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Your data matches 66 different statistics following compositions of up to 3 maps.
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Matching statistic: St000628
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00130: Permutations —descent tops⟶ Binary words
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000628: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,2] => 0 => 0
[2,1] => 1 => 0
[1,2,3] => 00 => 0
[1,3,2] => 01 => 1
[2,1,3] => 10 => 1
[2,3,1] => 01 => 1
[3,1,2] => 01 => 1
[3,2,1] => 11 => 0
[1,2,3,4] => 000 => 0
[1,2,4,3] => 001 => 1
[1,3,2,4] => 010 => 1
[1,3,4,2] => 001 => 1
[1,4,2,3] => 001 => 1
[1,4,3,2] => 011 => 1
[2,1,3,4] => 100 => 1
[2,1,4,3] => 101 => 1
[2,3,1,4] => 010 => 1
[2,3,4,1] => 001 => 1
[2,4,1,3] => 001 => 1
[2,4,3,1] => 011 => 1
[3,1,2,4] => 010 => 1
[3,1,4,2] => 011 => 1
[3,2,1,4] => 110 => 1
[3,2,4,1] => 011 => 1
[3,4,1,2] => 001 => 1
[3,4,2,1] => 101 => 1
[4,1,2,3] => 001 => 1
[4,1,3,2] => 011 => 1
[4,2,1,3] => 101 => 1
[4,2,3,1] => 011 => 1
[4,3,1,2] => 011 => 1
[4,3,2,1] => 111 => 0
[1,2,3,4,5] => 0000 => 0
[1,2,3,5,4] => 0001 => 1
[1,2,4,3,5] => 0010 => 1
[1,2,4,5,3] => 0001 => 1
[1,2,5,3,4] => 0001 => 1
[1,2,5,4,3] => 0011 => 2
[1,3,2,4,5] => 0100 => 1
[1,3,2,5,4] => 0101 => 1
[1,3,4,2,5] => 0010 => 1
[1,3,4,5,2] => 0001 => 1
[1,3,5,2,4] => 0001 => 1
[1,3,5,4,2] => 0011 => 2
[1,4,2,3,5] => 0010 => 1
[1,4,2,5,3] => 0011 => 2
[1,4,3,2,5] => 0110 => 1
[1,4,3,5,2] => 0011 => 2
[1,4,5,2,3] => 0001 => 1
[1,4,5,3,2] => 0101 => 1
Description
The balance of a binary word.
The balance of a word is the smallest number $q$ such that the word is $q$-balanced [1].
A binary word $w$ is $q$-balanced if for any two factors $u$, $v$ of $w$ of the same length, the difference between the number of ones in $u$ and $v$ is at most $q$.
Matching statistic: St000668
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000668: Integer partitions ⟶ ℤResult quality: 75% ●values known / values provided: 84%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,1,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[4,2,3,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,5,1,3,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,1,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,4,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,1,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,4,5,2,1] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[3,5,1,2,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,1,4,2] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,1,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,5,2,4,1] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,2,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,1,2,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,1,3,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,1,5,3,2] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,3,5] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,2,1,5,3] => [1,4,5,3,2] => [3,1,1]
=> [1,1]
=> 1
[4,2,3,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,2,5,3,1] => [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[4,3,1,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,3,2,5,1] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[4,5,1,2,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,1,3,2] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,2,1,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,2,3,1] => [1,4,3,2,5] => [3,1,1]
=> [1,1]
=> 1
[4,5,3,1,2] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,5,3,2,1] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,1,2,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,1,3,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,1,4,2,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,2,1,3,4] => [1,5,4,3,2] => [2,1,1,1]
=> [1,1,1]
=> 1
[5,2,1,4,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,2,3,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,2,4,1,3] => [1,5,3,4,2] => [3,1,1]
=> [1,1]
=> 1
[5,3,1,2,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,3,2,1,4] => [1,5,4,2,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,2,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,1,3,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,1,3] => [1,5,3,2,4] => [3,1,1]
=> [1,1]
=> 1
[5,4,2,3,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,1,2] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[5,4,3,2,1] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,2,6,3,4,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,3,6,5,4] => [4,1,1]
=> [1,1]
=> 1
Description
The least common multiple of the parts of the partition.
