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Matching statistic: St001425
St001425: Decorated permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => 0
[-] => 1
[+,+] => 0
[-,+] => 1
[+,-] => 1
[-,-] => 2
[2,1] => 0
[+,+,+] => 0
[-,+,+] => 1
[+,-,+] => 1
[+,+,-] => 1
[-,-,+] => 2
[-,+,-] => 2
[+,-,-] => 2
[-,-,-] => 3
[+,3,2] => 0
[-,3,2] => 1
[2,1,+] => 0
[2,1,-] => 1
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 0
[3,-,1] => 1
[+,+,+,+] => 0
[-,+,+,+] => 1
[+,-,+,+] => 1
[+,+,-,+] => 1
[+,+,+,-] => 1
[-,-,+,+] => 2
[-,+,-,+] => 2
[-,+,+,-] => 2
[+,-,-,+] => 2
[+,-,+,-] => 2
[+,+,-,-] => 2
[-,-,-,+] => 3
[-,-,+,-] => 3
[-,+,-,-] => 3
[+,-,-,-] => 3
[-,-,-,-] => 4
[+,+,4,3] => 0
[-,+,4,3] => 1
[+,-,4,3] => 1
[-,-,4,3] => 2
[+,3,2,+] => 0
[-,3,2,+] => 1
[+,3,2,-] => 1
[-,3,2,-] => 2
[+,3,4,2] => 0
[-,3,4,2] => 1
[+,4,2,3] => 0
Description
The number of negatively decorated fixed points of a decorated permutation.
Matching statistic: St001426
St001426: Decorated permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[+] => 1
[-] => 0
[+,+] => 2
[-,+] => 1
[+,-] => 1
[-,-] => 0
[2,1] => 0
[+,+,+] => 3
[-,+,+] => 2
[+,-,+] => 2
[+,+,-] => 2
[-,-,+] => 1
[-,+,-] => 1
[+,-,-] => 1
[-,-,-] => 0
[+,3,2] => 1
[-,3,2] => 0
[2,1,+] => 1
[2,1,-] => 0
[2,3,1] => 0
[3,1,2] => 0
[3,+,1] => 1
[3,-,1] => 0
[+,+,+,+] => 4
[-,+,+,+] => 3
[+,-,+,+] => 3
[+,+,-,+] => 3
[+,+,+,-] => 3
[-,-,+,+] => 2
[-,+,-,+] => 2
[-,+,+,-] => 2
[+,-,-,+] => 2
[+,-,+,-] => 2
[+,+,-,-] => 2
[-,-,-,+] => 1
[-,-,+,-] => 1
[-,+,-,-] => 1
[+,-,-,-] => 1
[-,-,-,-] => 0
[+,+,4,3] => 2
[-,+,4,3] => 1
[+,-,4,3] => 1
[-,-,4,3] => 0
[+,3,2,+] => 2
[-,3,2,+] => 1
[+,3,2,-] => 1
[-,3,2,-] => 0
[+,3,4,2] => 1
[-,3,4,2] => 0
[+,4,2,3] => 1
Description
The number of positively decorated fixed points of a decorated permutation.
