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Matching statistic: St000752
St000752: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 1
[2,1]
=> 0
[1,1,1]
=> 0
[4]
=> 2
[3,1]
=> 1
[2,2]
=> 0
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 2
[3,2]
=> 1
[3,1,1]
=> 1
[2,2,1]
=> 0
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 1
[5,1]
=> 0
[4,2]
=> 2
[4,1,1]
=> 2
[3,3]
=> 0
[3,2,1]
=> 1
[3,1,1,1]
=> 1
[2,2,2]
=> 0
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 2
[6,1]
=> 1
[5,2]
=> 0
[5,1,1]
=> 0
[4,3]
=> 3
[4,2,1]
=> 2
[4,1,1,1]
=> 2
[3,3,1]
=> 0
[3,2,2]
=> 1
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 1
[2,2,2,1]
=> 0
[2,2,1,1,1]
=> 0
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 3
[7,1]
=> 2
[6,2]
=> 1
[6,1,1]
=> 1
[5,3]
=> 1
[5,2,1]
=> 0
Description
The Grundy value for the game 'Couples are forever' on an integer partition.
Two players alternately choose a part of the partition greater than two, and split it into two parts. The player facing a partition with all parts at most two looses.
Matching statistic: St001274
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00032: Dyck paths —inverse zeta map⟶ Dyck paths
St001274: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 80%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[5]
=> []
=> []
=> []
=> ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[6]
=> []
=> []
=> []
=> ? = 2
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[7]
=> []
=> []
=> []
=> ? ∊ {2,3}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {2,3}
[8]
=> []
=> []
=> []
=> ? ∊ {1,1,3}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 1
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 0
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,3}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,3}
[9]
=> []
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {1,1,2,2,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {1,1,2,2,3}
[10]
=> []
=> []
=> []
=> ? ∊ {0,0,1,2,2,3,3,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,3,3,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,2,2,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,3,3,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,2,2,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,2,2,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,2,2,3,3,3}
[11]
=> []
=> []
=> []
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,1,1,1,1,2,2,3,3,3,3,3}
[12]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[6,6]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[6,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[5,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[5,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[4,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,0,0,1,1,1,2,2,2,3,3,3,3,3,3,3,3,4}
Description
The number of indecomposable injective modules with projective dimension equal to two.
Matching statistic: St001233
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 80%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001233: Dyck paths ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 80%
Values
[1]
=> []
=> ?
=> ?
=> ? = 0
[2]
=> []
=> ?
=> ?
=> ? ∊ {0,0}
[1,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0}
[3]
=> []
=> ?
=> ?
=> ? ∊ {0,1}
[2,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2}
[3,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2}
[2,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2}
[2,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2}
[4,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2}
[3,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2}
[3,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[6]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2}
[5,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,2}
[4,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,2}
[4,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,2}
[3,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[7]
=> []
=> ?
=> ?
=> ? ∊ {0,1,2,2}
[6,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,2,2}
[5,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,2,2}
[5,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,2,2}
[4,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
[8]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,2,3}
[7,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3}
[6,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3}
[6,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3}
[5,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,2,2,3}
[4,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,2,3}
[9]
=> []
=> ?
=> ?
=> ? ∊ {0,1,1,2,2,2,3}
[8,1]
=> [1]
=> []
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[7,2]
=> [2]
=> []
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[7,1,1]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,3]
=> [3]
=> []
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[6,2,1]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5,4]
=> [4]
=> []
=> []
=> ? ∊ {0,1,1,2,2,2,3}
[5,3,1]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[4,4,1]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 0
[4,3,2]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,2,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,1,1,2,2,2,3}
[10]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[9,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[8,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[7,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[6,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[5,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,2,2,3,3,3,3}
[11]
=> []
=> ?
=> ?
=> ? ∊ {0,0,1,2,2,2,2,2,3,3,3,3,3}
[10,1]
=> [1]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,3,3,3,3,3}
[9,2]
=> [2]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,3,3,3,3,3}
[8,3]
=> [3]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,3,3,3,3,3}
[7,4]
=> [4]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,3,3,3,3,3}
[6,5]
=> [5]
=> []
=> []
=> ? ∊ {0,0,1,2,2,2,2,2,3,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,2,2,2,2,3,3,3,3,3}
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,1,2,2,2,2,2,3,3,3,3,3}
Description
The number of indecomposable 2-dimensional modules with projective dimension one.