Matching statistic: St000708
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000708: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> [2]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> [2]
=> 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [4,2]
=> [2]
=> 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 2
Description
The product of the parts of an integer partition.
Matching statistic: St000933
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00223: Permutations —runsort⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000933: Integer partitions ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[2,1] => [1,2] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [1,3,4,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,4,2,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,2,4,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,3,2,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [5]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,5,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,4,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,5,3] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,2,5,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,3,4,2,5] => [1,3,4,2,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,3,5,4,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[1,4,2,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[1,4,3,5,2] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,4,5,3,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[1,5,2,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[1,5,3,4,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [1,3,4,5,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,1,4,3,5] => [1,4,2,3,5] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,1,5,3,4] => [1,5,2,3,4] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2}
[2,3,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[2,4,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[2,4,1,5,3] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,4,3,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[2,5,1,4,3] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,3,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[2,5,4,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[3,1,4,2,5] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[3,1,4,5,2] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,1,5,2,4] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,1,4,5] => [1,4,5,2,3] => [3,2]
=> [2]
=> 2
[3,2,4,1,5] => [1,5,2,4,3] => [3,1,1]
=> [1,1]
=> 1
[3,2,5,1,4] => [1,4,2,5,3] => [3,2]
=> [2]
=> 2
[4,1,3,2,5] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[4,1,3,5,2] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,1,3,5] => [1,3,5,2,4] => [3,2]
=> [2]
=> 2
[4,2,5,1,3] => [1,3,2,5,4] => [3,2]
=> [2]
=> 2
[1,2,4,3,6,5] => [1,2,4,3,6,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,3,5] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,4,6,5,3] => [1,2,4,6,3,5] => [4,2]
=> [2]
=> 2
[1,2,5,3,6,4] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,4,3,6] => [1,2,5,3,6,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,3,4] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,5,6,4,3] => [1,2,5,6,3,4] => [4,2]
=> [2]
=> 2
[1,2,6,3,5,4] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [1,2,6,3,5,4] => [4,1,1]
=> [1,1]
=> 1
[1,3,2,4,6,5] => [1,3,2,4,6,5] => [4,2]
=> [2]
=> 2
[1,3,2,5,4,6] => [1,3,2,5,4,6] => [4,2]
=> [2]
=> 2
[1,3,2,5,6,4] => [1,3,2,5,6,4] => [4,2]
=> [2]
=> 2
[1,3,2,6,4,5] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,2,6,5,4] => [1,3,2,6,4,5] => [4,2]
=> [2]
=> 2
[1,3,4,2,6,5] => [1,3,4,2,6,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,2,5] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,4,6,5,2] => [1,3,4,6,2,5] => [4,2]
=> [2]
=> 2
[1,3,5,2,4,6] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,2,6,4] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,2,6] => [1,3,5,2,6,4] => [4,2]
=> [2]
=> 2
[1,3,5,4,6,2] => [1,3,5,2,4,6] => [4,2]
=> [2]
=> 2
[1,3,5,6,2,4] => [1,3,5,6,2,4] => [4,2]
=> [2]
=> 2
Description
The number of multipartitions of sizes given by an integer partition.
This is, for $\lambda = (\lambda_1,\ldots,\lambda_n)$, this is the number of $n$-tuples $(\lambda^{(1)},\ldots,\lambda^{(n)})$ of partitions $\lambda^{(i)}$ such that $\lambda^{(i)} \vdash \lambda_i$.
