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Your data matches 259 different statistics following compositions of up to 3 maps.
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Matching statistic: St001906
St001906: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => 0
[1,2] => 0
[2,1] => 0
[1,2,3] => 0
[1,3,2] => 0
[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] => 0
[1,3,2,4] => 0
[1,3,4,2] => 0
[1,4,2,3] => 0
[1,4,3,2] => 0
[2,1,3,4] => 0
[2,1,4,3] => 0
[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] => 1
[3,4,2,1] => 0
[4,1,2,3] => 0
[4,1,3,2] => 0
[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] => 0
[1,2,4,3,5] => 0
[1,2,4,5,3] => 0
[1,2,5,3,4] => 0
[1,2,5,4,3] => 0
[1,3,2,4,5] => 0
[1,3,2,5,4] => 0
[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] => 1
Description
Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation.
Let $\pi$ be a permutation. Its total displacement [[St000830]] is $D(\pi) = \sum_i |\pi(i) - i|$, and its absolute length [[St000216]] is the minimal number $T(\pi)$ of transpositions whose product is $\pi$. Finally, let $I(\pi)$ be the number of inversions [[St000018]] of $\pi$.
This statistic equals $\left(D(\pi)-T(\pi)-I(\pi)\right)/2$.
Diaconis and Graham [1] proved that this statistic is always nonnegative.
Matching statistic: St000940
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 92%●distinct values known / distinct values provided: 50%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000940: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 92%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0
[1,2] => [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [1,1]
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3]
=> [3]
=> []
=> ? ∊ {0,0}
[1,3,2] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,3,1] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,2] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1] => [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0}
[1,2,3,4] => [4]
=> [2,2]
=> [2]
=> 0
[1,2,4,3] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,3,1,4] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,2,4] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,4] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,4,2,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,3] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,2,3,4,5] => [5]
=> [5]
=> []
=> ? ∊ {0,1}
[1,2,3,5,4] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,4,3,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,4,5,3] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,5,3,4] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,3,2,4,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,2,5,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,2,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,5,2,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,2,3,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,4,2,5,3] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,5,2,3] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,2,3,4] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,1,3,5,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,4,3,5] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,4,5,3] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,5,3,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,3,1,4,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,4,1,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,1}
[1,2,3,4,6,5] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,5,4,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,5,6,4] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,6,4,5] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,3,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,5,3,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,5,6,3] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,5,3,4,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,6,3,4,5] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,2,4,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,2,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,5,2,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,5,6,2] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,2,3,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,5,2,3,4,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,6,2,3,4,5] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,3,4,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,4,3,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,4,6,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,5,3,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,5,6,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,1,4,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,1,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,5,1,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,5,6,1] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,3,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,6,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,1,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,6,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,6,1,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,2,4,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,4,2,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of characters of the symmetric group whose value on the partition is zero.
The maximal value for any given size is recorded in [2].
Matching statistic: St001124
(load all 26 compositions to match this statistic)
(load all 26 compositions to match this statistic)
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 92%●distinct values known / distinct values provided: 50%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001124: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 92%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0
[1,2] => [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [1,1]
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [3]
=> [3]
=> []
=> ? ∊ {0,0}
[1,3,2] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,1,3] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[2,3,1] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,1,2] => [2,1]
=> [1,1,1]
=> [1,1]
=> 0
[3,2,1] => [1,1,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0}
[1,2,3,4] => [4]
=> [2,2]
=> [2]
=> 0
[1,2,4,3] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,3,1,4] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,2,4] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,4] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [2,2]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[3,4,2,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [3,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,3] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [2,1,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,1}
[1,2,3,4,5] => [5]
=> [5]
=> []
=> ? ∊ {0,1}
[1,2,3,5,4] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,4,3,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,4,5,3] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,5,3,4] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,2,5,4,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,3,2,4,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,2,5,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,2,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,4,5,2] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,3,5,2,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,2,3,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,4,2,5,3] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,4,5,2,3] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,2,3,4] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[1,5,2,4,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [3,2]
=> [2]
=> 0
[1,5,4,3,2] => [2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,4,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,1,3,5,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,4,3,5] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,4,5,3] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,5,3,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,5,4,3] => [2,2,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,3,1,4,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,1,5,4] => [3,2]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,4,1,5] => [4,1]
=> [2,2,1]
=> [2,1]
=> 1
[5,4,3,2,1] => [1,1,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,1}
[1,2,3,4,6,5] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,5,4,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,5,6,4] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,3,6,4,5] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,3,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,5,3,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,4,5,6,3] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,5,3,4,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,2,6,3,4,5] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,2,4,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,2,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,5,2,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,3,4,5,6,2] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,4,2,3,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,5,2,3,4,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[1,6,2,3,4,5] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,3,4,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,4,3,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,4,6,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,5,3,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,1,5,6,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,1,4,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,1,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,5,1,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,5,6,1] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,3,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,6,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,1,3,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,6,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,6,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,6,1,3,4] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,2,4,5,6] => [5,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[3,1,4,2,6,5] => [3,3]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The multiplicity of the standard 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 $g_{\lambda,\lambda}^{(n-1)1}$, for $\lambda\vdash n > 1$. For $n\leq1$ the statistic is undefined.