Matching statistic: St001498
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00255: Decorated permutations —lower permutation⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 80%
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 80%
Values
[+] => [1] => [1,0]
=> [1,0]
=> ? ∊ {0,1}
[-] => [1] => [1,0]
=> [1,0]
=> ? ∊ {0,1}
[+,+] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0,1,2}
[-,+] => [2,1] => [1,1,0,0]
=> [1,0,1,0]
=> 1
[+,-] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0,1,2}
[-,-] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0,1,2}
[2,1] => [1,2] => [1,0,1,0]
=> [1,1,0,0]
=> ? ∊ {0,0,1,2}
[+,+,+] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,2,2,3}
[-,+,+] => [2,3,1] => [1,1,0,1,0,0]
=> [1,1,0,0,1,0]
=> 2
[+,-,+] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[+,+,-] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,2,2,3}
[-,-,+] => [3,1,2] => [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> 1
[-,+,-] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[+,-,-] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,2,2,3}
[-,-,-] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,2,2,3}
[+,3,2] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,2,2,3}
[-,3,2] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[2,1,+] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,1,-] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,2,2,3}
[2,3,1] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,2,2,3}
[3,1,2] => [1,2,3] => [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,0,1,1,2,2,3}
[3,+,1] => [2,1,3] => [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0
[3,-,1] => [1,3,2] => [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[+,+,+,+] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[-,+,+,+] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 3
[+,-,+,+] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[+,+,-,+] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[+,+,+,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[-,-,+,+] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,0]
=> 2
[-,+,-,+] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[-,+,+,-] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[+,-,-,+] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[+,-,+,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[+,+,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[-,-,-,+] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> 1
[-,-,+,-] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[-,+,-,-] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[+,-,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[+,+,4,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[-,+,4,3] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[+,-,4,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[-,-,4,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[+,3,2,+] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[-,3,2,+] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0]
=> 2
[+,3,2,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[-,3,2,-] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[+,3,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[-,3,4,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[+,4,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[-,4,2,3] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0
[+,4,+,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[-,4,+,2] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[+,4,-,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[-,4,-,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,+,+] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[2,1,-,+] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,1,+,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,1,-,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[2,1,4,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,3,1,+] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,3,1,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[2,3,4,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[2,4,1,3] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[2,4,+,1] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1
[2,4,-,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,+] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,2,-] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[3,1,4,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[3,+,1,+] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,-,1,+] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[3,+,1,-] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,-,1,-] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,+,4,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[3,-,4,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[3,4,1,2] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[3,4,2,1] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0
[4,1,2,3] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,3,3,4}
[4,1,+,2] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 0
[4,1,-,2] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
Description
The normalised height of a Nakayama algebra with magnitude 1.
We use the bijection (see code) suggested by Christian Stump, to have a bijection between such Nakayama algebras with magnitude 1 and Dyck paths. The normalised height is the height of the (periodic) Dyck path given by the top of the Auslander-Reiten quiver. Thus when having a CNakayama algebra it is the Loewy length minus the number of simple modules and for the LNakayama algebras it is the usual height.
Matching statistic: St001600
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001600: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001600: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,1}
[-] => [1] => [1]
=> []
=> ? ∊ {0,1}
[+,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,2}
[-,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,2}
[+,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,2}
[-,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,2}
[2,1] => [2,1] => [1,1]
=> [1]
=> 1
[+,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[-,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[+,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[+,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[-,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[-,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[+,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[-,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> 1
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[+,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[-,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[+,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[-,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[+,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[-,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[+,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[-,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 1
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 1
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,+,+] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,-,+] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,+,-] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,-,-] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,+] => [2,3,1,4] => [3,1]
=> [1]
=> 1
[2,3,1,-] => [2,3,1,4] => [3,1]
=> [1]
=> 1
[2,3,4,1] => [2,3,4,1] => [3,1]
=> [1]
=> 1
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,2,+] => [3,1,2,4] => [3,1]
=> [1]
=> 1
[3,1,2,-] => [3,1,2,4] => [3,1]
=> [1]
=> 1
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [4,1,2,3] => [3,1]
=> [1]
=> 1
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on simple graphs.