Matching statistic: St000259
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Mp00159: Permutations —Demazure product with inverse⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000259: Graphs ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 100%
Values
[1,2] => [1] => [1] => ([],1)
=> 0
[2,1] => [1] => [1] => ([],1)
=> 0
[1,2,3] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1}
[1,3,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1}
[2,1,3] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[2,3,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[3,1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1}
[3,2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[1,2,3,4] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,3,4,1] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[2,4,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,2,4] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,1,4,2] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,1,4] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,2,4,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,4,1,2] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[3,4,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [1,3,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,3,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3,1,2] => [3,1,2] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[4,3,2,1] => [3,2,1] => [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 1
[1,2,3,4,5] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,3,5,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,4,3,5] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,4,5,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,5,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,2,5,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2,4,5] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,2,5,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,4,2,5] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,4,5,2] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,5,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,3,5,4,2] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,4,2,3,5] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,4,2,5,3] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,4,3,2,5] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,4,3,5,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,4,5,2,3] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,4,5,3,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,5,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,5,2,4,3] => [1,2,4,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,5,3,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,5,3,4,2] => [1,3,4,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,5,4,2,3] => [1,4,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[1,5,4,3,2] => [1,4,3,2] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,3,4,5] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,3,5,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,4,3,5] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,4,5,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,5,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,1,5,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,1,4,5] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,1,5,4] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,4,1,5] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,4,5,1] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,5,1,4] => [2,3,1,4] => [3,2,1,4] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,3,5,4,1] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,4,1,3,5] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,1,5,3] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,3,1,5] => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,3,5,1] => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,4,5,1,3] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,4,5,3,1] => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[2,5,1,3,4] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,1,4,3] => [2,1,4,3] => [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2}
[2,5,3,4,1] => [2,3,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,5,4,1,3] => [2,4,1,3] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[2,5,4,3,1] => [2,4,3,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,1,4,2,5] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,4,5,2] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,1,5,4,2] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1,5] => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,4,5,1] => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,2,5,4,1] => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,4,1,2,5] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,1,5,2] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,2,1,5] => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,2,5,1] => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,5,1,2] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,4,5,2,1] => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,5,1,4,2] => [3,1,4,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,5,2,4,1] => [3,2,4,1] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,5,4,1,2] => [3,4,1,2] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[3,5,4,2,1] => [3,4,2,1] => [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,2,3,5] => [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,2,5,3] => [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,2,5] => [4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,3,5,2] => [4,1,3,2] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,1,5,2,3] => [4,1,2,3] => [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
Description
The diameter of a connected graph.
This is the greatest distance between any pair of vertices.
Matching statistic: St001878
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00061: Permutations —to increasing tree⟶ Binary trees
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Mp00013: Binary trees —to poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001878: Lattices ⟶ ℤResult quality: 50% ●values known / values provided: 53%●distinct values known / distinct values provided: 50%
Values
[1,2] => [.,[.,.]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0}
[2,1] => [[.,.],.]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0}
[1,2,3] => [.,[.,[.,.]]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2] => [.,[[.,.],.]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,0}
[2,3,1] => [[.,[.,.]],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2] => [[.,.],[.,.]]
=> ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,0}
[3,2,1] => [[[.,.],.],.]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4] => [.,[.,[.,[.,.]]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,4,3] => [.,[.,[[.,.],.]]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,3,2,4] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [.,[[.,[.,.]],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,4,2,3] => [.,[[.,.],[.,.]]]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [.,[[[.,.],.],.]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,3,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [[.,[.,[.,.]]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[2,4,1,3] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [[.,[[.,.],.]],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,4] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [[.,[.,.]],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [[[.,[.,.]],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,2,3] => [[.,.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [[.,.],[[.,.],.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [[[.,.],[.,.]],.]
=> ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [[[.,.],.],[.,.]]