It follows from [3, Prop.4.1] (or, slightly easier from [3, Thm.4.2]) that this is one less than [[St000159]], the number of distinct parts of the partition.
Matching statistic: St000938
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 88%●distinct values known / distinct values provided: 75%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000938: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 88%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [2]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0}
[1,3,2] => [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [3]
=> [1,1,1]
=> [1,1]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,2]
=> [2,2]
=> [2]
=> 0
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [2,2]
=> [2,2]
=> [2]
=> 0
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [5]
=> []
=> ? ∊ {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}
[1,2,3,5,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,2,4,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,2,4,5,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,2,5,3,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,2,5,4,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,3,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[1,3,4,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,3,4,5,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,3,5,2,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,3,5,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,4,2,5,3] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[1,4,3,2,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,4,5,3,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,5,2,4,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,4,3,2] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[2,1,3,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[2,1,4,5,3] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[2,1,5,3,4] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[2,1,5,4,3] => [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[2,3,1,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[2,3,1,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[2,3,4,1,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[2,3,4,5,1] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[2,3,5,1,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[2,3,5,4,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,1,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[2,4,1,5,3] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[2,4,3,1,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,5,1,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[2,4,5,3,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,1,3,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[2,5,1,4,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,4,1,3] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,4,3,1] => [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,1,2,4,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[3,1,2,5,4] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[3,1,4,2,5] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[3,1,4,5,2] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[3,1,5,2,4] => [2,2,1]
=> [3,2]
=> [2]
=> 0
[3,1,5,4,2] => [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[3,2,1,4,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[3,2,1,5,4] => [3,2]
=> [2,2,1]
=> [2,1]
=> 1
[3,2,4,1,5] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[3,2,4,5,1] => [3,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[3,4,1,2,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[3,4,5,1,2] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[3,5,1,2,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[4,1,2,3,5] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[4,5,1,2,3] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[5,1,2,3,4] => [2,1,1,1]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1}
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3}
Description
The number of zeros of the symmetric group character corresponding to the partition.
For example, the character values of the irreducible representation $S^{(2,2)}$ are $2$ on the conjugacy classes $(4)$ and $(2,2)$, $0$ on the conjugacy classes $(3,1)$ and $(1,1,1,1)$, and $-1$ on the conjugacy class $(2,1,1)$. Therefore, the statistic on the partition $(2,2)$ is $2$.