Matching statistic: St001606
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Mp00060: Permutations —Robinson-Schensted tableau shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001606: Integer partitions ⟶ ℤResult quality: 60% ●values known / values provided: 66%●distinct values known / distinct values provided: 60%
Values
[+] => [1] => [1]
=> []
=> ? ∊ {0,1}
[-] => [1] => [1]
=> []
=> ? ∊ {0,1}
[+,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,2}
[-,+] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,2}
[+,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,2}
[-,-] => [1,2] => [2]
=> []
=> ? ∊ {0,0,1,2}
[2,1] => [2,1] => [1,1]
=> [1]
=> 1
[+,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[-,+,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[+,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[+,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[-,-,+] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[-,+,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[+,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[-,-,-] => [1,2,3] => [3]
=> []
=> ? ∊ {0,0,0,0,2,2,2,3}
[+,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[-,3,2] => [1,3,2] => [2,1]
=> [1]
=> 1
[2,1,+] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,1,-] => [2,1,3] => [2,1]
=> [1]
=> 1
[2,3,1] => [2,3,1] => [2,1]
=> [1]
=> 1
[3,1,2] => [3,1,2] => [2,1]
=> [1]
=> 1
[3,+,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[3,-,1] => [3,2,1] => [1,1,1]
=> [1,1]
=> 0
[+,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,+,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,-,+,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,+,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,+,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,-,-,+] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,-,+,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,+,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [4]
=> []
=> ? ∊ {0,1,1,2,2,2,2,2,2,2,2,3,3,3,3,4}
[+,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[-,+,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[+,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[-,-,4,3] => [1,2,4,3] => [3,1]
=> [1]
=> 1
[+,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[-,3,2,+] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[+,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[-,3,2,-] => [1,3,2,4] => [3,1]
=> [1]
=> 1
[+,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 1
[-,3,4,2] => [1,3,4,2] => [3,1]
=> [1]
=> 1
[+,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[-,4,2,3] => [1,4,2,3] => [3,1]
=> [1]
=> 1
[+,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,+,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[+,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[-,4,-,2] => [1,4,3,2] => [2,1,1]
=> [1,1]
=> 0
[2,1,+,+] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,-,+] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,+,-] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,-,-] => [2,1,3,4] => [3,1]
=> [1]
=> 1
[2,1,4,3] => [2,1,4,3] => [2,2]
=> [2]
=> 2
[2,3,1,+] => [2,3,1,4] => [3,1]
=> [1]
=> 1
[2,3,1,-] => [2,3,1,4] => [3,1]
=> [1]
=> 1
[2,3,4,1] => [2,3,4,1] => [3,1]
=> [1]
=> 1
[2,4,1,3] => [2,4,1,3] => [2,2]
=> [2]
=> 2
[2,4,+,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[2,4,-,1] => [2,4,3,1] => [2,1,1]
=> [1,1]
=> 0
[3,1,2,+] => [3,1,2,4] => [3,1]
=> [1]
=> 1
[3,1,2,-] => [3,1,2,4] => [3,1]
=> [1]
=> 1
[3,1,4,2] => [3,1,4,2] => [2,2]
=> [2]
=> 2
[3,+,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,+] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,-,1,-] => [3,2,1,4] => [2,1,1]
=> [1,1]
=> 0
[3,+,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,-,4,1] => [3,2,4,1] => [2,1,1]
=> [1,1]
=> 0
[3,4,1,2] => [3,4,1,2] => [2,2]
=> [2]
=> 2
[3,4,2,1] => [3,4,2,1] => [2,1,1]
=> [1,1]
=> 0
[4,1,2,3] => [4,1,2,3] => [3,1]
=> [1]
=> 1
[4,1,+,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
[4,1,-,2] => [4,1,3,2] => [2,1,1]
=> [1,1]
=> 0
Description
The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on set partitions.