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [[[[.,.],.],.],.]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 1
[1,2,3,4,5] => [.,[.,[.,[.,[.,.]]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,5,4] => [.,[.,[.,[[.,.],.]]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,4,3,5] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [.,[.,[[.,[.,.]],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,5,3,4] => [.,[.,[[.,.],[.,.]]]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [.,[.,[[[.,.],.],.]]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,2,4,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [.,[[.,[.,[.,.]]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,5,2,4] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [.,[[.,[[.,.],.]],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,4,2,3,5] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [.,[[.,[.,.]],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [.,[[[.,[.,.]],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,5,2,3,4] => [.,[[.,.],[.,[.,.]]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [.,[[.,.],[[.,.],.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [.,[[[.,.],[.,.]],.]]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [.,[[[.,.],.],[.,.]]]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [.,[[[[.,.],.],.],.]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,1,3,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,3,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,4,3,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,1,5,3,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,1,5,4,3] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,1,4,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,1,5,4] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,3,4,1,5] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [[.,[.,[.,[.,.]]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,3,5,1,4] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,4,1] => [[.,[.,[[.,.],.]]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,4,1,3,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,1,5,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,4,3,1,5] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,3,5,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,4,5,1,3] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,3,1] => [[.,[[.,[.,.]],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[2,5,1,3,4] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,1,4,3] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[2,5,3,1,4] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,3,4,1] => [[.,[[.,.],[.,.]]],.]
=> ([(0,4),(1,4),(2,3),(4,2)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[2,5,4,1,3] => [[.,[[.,.],.]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,3,1] => [[.,[[[.,.],.],.]],.]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[3,1,2,4,5] => [[.,.],[.,[.,[.,.]]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,2,5,4] => [[.,.],[.,[[.,.],.]]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,4,5,2] => [[.,.],[[.,[.,.]],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,2,4] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,1,5,4,2] => [[.,.],[[[.,.],.],.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,2,1,4,5] => [[[.,.],.],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,1,5,4] => [[[.,.],.],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,2,4,1,5] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,4,5,1] => [[[.,.],[.,[.,.]]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,5,1,4] => [[[.,.],[.,.]],[.,.]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,2,5,4,1] => [[[.,.],[[.,.],.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,1,2,5] => [[.,[.,.]],[.,[.,.]]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,5,2] => [[.,[.,.]],[[.,.],.]]
=> ([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,2,1,5] => [[[.,[.,.]],.],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,4,2,5,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[3,4,5,1,2] => [[.,[.,[.,.]]],[.,.]]
=> ([(0,4),(1,2),(2,3),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[3,5,2,4,1] => [[[.,[.,.]],[.,.]],.]
=> ([(0,4),(1,2),(2,4),(4,3)],5)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,3,2,5] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
[4,1,5,2,3] => [[.,.],[[.,.],[.,.]]]
=> ([(0,4),(1,3),(2,3),(3,4)],5)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2}
Description
The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L.
Matching statistic: St001630
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00252: Permutations —restriction⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Mp00065: Permutations —permutation poset⟶ Posets
Mp00206: Posets —antichains of maximal size⟶ Lattices
St001630: Lattices ⟶ ℤResult quality: 35% ●values known / values provided: 35%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1] => ([],1)
=> ([],1)
=> ? ∊ {0,0}
[2,1] => [1] => ([],1)
=> ([],1)
=> ? ∊ {0,0}
[1,2,3] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [1,2] => ([(0,1)],2)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [2,1] => ([],2)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,4,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,2,4] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,2,3] => ([(0,2),(2,1)],3)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,3,2] => [1,3,2] => ([(0,1),(0,2)],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [2,1,3] => ([(0,2),(1,2)],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,3,1] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [3,1,2] => ([(1,2)],3)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [3,2,1] => ([],3)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,2,3,5,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,2,4,3,5] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,2,5,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[1,5,2,4,3] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,3,2,4] => ([(0,1),(0,2),(1,3),(2,3)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> ([(0,1)],2)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,4,3,2] => ([(0,1),(0,2),(0,3)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [2,1,3,4] => ([(0,3),(1,3),(3,2)],4)
=> ([],1)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,4,5,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,3,5,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,3,5] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,1,5,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,4,5,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[2,5,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,2,5] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,4,5,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,1,5,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,4,1,2,5] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,1,5,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,4,5,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[3,5,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[3,5,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[4,1,2,3,5] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,2,5,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[4,1,5,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[4,5,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[5,1,2,3,4] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(2,1),(3,2)],4)