Matching statistic: St000934
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Mp00321: Integer partitions —2-conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000934: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 86%●distinct values known / distinct values provided: 75%
Values
[1] => [1]
=> [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2]
=> [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0}
[2,1,3] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0}
[2,3,1] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0}
[3,1,2] => [2,1]
=> [3]
=> []
=> ? ∊ {0,0,0,0,0}
[3,2,1] => [3]
=> [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,4,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[1,3,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[1,3,4,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[1,4,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[1,4,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,1,4,3] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,3,1,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,3,4,1] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,4,1,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[2,4,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,1,2,4] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[3,1,4,2] => [2,2]
=> [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[3,2,1,4] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[3,4,2,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [2,1,1]
=> [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0}
[4,1,3,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,1,3] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,2,3,1] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [3,1]
=> [2,1,1]
=> [1,1]
=> 0
[4,3,2,1] => [4]
=> [2,2]
=> [2]
=> 1
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,4,5,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[1,3,4,2,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,2,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[1,4,3,2,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,2,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,4,5,3,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [4,1]
=> [3,2]
=> [2]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[2,1,4,3,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[2,1,4,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[2,1,5,3,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[2,1,5,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,1,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[2,3,4,1,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,4,5,1] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,1,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,3,5,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,3,5] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,1,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[2,4,3,1,5] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,5,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,5,1,3] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,4,5,3,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,1,3,4] => [2,1,1,1]
=> [3,1,1]
=> [1,1]
=> 0
[2,5,1,4,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,1,4] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,3,4,1] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[2,5,4,1,3] => [3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> 0
[3,1,2,5,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[3,1,4,2,5] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[3,1,4,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[3,1,5,2,4] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[3,1,5,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,1,5,4] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,5,4,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,1,5,2] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[4,1,2,5,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[4,1,5,2,3] => [2,2,1]
=> [5]
=> []
=> ? ∊ {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}
[4,1,5,3,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,2,1,5,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,2,5,3,1] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,3,1,5,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[5,2,1,4,3] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[5,3,1,4,2] => [3,2]
=> [4,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,3}
[1,2,5,3,6,4] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,3}
[1,3,2,4,6,5] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,3}
[1,3,2,5,4,6] => [2,2,1,1]
=> [5,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,2,2,2,3}
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: St001604
(load all 38 compositions to match this statistic)
(load all 38 compositions to match this statistic)
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001604: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 85%●distinct values known / distinct values provided: 50%
Values
[1] => [1] => [1]
=> []
=> ? = 0
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[2,1] => [2,1] => [1,1]
=> [1]
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,0,0}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0}
[2,3,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0}
[3,1,2] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0}
[3,2,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,3,4,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[1,4,3,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,3,4] => [2,1,3,4] => [3,1]
=> [1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,3,4,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,1,3] => [3,4,1,2] => [2,2]
=> [2]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[2,4,3,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[3,1,2,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,1,4,2] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,1,4] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,2,4,1] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[3,4,1,2] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[3,4,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[4,1,2,3] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,1,3,2] => [4,2,3,1] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1}
[4,2,1,3] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[4,2,3,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,1,2] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[4,3,2,1] => [4,3,2,1] => [1,1,1,1]
=> [1,1,1]
=> 0
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {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}
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {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}
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {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}
[1,2,4,5,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,2,5,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,2,5,4,3] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {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}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> ? ∊ {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}
[1,3,4,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,3,4,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,3,5,2,4] => [1,4,5,2,3] => [3,2]
=> [2]
=> ? ∊ {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}
[1,3,5,4,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,4,2,5,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,4,3,2,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,4,3,5,2] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,4,5,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,5,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,2,3,4] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,5,2,4,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[1,5,3,2,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,3,4,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,3,2] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,4,5] => [2,1,3,4,5] => [4,1]
=> [1]
=> ? ∊ {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}
[2,1,3,5,4] => [2,1,3,5,4] => [3,2]
=> [2]
=> ? ∊ {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}
[2,1,4,3,5] => [2,1,4,3,5] => [3,2]
=> [2]
=> ? ∊ {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}
[2,1,4,5,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,3,4] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,1,5,4,3] => [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 0
[2,3,1,4,5] => [3,2,1,4,5] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[2,3,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[2,3,4,1,5] => [4,2,3,1,5] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[2,3,4,5,1] => [5,2,3,4,1] => [3,1,1]
=> [1,1]
=> ? ∊ {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}
[2,3,5,1,4] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 0
[2,3,5,4,1] => [5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,1,3,5] => [3,4,1,2,5] => [3,2]
=> [2]
=> ? ∊ {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}
[2,4,1,5,3] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0
[2,4,3,1,5] => [4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,3,5,1] => [5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,4,5,1,3] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0
[2,4,5,3,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,5,1,3,4] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0
[2,5,1,4,3] => [3,5,1,4,2] => [2,2,1]
=> [2,1]
=> 0
[2,5,3,1,4] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0
[2,5,3,4,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[2,5,4,1,3] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0
[2,5,4,3,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,1,2,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,1,5,2,4] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 0
[3,1,5,4,2] => [5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,2,1,5,4] => [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 0
[3,2,5,1,4] => [4,2,5,1,3] => [2,2,1]
=> [2,1]
=> 0
[3,2,5,4,1] => [5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,4,1,2,5] => [4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,4,1,5,2] => [5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,4,2,1,5] => [4,3,2,1,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,4,2,5,1] => [5,3,2,4,1] => [2,1,1,1]
=> [1,1,1]
=> 0
[3,4,5,1,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,4,5,2,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,5,1,2,4] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0
[3,5,1,4,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,5,2,1,4] => [4,5,3,1,2] => [2,2,1]
=> [2,1]
=> 0
[3,5,2,4,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,5,4,1,2] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[3,5,4,2,1] => [5,4,3,2,1] => [1,1,1,1,1]
=> [1,1,1,1]
=> 0
[4,1,5,2,3] => [5,2,4,3,1] => [2,1,1,1]
=> [1,1,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons.