Matching statistic: St000456
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 50%●distinct values known / distinct values provided: 40%
Mp00175: Permutations —inverse Foata bijection⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 40% ●values known / values provided: 50%●distinct values known / distinct values provided: 40%
Values
[+] => [1] => [1] => ([],1)
=> ? ∊ {0,1}
[-] => [1] => [1] => ([],1)
=> ? ∊ {0,1}
[+,+] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,2}
[-,+] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,2}
[-,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,2}
[2,1] => [2,1] => [2,1] => ([(0,1)],2)
=> 1
[+,+,+] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[-,+,+] => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[+,-,+] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[+,+,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[-,-,+] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[-,+,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[+,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[-,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[+,3,2] => [1,3,2] => [3,1,2] => ([(0,2),(1,2)],3)
=> 1
[-,3,2] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[2,1,+] => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[2,1,-] => [2,1,3] => [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[2,3,1] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[3,1,2] => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,+,1] => [2,3,1] => [2,3,1] => ([(0,2),(1,2)],3)
=> 1
[3,-,1] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,2,2,2,3}
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,+,+,+] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,-,+,+] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[+,+,-,+] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,-,+,+] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[-,+,-,+] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[-,+,+,-] => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[+,-,-,+] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[+,-,+,-] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,-,-,+] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,-,+,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,+,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[+,+,4,3] => [1,2,4,3] => [4,1,2,3] => ([(0,3),(1,3),(2,3)],4)
=> 1
[-,+,4,3] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[+,-,4,3] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,-,4,3] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[+,3,2,+] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[-,3,2,+] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[+,3,2,-] => [1,3,2,4] => [3,1,2,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,3,2,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[+,3,4,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,3,4,2] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[+,4,2,3] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[-,4,2,3] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[+,4,+,2] => [1,3,4,2] => [3,4,1,2] => ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
[-,4,+,2] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[+,4,-,2] => [1,4,2,3] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[-,4,-,2] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[2,1,+,+] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[2,1,-,+] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,1,+,-] => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[2,1,-,-] => [2,1,3,4] => [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[2,1,4,3] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[2,3,1,+] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,3,1,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[2,3,4,1] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[2,4,1,3] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,4,+,1] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[2,4,-,1] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[3,1,2,+] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,1,2,-] => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[3,1,4,2] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,+,1,+] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[3,-,1,+] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,+,1,-] => [2,3,1,4] => [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[3,-,1,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[3,+,4,1] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[3,-,4,1] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[3,4,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[3,4,2,1] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[4,1,2,3] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,+,2] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,1,-,2] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,+,1,3] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,-,1,3] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[4,+,+,1] => [2,3,4,1] => [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 1
[4,-,+,1] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[4,+,-,1] => [2,4,1,3] => [4,2,1,3] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
[4,-,-,1] => [4,1,2,3] => [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,3,3,3,3,4}
[4,3,1,2] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
[4,3,2,1] => [3,4,1,2] => [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1
Description
The monochromatic index of a connected graph.
This is the maximal number of colours such that there is a colouring of the edges where any two vertices can be joined by a monochromatic path.
For example, a circle graph other than the triangle can be coloured with at most two colours: one edge blue, all the others red.
Matching statistic: St001199