=> 2
[5,2,3,4,1] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[5,2,4,1,3] => [2,4,1,3] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[5,3,1,4,2] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[5,3,4,1,2] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
[5,4,1,2,3] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ([(0,2),(2,1)],3)
=> 1
[1,2,3,4,5,6] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,4,6,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,3,6,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,2,6,3,4,5] => [1,2,3,4,5] => ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> 1
[1,3,4,5,2,6] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,5,6,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,4,6,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,2,4,6] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,2,6,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,5,6,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,6,4,5,2] => [1,3,4,5,2] => ([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,3,6,5,2,4] => [1,3,5,2,4] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,5,3,6] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,5,6,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,2,6,5,3] => [1,4,2,5,3] => ([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(0,2),(2,1)],3)
=> 1
[1,4,5,2,3,6] => [1,4,5,2,3] => ([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
Matching statistic: St000260
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 75%
Mp00126: Permutations —cactus evacuation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000260: Graphs ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 75%
Values
[1,2] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
[2,1] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0}
[1,2,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1}
[1,3,2] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1}
[2,1,3] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1}
[2,3,1] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,1,1}
[3,1,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[3,2,1] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[1,2,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[1,4,3,2] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,3,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,4,3,1] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [1,3,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [1,3,4,2] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [1,3,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,1,3,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [1,4,3,2] => [4,3,1,2] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
[4,2,3,1] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,2,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,2,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,3,5,4,2] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[1,4,2,3,5] => [1,2,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,2,4,5,3] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,2,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,2,4,5,3] => [4,1,2,3,5] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,2,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,2,4,3,5] => [1,4,2,3,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,2,4,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,5,3,4,2] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,2,3] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [1,2,5,3,4] => [1,5,2,3,4] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,3,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,5,3] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,5,3,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,1,5,4,3] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,3,1,4,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,1,5,4] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,1,5] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,4,5,1] => [1,2,3,4,5] => [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,3,5,1,4] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,3,5,4,1] => [1,2,3,5,4] => [5,1,2,3,4] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
[2,5,1,3,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[2,5,3,1,4] => [1,2,5,4,3] => [5,4,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,5,2,4] => [1,3,5,4,2] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,1,5,4,2] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,2,5,1,4] => [1,3,5,4,2] => [5,3,1,2,4] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[3,2,5,4,1] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,4,5,1,2] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,4,5,2,1] => [1,3,5,2,4] => [3,5,1,2,4] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[3,5,1,2,4] => [1,3,2,5,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[3,5,1,4,2] => [1,3,2,5,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[3,5,2,1,4] => [1,3,2,5,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[3,5,2,4,1] => [1,3,2,5,4] => [3,1,5,2,4] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
[4,1,3,5,2] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,2,3,5,1] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,3,1,5,2] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,3,2,5,1] => [1,4,5,2,3] => [4,5,1,2,3] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
[4,5,1,2,3] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[4,5,2,1,3] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[4,5,3,1,2] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[4,5,3,2,1] => [1,4,2,5,3] => [4,1,5,2,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
[5,1,2,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,1,4,2,3] => [1,5,3,4,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,1,3,4] => [1,5,4,3,2] => [5,4,3,1,2] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,2,4,1,3] => [1,5,3,4,2] => [5,1,3,2,4] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,1,3,2] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,2,3,1] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,3,1,2] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[5,4,3,2,1] => [1,5,2,4,3] => [5,1,4,2,3] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
[1,2,3,6,4,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,2,3,6,5,4] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,2,4,6,3,5] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
[1,2,4,6,5,3] => [1,2,3,4,6,5] => [6,1,2,3,4,5] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1
Description
The radius of a connected graph.
This is the minimum eccentricity of any vertex.