Equivalently, this is the multiplicity of the irreducible representation corresponding to a partition in the cycle index of the dihedral group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000225
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000225: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2]
=> []
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,2,1] => [2,1]
=> [1]
=> 0
[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] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,3,1] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,3,2] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,2,1] => [2,2]
=> [2]
=> 0
[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] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [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]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [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] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,6,4,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,6,5,1,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,3,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,6,1,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,3,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,5,3,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,6,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,6,3,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
Description
Difference between largest and smallest parts in a partition.
Matching statistic: St000944
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000944: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> 0
[2,1] => [2]
=> []
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> 0
[2,1,3] => [2,1]
=> [1]
=> 0
[2,3,1] => [3]
=> []
=> ? ∊ {0,0}
[3,1,2] => [3]
=> []
=> ? ∊ {0,0}
[3,2,1] => [2,1]
=> [1]
=> 0
[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] => [3,1]
=> [1]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> 0
[2,3,4,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,3,1] => [3,1]
=> [1]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> 0
[3,1,4,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> 0
[3,4,2,1] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,3,2] => [3,1]
=> [1]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> 0
[4,3,1,2] => [4]
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,2,1] => [2,2]
=> [2]
=> 0
[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] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [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]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> 0
[1,5,2,3,4] => [4,1]
=> [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] => [2,1,1,1]
=> [1,1,1]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[2,3,4,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,5,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,5,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,5,1,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,5,4,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,5,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,2,5,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,5,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,5,2,1,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,5,4,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,2,5,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,5,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,1,5,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,5,2,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,5,1,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,5,2,3,1] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,2,3,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,4,2,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,3,1,2,4] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,3,4,1,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,4,1,3,2] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,4,2,1,3] => [5]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,6,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,1,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,6,4,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,6,1,4,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,6,5,1,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,5,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,6,3,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,3,6,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,6,1,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,3,1,5] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,5,3,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,6,4] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,6,4,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,1,6,3] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,6,3,1] => [6]
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The 3-degree of an integer partition.
For an integer partition $\lambda$, this is given by the exponent of 3 in the Gram determinant of the integal Specht module of the symmetric group indexed by $\lambda$.
This stupid comment should not be accepted as an edit!
Matching statistic: St000117
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000117: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> []
=> ? = 0
[1,2] => [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1] => [2]
=> []
=> []
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,2] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,3] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,3,1] => [3]
=> []
=> []
=> ? ∊ {0,0}
[3,1,2] => [3]
=> []
=> []
=> ? ∊ {0,0}
[3,2,1] => [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,3,4,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,1,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[2,4,3,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,4,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,4,2,1] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,2,3] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,1,3,2] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,3,1,2] => [4]
=> []
=> []
=> ? ∊ {0,0,0,0,0,1}
[4,3,2,1] => [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 0
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,4,5,3,2] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[1,5,4,2,3] => [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 0
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[2,3,4,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,5,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,1,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,4,5,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,5,1,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,5,4,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,4,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,1,5,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,2,5,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,4,5,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,5,2,1,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[3,5,4,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,2,5,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,1,5,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,1,5,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,3,5,2,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,5,1,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[4,5,2,3,1] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,2,3,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,1,4,2,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,3,1,2,4] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,3,4,1,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,4,1,3,2] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[5,4,2,1,3] => [5]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1}
[2,3,4,5,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,6,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,1,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,6,4,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,6,1,4,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,6,5,1,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,5,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,6,3,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,3,6,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,6,1,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,3,1,5] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,5,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,6,4] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,6,4,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,1,6,3] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,6,3,1] => [6]
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The number of centered tunnels of a Dyck path.