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Mp00064: Permutations —reverse⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 45% ●values known / values provided: 45%●distinct values known / distinct values provided: 60%
Values
[+] => [1] => [1] => [1,0]
=> ? ∊ {0,1}
[-] => [1] => [1] => [1,0]
=> ? ∊ {0,1}
[+,+] => [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,1,2}
[-,+] => [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,1,2}
[+,-] => [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,1,2}
[-,-] => [1,2] => [2,1] => [1,1,0,0]
=> ? ∊ {0,0,1,2}
[2,1] => [2,1] => [1,2] => [1,0,1,0]
=> 1
[+,+,+] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,+,+] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[+,-,+] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[+,+,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,-,+] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,+,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[+,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,-,-] => [1,2,3] => [3,2,1] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[+,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[-,3,2] => [1,3,2] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[2,1,+] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[2,1,-] => [2,1,3] => [3,1,2] => [1,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[2,3,1] => [2,3,1] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[3,1,2] => [3,1,2] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[3,+,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[3,-,1] => [3,2,1] => [1,2,3] => [1,0,1,0,1,0]
=> 2
[+,+,+,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,+,+,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,-,+,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,+,-,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,+,+,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,-,+,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,+,-,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,+,+,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,-,-,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,-,+,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,-,-,+] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,-,+,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,+,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,+,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[-,+,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[+,-,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[-,-,4,3] => [1,2,4,3] => [3,4,2,1] => [1,1,1,0,1,0,0,0]
=> 1
[+,3,2,+] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,3,2,+] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,3,2,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[-,3,2,-] => [1,3,2,4] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[+,3,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[-,3,4,2] => [1,3,4,2] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[+,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[-,4,2,3] => [1,4,2,3] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[+,4,+,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[-,4,+,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[+,4,-,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[-,4,-,2] => [1,4,3,2] => [2,3,4,1] => [1,1,0,1,0,1,0,0]
=> 1
[2,1,+,+] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[2,1,-,+] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[2,1,+,-] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[2,1,-,-] => [2,1,3,4] => [4,3,1,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[2,1,4,3] => [2,1,4,3] => [3,4,1,2] => [1,1,1,0,1,0,0,0]
=> 1
[2,3,1,+] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[2,3,1,-] => [2,3,1,4] => [4,1,3,2] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[2,3,4,1] => [2,3,4,1] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => [1,1,1,0,0,1,0,0]
=> 1
[2,4,+,1] => [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[2,4,-,1] => [2,4,3,1] => [1,3,4,2] => [1,0,1,1,0,1,0,0]
=> 2
[3,1,2,+] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[3,1,2,-] => [3,1,2,4] => [4,2,1,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[3,1,4,2] => [3,1,4,2] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[3,+,1,+] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[3,-,1,+] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[3,+,1,-] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[3,-,1,-] => [3,2,1,4] => [4,1,2,3] => [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,2,2,3,3,3,4}
[3,+,4,1] => [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,-,4,1] => [3,2,4,1] => [1,4,2,3] => [1,0,1,1,1,0,0,0]
=> 1
[3,4,1,2] => [3,4,1,2] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[3,4,2,1] => [3,4,2,1] => [1,2,4,3] => [1,0,1,0,1,1,0,0]
=> 2
[4,1,2,3] => [4,1,2,3] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,1,+,2] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,1,-,2] => [4,1,3,2] => [2,3,1,4] => [1,1,0,1,0,0,1,0]
=> 2
[4,+,1,3] => [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,-,1,3] => [4,2,1,3] => [3,1,2,4] => [1,1,1,0,0,0,1,0]
=> 1
[4,+,+,1] => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[4,-,+,1] => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[4,+,-,1] => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[4,-,-,1] => [4,2,3,1] => [1,3,2,4] => [1,0,1,1,0,0,1,0]
=> 2
[4,3,1,2] => [4,3,1,2] => [2,1,3,4] => [1,1,0,0,1,0,1,0]
=> 2
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => [1,0,1,0,1,0,1,0]
=> 3
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001118
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 60%
Mp00090: Permutations —cycle-as-one-line notation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St001118: Graphs ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 60%
Values
[+] => [1] => [1] => ([],1)
=> ? ∊ {0,1}
[-] => [1] => [1] => ([],1)