Matching statistic: St001605
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Mp00204: Permutations —LLPS⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001605: Integer partitions ⟶ ℤResult quality: 28% ●values known / values provided: 28%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,2] => [1,1]
=> [1]
=> ? ∊ {0,0}
[2,1] => [2,1] => [2]
=> []
=> ? ∊ {0,0}
[1,2,3] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [1,3,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1,3] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [3,2,1] => [3]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,4,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [1,4,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,4,3] => [2,2]
=> [2]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [2,4,1,3] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [2,4,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1,4,2] => [2,2]
=> [2]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [3,2,1,4] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,2,4,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [3,4,1,2] => [2,1,1]
=> [1,1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [3,4,2,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [4,1,3,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [4,2,1,3] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [4,2,3,1] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [4,3,1,2] => [3,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [4,3,2,1] => [4]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,3,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,4,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,4,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,4,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,2,5,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,2,5] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,3,5,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [1,5,4,3,2] => [4,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5,6] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1
[3,2,1,4,6,5] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1
[3,2,1,5,4,6] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1
[3,2,1,5,6,4] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1
[3,2,1,6,4,5] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1
[3,2,1,6,5,4] => [3,2,1,6,5,4] => [3,3]
=> [3]
=> 1
[3,2,4,1,5,6] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,4,1,6,5] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,4,5,1,6] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,4,6,1,5] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,5,1,4,6] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,5,1,6,4] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,5,4,1,6] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,5,6,1,4] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,6,1,4,5] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,6,1,5,4] => [3,2,6,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,6,4,1,5] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1
[3,2,6,5,1,4] => [3,2,6,5,1,4] => [3,2,1]
=> [2,1]
=> 1
[3,4,2,1,5,6] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,4,2,1,6,5] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,4,2,5,1,6] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2
[3,4,2,6,1,5] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2
[3,5,2,1,4,6] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,5,2,1,6,4] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,5,2,4,1,6] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2
[3,5,2,6,1,4] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2
[3,6,2,1,4,5] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,6,2,1,5,4] => [3,6,2,1,5,4] => [3,2,1]
=> [2,1]
=> 1
[3,6,2,4,1,5] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2
[3,6,2,5,1,4] => [3,6,2,5,1,4] => [3,1,1,1]
=> [1,1,1]
=> 2
[4,2,1,3,5,6] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1
[4,2,1,3,6,5] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1
[4,2,1,5,3,6] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1
[4,2,1,5,6,3] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1
[4,2,1,6,3,5] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1
[4,2,1,6,5,3] => [4,2,1,6,5,3] => [3,3]
=> [3]
=> 1
[4,2,3,1,5,6] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,3,1,6,5] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,3,5,1,6] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,3,6,1,5] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,5,1,3,6] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,5,1,6,3] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,5,3,1,6] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,5,6,1,3] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,6,1,3,5] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,6,1,5,3] => [4,2,6,1,5,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,6,3,1,5] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1
[4,2,6,5,1,3] => [4,2,6,5,1,3] => [3,2,1]
=> [2,1]
=> 1
[4,3,1,2,5,6] => [4,3,1,6,5,2] => [3,3]
=> [3]
=> 1
[4,3,1,2,6,5] => [4,3,1,6,5,2] => [3,3]
=> [3]
=> 1
Description
The number of colourings of a cycle such that the multiplicities of colours are given by a partition.
Two colourings are considered equal, if they are obtained by an action of the cyclic group.
This statistic is only defined for partitions of size at least 3, to avoid ambiguity.
Matching statistic: St000704
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000704: Integer partitions ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 50%
Values
[1,2] => [1,1]
=> [1]
=> []
=> ? ∊ {0,0}
[2,1] => [2]
=> []
=> ?
=> ? ∊ {0,0}
[1,2,3] => [1,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1}
[1,3,2] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,1,3] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1}
[2,3,1] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1}
[3,1,2] => [3]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1}
[3,2,1] => [2,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1}
[1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,3] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,2,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,3,4,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,2,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,4,3,2] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,3,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,1,4,3] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,1,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,3,4,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,1,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[2,4,3,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,2,4] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,1,4,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,1,4] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,2,4,1] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,1,2] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[3,4,2,1] => [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,2,3] => [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,1,3,2] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,1,3] => [3,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,2,3,1] => [2,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,1,2] => [4]
=> []
=> ?