A tunnel is a pair (a,b) where a is the position of an open parenthesis and b is the position of the matching close parenthesis. If a+b==n then the tunnel is called centered.
Matching statistic: St000143
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000143: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 82%●distinct values known / distinct values provided: 50%
Values
[1] => [1]
=> []
=> ?
=> ? = 0
[1,2] => [1,1]
=> [1]
=> []
=> 0
[2,1] => [2]
=> []
=> ?
=> ? = 0
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2] => [2,1]
=> [1]
=> []
=> 0
[2,1,3] => [2,1]
=> [1]
=> []
=> 0
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0}
[3,2,1] => [2,1]
=> [1]
=> []
=> 0
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,4,2] => [3,1]
=> [1]
=> []
=> 0
[1,4,2,3] => [3,1]
=> [1]
=> []
=> 0
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[2,1,4,3] => [2,2]
=> [2]
=> []
=> 0
[2,3,1,4] => [3,1]
=> [1]
=> []
=> 0
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> 0
[3,1,2,4] => [3,1]
=> [1]
=> []
=> 0
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[3,2,4,1] => [3,1]
=> [1]
=> []
=> 0
[3,4,1,2] => [2,2]
=> [2]
=> []
=> 0
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> 0
[4,2,1,3] => [3,1]
=> [1]
=> []
=> 0
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> 0
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> 0
[1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,5,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,2,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> 0
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> 0
[1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,5,2] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> 0
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> 0
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> 0
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[2,1,3,5,4] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> 0
[2,3,4,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,5,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,1,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,4,5,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,5,1,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,5,4,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,4,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,1,5,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,2,5,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,4,5,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,5,2,1,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[3,5,4,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,1,2,5,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,1,5,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,3,1,5,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,3,5,2,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,5,1,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[4,5,2,3,1] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[5,1,2,3,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[5,1,4,2,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[5,3,1,2,4] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[5,3,4,1,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[5,4,1,3,2] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[5,4,2,1,3] => [5]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1}
[2,3,4,5,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,4,6,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,1,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,5,6,4,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,6,1,4,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,3,6,5,1,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,5,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,1,6,3,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,3,6,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,5,6,1,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,3,1,5] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,4,6,5,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,3,6,4] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,1,6,4,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,1,6,3] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
[2,5,4,6,3,1] => [6]
=> []
=> ?
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,3}
Description
The largest repeated part of a partition.
If the parts of the partition are all distinct, the value of the statistic is defined to be zero.
The following 249 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000150The floored half-sum of the multiplicities of a partition. St000175Degree of the polynomial counting the number of semistandard Young tableaux when stretching the shape. St000257The number of distinct parts of a partition that occur at least twice. St000292The number of ascents of a binary word. St000481The number of upper covers of a partition in dominance order. St000547The number of even non-empty partial sums of an integer partition. St000628The balance of a binary word. St000691The number of changes of a binary word. St000966Number of peaks minus the global dimension of the corresponding LNakayama algebra. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum in the Nakayama algebra corresponding to the Dyck path. St001036The number of inner corners of the parallelogram polyomino associated with the Dyck path. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001423The number of distinct cubes in a binary word. St001524The degree of symmetry of a binary word. St001584The area statistic between a Dyck path and its bounce path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001695The natural comajor index of a standard Young tableau. St001712The number of natural descents of a standard Young tableau. St001730The number of times the path corresponding to a binary word crosses the base line. 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. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001280The number of parts of an integer partition that are at least two. 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. 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. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the 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. St000455The second largest eigenvalue of a graph if it is integral. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St000379The number of Hamiltonian cycles in a graph. St001393The induced matching number of a graph. St001261The Castelnuovo-Mumford regularity of a graph. St000929The constant term of the character polynomial of an integer partition. St001651The Frankl number of a lattice. 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. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001199The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000031The number of cycles in the cycle decomposition of a permutation. St001498The normalised height of a Nakayama algebra with magnitude 1. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001964The interval resolution global dimension 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. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000478Another weight of a partition according to Alladi. St000376The bounce deficit of a Dyck path. 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. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001502The global dimension minus the dominant dimension of magnitude 1 Nakayama algebras. St001570The minimal number of edges to add to make a graph Hamiltonian. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St000370The genus of a graph. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001793The difference between the clique number and the chromatic number of a graph. St001877Number of indecomposable injective modules with projective dimension 2. 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. St001875The number of simple modules with projective dimension at most 1. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001175The size of a partition minus the hook length of the base cell. St001307The number of induced stars on four vertices in a graph. St000699The toughness times the least common multiple of 1,. St000567The sum of the products of all pairs of parts. St000936The number of even values of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St000781The number of proper colouring schemes of a Ferrers diagram. St000322The skewness of a graph. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. 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. St001714The number of subpartitions of an integer partition that do not dominate the conjugate subpartition. St001178Twelve times the variance of the major index among all standard Young tableaux of a partition. St001541The Gini index of an integer partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001578The minimal number of edges to add or remove to make a graph a line graph. St001577The minimal number of edges to add or remove to make a graph a cograph. St001871The number of triconnected components of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001371The length of the longest Yamanouchi prefix of a binary word. St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001490The number of connected components of a skew partition. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000068The number of minimal elements in a poset. St001330The hat guessing number of a graph. St000759The smallest missing part in an integer partition. St000897The number of different multiplicities of parts of an integer partition. St000752The Grundy value for the game 'Couples are forever' on an integer partition. St000475The number of parts equal to 1 in a partition. St000842The breadth of a permutation. St001301The first Betti number of the order complex associated with the poset. St000908The length of the shortest maximal antichain in a poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. 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. St000901The cube of the number of standard Young tableaux with shape given by 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. St001934The number of monotone factorisations of genus zero of a permutation of given cycle type. St000181The number of connected components of the Hasse diagram for the poset. St001568The smallest positive integer that does not appear twice in the partition. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000069The number of maximal elements of a poset. St001621The number of atoms of a lattice. St001846The number of elements which do not have a complement in the lattice. St001820The size of the image of the pop stack sorting operator. St001249Sum of the odd parts of a partition. St001383The BG-rank of an integer partition. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000256The number of parts from which one can substract 2 and still get an integer partition. St000629The defect of a binary word. St001868The number of alignments of type NE of a signed permutation. St000022The number of fixed points of a permutation. St000731The number of double exceedences of a permutation. 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. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St001862The number of crossings of a signed permutation. St001867The number of alignments of type EN of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St000632The jump number of the 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". St000298The order dimension or Dushnik-Miller dimension of a poset. St000307The number of rowmotion orbits of a poset. St001268The size of the largest ordinal summand in the poset. St001399The distinguishing number of a poset. St001510The number of self-evacuating linear extensions of a finite poset. St001533The largest coefficient of the Poincare polynomial of the poset cone. St001779The order of promotion on the set of linear extensions of a poset. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000323The minimal crossing number of a graph. St000351The determinant of the adjacency matrix of a graph. St000368The Altshuler-Steinberg determinant of a graph. St000403The Szeged index minus the Wiener index of a graph. St000449The number of pairs of vertices of a graph with distance 4. St000671The maximin edge-connectivity for choosing a subgraph. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001119The length of a shortest maximal path in a graph. St001271The competition number of a graph. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs in a graph. St001323The independence gap of a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001357The maximal degree of a regular spanning subgraph of a graph. St001395The number of strictly unfriendly partitions of a graph. St001702The absolute value of the determinant of the adjacency matrix of a graph. St001794Half the number of sets of vertices in a graph which are dominating and non-blocking. St001797The number of overfull subgraphs of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St001316The domatic number of a graph. St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001496The number of graphs with the same Laplacian spectrum as the given graph. St000447The number of pairs of vertices of a graph with distance 3. St000552The number of cut vertices of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000553The number of blocks of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001363The Euler characteristic of a graph according to Knill. St001739The number of graphs with the same edge polytope as the given graph. St001740The number of graphs with the same symmetric edge polytope as the given graph. St001776The degree of the minimal polynomial of the largest Laplacian eigenvalue of a graph. St001829The common independence number of a graph. St001281The normalized isoperimetric number of a graph. St001549The number of restricted non-inversions between exceedances. St001811The Castelnuovo-Mumford regularity of a permutation. St000805The number of peaks of the associated bargraph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St001768The number of reduced words of a signed permutation. St000264The girth of a graph, which is not a tree. St000095The number of triangles of a graph. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000274The number of perfect matchings of a graph. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by $4$. St000310The minimal degree of a vertex of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St001518The number of graphs with the same ordinary spectrum as the given graph. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001765The number of connected components of the friends and strangers graph.
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