=> ? ∊ {0,1}
[+,+] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,2}
[-,+] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,2}
[+,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,2}
[-,-] => [1,2] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,2}
[2,1] => [2,1] => [1,2] => ([],2)
=> ? ∊ {0,0,1,1,2}
[+,+,+] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[-,+,+] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[+,-,+] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[+,+,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[-,-,+] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[-,+,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[+,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[-,-,-] => [1,2,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[+,3,2] => [1,3,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[-,3,2] => [1,3,2] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[2,1,+] => [2,1,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[2,1,-] => [2,1,3] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[2,3,1] => [2,3,1] => [1,2,3] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,1,2,2,2,3}
[3,1,2] => [3,1,2] => [1,3,2] => ([(1,2)],3)
=> 1
[3,+,1] => [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 1
[3,-,1] => [3,2,1] => [1,3,2] => ([(1,2)],3)
=> 1
[+,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,+,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,-,+,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,+,-,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,+,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,-,-,+] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,-,+,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,+,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,-,-,-] => [1,2,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,+,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,+,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,-,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,-,4,3] => [1,2,4,3] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,3,2,+] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,3,2,+] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,3,2,-] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,3,2,-] => [1,3,2,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,3,4,2] => [1,3,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[-,3,4,2] => [1,3,4,2] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[+,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[-,4,2,3] => [1,4,2,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[+,4,+,2] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
[-,4,+,2] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
[+,4,-,2] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
[-,4,-,2] => [1,4,3,2] => [1,2,4,3] => ([(2,3)],4)
=> 1
[2,1,+,+] => [2,1,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[2,1,-,+] => [2,1,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[2,1,+,-] => [2,1,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[2,1,-,-] => [2,1,3,4] => [1,2,3,4] => ([],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,3,4}
[2,4,1,3] => [2,4,1,3] => [1,2,4,3] => ([(2,3)],4)
=> 1
[2,4,+,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
[2,4,-,1] => [2,4,3,1] => [1,2,4,3] => ([(2,3)],4)
=> 1
[3,1,2,+] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[3,1,2,-] => [3,1,2,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[3,1,4,2] => [3,1,4,2] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,+,1,+] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[3,-,1,+] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[3,+,1,-] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[3,-,1,-] => [3,2,1,4] => [1,3,2,4] => ([(2,3)],4)
=> 1
[3,+,4,1] => [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,-,4,1] => [3,2,4,1] => [1,3,4,2] => ([(1,3),(2,3)],4)
=> 2
[3,4,1,2] => [3,4,1,2] => [1,3,2,4] => ([(2,3)],4)
=> 1
[3,4,2,1] => [3,4,2,1] => [1,3,2,4] => ([(2,3)],4)
=> 1
[4,1,2,3] => [4,1,2,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,1,+,2] => [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,1,-,2] => [4,1,3,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,+,1,3] => [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,-,1,3] => [4,2,1,3] => [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> 3
[4,+,+,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,-,+,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,+,-,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,-,-,1] => [4,2,3,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,3,1,2] => [4,3,1,2] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
[4,3,2,1] => [4,3,2,1] => [1,4,2,3] => ([(1,3),(2,3)],4)
=> 2
Description
The acyclic chromatic index of a graph.
An acyclic edge coloring of a graph is a proper colouring of the edges of a graph such that the union of the edges colored with any two given colours is a forest.
The smallest number of colours such that such a colouring exists is the acyclic chromatic index.
Matching statistic: St001632
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00256: Decorated permutations —upper permutation⟶ Permutations
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 60%
Mp00089: Permutations —Inverse Kreweras complement⟶ Permutations
Mp00065: Permutations —permutation poset⟶ Posets
St001632: Posets ⟶ ℤResult quality: 39% ●values known / values provided: 39%●distinct values known / distinct values provided: 60%
Values
[+] => [1] => [1] => ([],1)
=> ? ∊ {0,1}
[-] => [1] => [1] => ([],1)
=> ? ∊ {0,1}
[+,+] => [1,2] => [2,1] => ([],2)
=> ? ∊ {0,0,2}
[-,+] => [2,1] => [1,2] => ([(0,1)],2)
=> 1
[+,-] => [1,2] => [2,1] => ([],2)
=> ? ∊ {0,0,2}
[-,-] => [1,2] => [2,1] => ([],2)
=> ? ∊ {0,0,2}
[2,1] => [2,1] => [1,2] => ([(0,1)],2)
=> 1