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[4,3,2,1] => [2,2]
=> [2]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1}
[1,2,3,4,5] => [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]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,2,5,3,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[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]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,2,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,4,5,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,2,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,3,5,4,2] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,3,5] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,2,5,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[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]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,2,3] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,4,5,3,2] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,3,4] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,2,4,3] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,2,4] => [3,1,1]
=> [1,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,3,4,2] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,4,2,3] => [4,1]
=> [1]
=> []
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[1,5,4,3,2] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[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]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[2,1,4,3,5] => [2,2,1]
=> [2,1]
=> [1]
=> ? ∊ {0,0,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2,2}
[3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[4,2,3,1,5] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[5,2,3,4,1] => [2,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,4,5,6] => [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,2,3,4,6,5] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,4,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,3,5,6,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,6,4,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,3,6,5,4] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,4,3,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,4,5,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,4,6,5,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,5,3,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,5,4,3,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,5,4,6,3] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,5,6,3,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,2,6,3,5,4] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,6,4,3,5] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,2,6,4,5,3] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,2,6,5,4,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,4,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,3,2,4,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,5,4,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,2,6,5,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,3,4,2,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,5,4,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,3,6,4,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,2,3,5,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,2,5,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,4,3,2,6,5] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,3,5,2,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,3,6,5,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,4,5,2,3,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,4,6,2,5,3] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,5,2,4,3,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,3,2,4,6] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,3,4,2,6] => [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> 1
[1,5,3,4,6,2] => [3,1,1,1]
=> [1,1,1]
=> [1,1]
=> 1
[1,5,3,6,2,4] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
[1,5,4,3,2,6] => [2,2,1,1]
=> [2,1,1]
=> [1,1]
=> 1
Description
The number of semistandard tableaux on a given integer partition with minimal maximal entry.
This is, for an integer partition $\lambda = (\lambda_1 > \cdots > \lambda_k > 0)$, the number of [[SemistandardTableaux|semistandard tableaux]] of shape $\lambda$ with maximal entry $k$.
Equivalently, this is the evaluation $s_\lambda(1,\ldots,1)$ of the Schur function $s_\lambda$ in $k$ variables, or, explicitly,
$$ \prod_{(i,j) \in L} \frac{k + j - i}{ \operatorname{hook}(i,j) }$$
where the product is over all cells $(i,j) \in L$ and $\operatorname{hook}(i,j)$ is the hook length of a cell.
See [Theorem 6.3, 1] for details.
The following 56 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000707The product of the factorials of the parts. St000815The number of semistandard Young tableaux of partition weight of given shape. St001128The exponens consonantiae of a partition. St001877Number of indecomposable injective modules with projective dimension 2. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St000454The largest eigenvalue of a graph if it is integral. St000455The second largest eigenvalue of a graph if it is integral. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001712The number of natural descents of a standard Young tableau. St001200The number of simple modules in $eAe$ with projective dimension at most 2 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St000872The number of very big descents of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St001489The maximum of the number of descents and the number of inverse descents. St001738The minimal order of a graph which is not an induced subgraph of the given graph. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St000527The width of the poset. St000257The number of distinct parts of a partition that occur at least twice. St000143The largest repeated part of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001335The cardinality of a minimal cycle-isolating set of a graph. St000845The maximal number of elements covered by an element in a poset. St000846The maximal number of elements covering an element of a poset. St001632The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset. St000535The rank-width of a graph. St000671The maximin edge-connectivity for choosing a subgraph. St001743The discrepancy of a graph. St001826The maximal number of leaves on a vertex of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000640The rank of the largest boolean interval in a poset. St001277The degeneracy of a graph. St001322The size of a minimal independent dominating set in a graph. St001339The irredundance number of a graph. St001358The largest degree of a regular subgraph of a graph. St001792The arboricity of a graph. St001116The game chromatic number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000785The number of distinct colouring schemes of a graph. St001029The size of the core of a graph. St001494The Alon-Tarsi number of a graph. St001829The common independence number of a graph. St000258The burning number of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St001582The grades of the simple modules corresponding to the points in the poset of the symmetric group under the Bruhat order. St001569The maximal modular displacement of a permutation. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000307The number of rowmotion orbits of a poset. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000310The minimal degree of a vertex of a graph. St000822The Hadwiger number of the graph. St001330The hat guessing number of a graph. St001642The Prague dimension of a graph. St001271The competition number of a graph.
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