,2,4] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,5,4,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,2,3,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,2,5,3] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,3,2,5] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,3,5,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
Number of simple modules that are 3-regular in the corresponding Nakayama algebra.
Matching statistic: St001940
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001940: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1,0]
=> []
=> ? = 0
[1,2] => [1,2] => [1,0,1,0]
=> [1]
=> 1
[2,1] => [2,1] => [1,1,0,0]
=> []
=> ? = 0
[1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> [1,1]
=> 0
[2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> [2]
=> 0
[2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> [1]
=> 1
[3,1,2] => [3,1,2] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1}
[3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1}
[1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,4,2,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 0
[2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [2,2]
=> 1
[2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,4,1,3] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> [1,1]
=> 0
[3,1,2,4] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,1,4,2] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,4,1,2] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[3,4,2,1] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> [1]
=> 1
[4,1,2,3] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,1,3,2] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,2,1,3] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,3,1,2] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,2,2}
[1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,5,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,5,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,2,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,2,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,5,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,4,5,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,2,3,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,2,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,3,2,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,4,2,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 0
[2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
[2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
[2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> 0
[5,1,2,3,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,2,4,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,3,2,4] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,3,4,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,4,2,3] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,1,4,3,2] => [5,1,4,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,1,3,4] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,1,4,3] => [5,2,1,4,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,3,1,4] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,4,1,3] => [5,2,4,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,1,2,4] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,1,4,2] => [5,3,1,4,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,2,1,4] => [5,3,2,1,4] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,4,1,2] => [5,3,4,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,1,2,3] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,1,3,2] => [5,4,1,3,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,2,1,3] => [5,4,2,1,3] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,2,3,1] => [5,4,2,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,3,1,2] => [5,4,3,1,2] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,2,2,2,2,2}
[6,1,2,3,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,3,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,4,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,4,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,5,3,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,2,5,4,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,2,4,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,2,5,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,4,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,4,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,5,2,4] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,3,5,4,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,2,3,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,2,5,3] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,3,2,5] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
[6,1,4,3,5,2] => [6,1,5,4,3,2] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,3,3}
Description
The number of distinct parts that are equal to their multiplicity in the integer partition.
The following 63 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000929The constant term of the character polynomial of an integer partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000478Another weight of a partition according to Alladi. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000225Difference between largest and smallest parts in a partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000749The smallest integer d such that the restriction of the representation corresponding to a partition of n to the symmetric group on n-d letters has a constituent of odd degree. St000944The 3-degree of an integer partition. St001280The number of parts of an integer partition that are at least two. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St001587Half of the largest even part of an integer partition. St001657The number of twos in an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St001498The normalised height of a Nakayama algebra with magnitude 1. St001570The minimal number of edges to add to make a graph Hamiltonian. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001541The Gini index of an integer partition. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St000455The second largest eigenvalue of a graph if it is integral. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St000941The number of characters of the symmetric group whose value on the partition is even. St001651The Frankl number of a lattice. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000661The number of rises of length 3 of a Dyck path. St000791The number of pairs of left tunnels, one strictly containing the other, of a Dyck path. St000931The number of occurrences of the pattern UUU in a Dyck path. St000980The number of boxes weakly below the path and above the diagonal that lie below at least two peaks. St001826The maximal number of leaves on a vertex of a graph. St000699The toughness times the least common multiple of 1,. St000781The number of proper colouring schemes of a Ferrers diagram. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000936The number of even values of the symmetric group character corresponding to the partition. St001525The number of symmetric hooks on the diagonal of a partition. St000506The number of standard desarrangement tableaux of shape equal to the given partition. St001440The number of standard Young tableaux whose major index is congruent one modulo the size of a given integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St000260The radius of a connected graph. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001249Sum of the odd parts of a partition. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001383The BG-rank of an integer partition. St001561The value of the elementary symmetric function evaluated at 1. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000454The largest eigenvalue of a graph if it is integral. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001621The number of atoms of a lattice.
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