[+,+,+] => [1,2,3] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,+,+] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[+,-,+] => [1,3,2] => [3,2,1] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[+,+,-] => [1,2,3] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,-,+] => [3,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,+,-] => [2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[+,-,-] => [1,2,3] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,-,-] => [1,2,3] => [2,3,1] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[+,3,2] => [1,3,2] => [3,2,1] => ([],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[-,3,2] => [3,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[2,1,+] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[2,1,-] => [2,1,3] => [1,3,2] => ([(0,1),(0,2)],3)
=> 2
[2,3,1] => [3,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[3,1,2] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,+,1] => [2,3,1] => [1,2,3] => ([(0,2),(2,1)],3)
=> 1
[3,-,1] => [3,1,2] => [3,1,2] => ([(1,2)],3)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3}
[+,+,+,+] => [1,2,3,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,+,+,+] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[+,-,+,+] => [1,3,4,2] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,+,-,+] => [1,2,4,3] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,+,+,-] => [1,2,3,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,-,+,+] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,+,-,+] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[-,+,+,-] => [2,3,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[+,-,-,+] => [1,4,2,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,-,+,-] => [1,3,2,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,-,-,+] => [4,1,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,-,+,-] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[-,+,-,-] => [2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[+,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [2,3,4,1] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,+,4,3] => [1,2,4,3] => [2,4,3,1] => ([(1,2),(1,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,+,4,3] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[+,-,4,3] => [1,4,2,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,-,4,3] => [4,1,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,3,2,+] => [1,3,4,2] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,3,2,+] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,3,2,-] => [1,3,2,4] => [3,2,4,1] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,3,2,-] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[+,3,4,2] => [1,4,2,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,3,4,2] => [4,1,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,4,2,3] => [1,3,4,2] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,4,2,3] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,4,+,2] => [1,3,4,2] => [4,2,3,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,4,+,2] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[+,4,-,2] => [1,4,2,3] => [3,4,2,1] => ([(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[-,4,-,2] => [4,1,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[2,1,+,+] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[2,1,-,+] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[2,1,+,-] => [2,3,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[2,1,-,-] => [2,1,3,4] => [1,3,4,2] => ([(0,2),(0,3),(3,1)],4)
=> 2
[2,1,4,3] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[2,3,1,+] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[2,3,1,-] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[2,3,4,1] => [4,1,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[2,4,1,3] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[2,4,+,1] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[2,4,-,1] => [4,1,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[3,1,2,+] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,1,2,-] => [2,3,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[3,1,4,2] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,+,1,+] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[3,-,1,+] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[3,+,1,-] => [2,3,1,4] => [1,2,4,3] => ([(0,3),(3,1),(3,2)],4)
=> 1
[3,-,1,-] => [3,1,2,4] => [3,1,4,2] => ([(0,3),(1,2),(1,3)],4)
=> 0
[3,+,4,1] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[3,-,4,1] => [4,1,2,3] => [3,4,1,2] => ([(0,3),(1,2)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[3,4,1,2] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[3,4,2,1] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[4,1,2,3] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,+,2] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,1,-,2] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
[4,+,1,3] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,-,1,3] => [3,4,1,2] => [4,1,2,3] => ([(1,2),(2,3)],4)
=> ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,1,1,1,2,2,3,3,3,3,4}
[4,+,+,1] => [2,3,4,1] => [1,2,3,4] => ([(0,3),(2,1),(3,2)],4)
=> 1
[4,+,-,1] => [2,4,1,3] => [1,4,2,3] => ([(0,2),(0,3),(3,1)],4)
=> 2
Description
The number of indecomposable injective modules $I$ with $dim Ext^1(I,A)=1$ for the incidence algebra A of a poset.
Matching statistic: St001491
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00253: Decorated permutations —permutation⟶ Permutations
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 60%
Mp00149: Permutations —Lehmer code rotation⟶ Permutations
Mp00114: Permutations —connectivity set⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 34% ●values known / values provided: 34%●distinct values known / distinct values provided: 60%
Values
[+] => [1] => [1] => => ? ∊ {0,1}
[-] => [1] => [1] => => ? ∊ {0,1}
[+,+] => [1,2] => [2,1] => 0 => ? ∊ {0,0,1,2}
[-,+] => [1,2] => [2,1] => 0 => ? ∊ {0,0,1,2}
[+,-] => [1,2] => [2,1] => 0 => ? ∊ {0,0,1,2}
[-,-] => [1,2] => [2,1] => 0 => ? ∊ {0,0,1,2}
[2,1] => [2,1] => [1,2] => 1 => 1
[+,+,+] => [1,2,3] => [2,3,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[-,+,+] => [1,2,3] => [2,3,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[+,-,+] => [1,2,3] => [2,3,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[+,+,-] => [1,2,3] => [2,3,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[-,-,+] => [1,2,3] => [2,3,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[-,+,-] => [1,2,3] => [2,3,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[+,-,-] => [1,2,3] => [2,3,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[-,-,-] => [1,2,3] => [2,3,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[+,3,2] => [1,3,2] => [2,1,3] => 01 => 1
[-,3,2] => [1,3,2] => [2,1,3] => 01 => 1
[2,1,+] => [2,1,3] => [3,2,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[2,1,-] => [2,1,3] => [3,2,1] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[2,3,1] => [2,3,1] => [3,1,2] => 00 => ? ∊ {0,0,0,0,0,0,1,1,1,2,3}
[3,1,2] => [3,1,2] => [1,3,2] => 10 => 1
[3,+,1] => [3,2,1] => [1,2,3] => 11 => 2
[3,-,1] => [3,2,1] => [1,2,3] => 11 => 2
[+,+,+,+] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,+,+,+] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,-,+,+] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,+,-,+] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,+,+,-] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,-,+,+] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,+,-,+] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,+,+,-] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,-,-,+] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,-,+,-] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,+,-,-] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,-,-,+] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,-,+,-] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,+,-,-] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,-,-,-] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,-,-,-] => [1,2,3,4] => [2,3,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,+,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[-,+,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[+,-,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[-,-,4,3] => [1,2,4,3] => [2,3,1,4] => 001 => 1
[+,3,2,+] => [1,3,2,4] => [2,4,3,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,3,2,+] => [1,3,2,4] => [2,4,3,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,3,2,-] => [1,3,2,4] => [2,4,3,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,3,2,-] => [1,3,2,4] => [2,4,3,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,3,4,2] => [1,3,4,2] => [2,4,1,3] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[-,3,4,2] => [1,3,4,2] => [2,4,1,3] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[+,4,2,3] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[-,4,2,3] => [1,4,2,3] => [2,1,4,3] => 010 => 1
[+,4,+,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[-,4,+,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[+,4,-,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[-,4,-,2] => [1,4,3,2] => [2,1,3,4] => 011 => 1
[2,1,+,+] => [2,1,3,4] => [3,2,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[2,1,-,+] => [2,1,3,4] => [3,2,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[2,1,+,-] => [2,1,3,4] => [3,2,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[2,1,-,-] => [2,1,3,4] => [3,2,4,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[2,1,4,3] => [2,1,4,3] => [3,2,1,4] => 001 => 1
[2,3,1,+] => [2,3,1,4] => [3,4,2,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[2,3,1,-] => [2,3,1,4] => [3,4,2,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[2,3,4,1] => [2,3,4,1] => [3,4,1,2] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[2,4,1,3] => [2,4,1,3] => [3,1,4,2] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[2,4,+,1] => [2,4,3,1] => [3,1,2,4] => 001 => 1
[2,4,-,1] => [2,4,3,1] => [3,1,2,4] => 001 => 1
[3,1,2,+] => [3,1,2,4] => [4,2,3,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[3,1,2,-] => [3,1,2,4] => [4,2,3,1] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[3,1,4,2] => [3,1,4,2] => [4,2,1,3] => 000 => ? ∊ {0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,1,1,2,2,2,2,2,2,2,2,2,2,3,3,3,4}
[4,1,2,3] => [4,1,2,3] => [1,3,4,2] => 100 => 1
[4,1,+,2] => [4,1,3,2] => [1,3,2,4] => 101 => 2
[4,1,-,2] => [4,1,3,2] => [1,3,2,4] => 101 => 2
[4,+,1,3] => [4,2,1,3] => [1,4,3,2] => 100 => 1
[4,-,1,3] => [4,2,1,3] => [1,4,3,2] => 100 => 1
[4,+,+,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,-,+,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,+,-,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,-,-,1] => [4,2,3,1] => [1,4,2,3] => 100 => 1
[4,3,1,2] => [4,3,1,2] => [1,2,4,3] => 110 => 1
[4,3,2,1] => [4,3,2,1] => [1,2,3,4] => 111 => 3
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
The following 23 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000478Another weight of a partition according to Alladi. St000566The number of ways to select a row of a Ferrers shape and two cells in this row. St000567The sum of the products of all pairs of parts. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000621The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is even. St000668The least common multiple of the parts of the partition. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St000934The 2-degree of an integer partition. St000937The number of positive values of the symmetric group character corresponding to the partition. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St001570The minimal number of edges to add to make a graph Hamiltonian. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001060The distinguishing index of a graph.
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