Matching statistic: St001556
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 63%●distinct values known / distinct values provided: 60%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
St001556: Permutations ⟶ ℤResult quality: 60% ●values known / values provided: 63%●distinct values known / distinct values provided: 60%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[5]
=> []
=> []
=> [] => ? = 1
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
[6]
=> []
=> []
=> [] => ? ∊ {0,2}
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {0,2}
[7]
=> []
=> []
=> [] => ? ∊ {0,2,3}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {0,2,3}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,2,3}
[8]
=> []
=> []
=> [] => ? ∊ {0,0,1,3,3}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,2] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [2,1,3] => 0
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [3,2,1,4] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,2,3] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => 1
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,3,2,1,5] => 1
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,3,1,4] => 0
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,1,4,3] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => 2
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,3,4] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,3,2,4] => 0
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {0,0,1,3,3}
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => 2
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => ? ∊ {0,0,1,3,3}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,1,3,3}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,1,3,3}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,1,1,2,3,3,3}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {0,0,1,1,2,3,3,3}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => ? ∊ {0,0,1,1,2,3,3,3}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,1,1,2,3,3,3}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => ? ∊ {0,0,1,1,2,3,3,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {0,0,1,1,2,3,3,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,1,1,2,3,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {0,0,1,1,2,3,3,3}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[5,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,1,6,5,4,3] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,3,6,5,4,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,5,7,6,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,7,8,6,5,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,10,9,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,2,2,2,2,3,3,3,3,3,3}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[6,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[5,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,5,3,2,1,6] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5,6,4,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5,4,6,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,4,6,5,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6,7,5,4,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,8,7,6,5,4,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,4,5,6,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6,5,7,4,3,2] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
Description
The number of inversions of the third entry of a permutation.
This is, for a permutation π of length n,
#{3<k≤n∣π(3)>π(k)}.
The number of inversions of the first entry is [[St000054]] and the number of inversions of the second entry is [[St001557]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001557
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 80%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
St001557: Permutations ⟶ ℤResult quality: 63% ●values known / values provided: 63%●distinct values known / distinct values provided: 80%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3]
=> []
=> []
=> [] => ? = 1
[2,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[5]
=> []
=> []
=> [] => ? = 2
[4,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[6]
=> []
=> []
=> [] => ? ∊ {0,2}
[5,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,2}
[7]
=> []
=> []
=> [] => ? ∊ {0,2,2}
[6,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 3
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,2,2}
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? ∊ {0,2,2}
[8]
=> []
=> []
=> [] => ? ∊ {0,0,2,2,3}
[7,1]
=> [1]
=> [1,0,1,0]
=> [2,1] => 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,3,2] => 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => 0
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => 0
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => 0
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => 0
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => 1
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => 2
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => 0
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => 0
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => 3
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,0,2,2,3}
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ? ∊ {0,0,2,2,3}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? ∊ {0,0,2,2,3}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => ? ∊ {0,0,2,2,3}
[9]
=> []
=> []
=> [] => ? ∊ {0,1,2,2,2,2,3,3}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,1,2,2,2,2,3,3}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ? ∊ {0,1,2,2,2,2,3,3}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? ∊ {0,1,2,2,2,2,3,3}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ? ∊ {0,1,2,2,2,2,3,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? ∊ {0,1,2,2,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => ? ∊ {0,1,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => ? ∊ {0,1,2,2,2,2,3,3}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[5,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[5,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[2,2,2,2,2]
=> [2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [2,6,1,3,4,5,7] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [2,8,1,3,4,5,6,7] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8,9,10] => ? ∊ {0,0,0,1,1,2,2,2,2,2,3,3,3,3,3}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[6,5]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[6,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[5,5,1]
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[5,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [2,7,1,3,4,5,6] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [2,1,3,4,5,6,7,8] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [2,1,7,3,4,5,6] => ? ∊ {0,0,0,0,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
Description
The number of inversions of the second entry of a permutation.
This is, for a permutation π of length n,
#{2<k≤n∣π(2)>π(k)}.
The number of inversions of the first entry is [[St000054]] and the number of inversions of the third entry is [[St001556]]. The sequence of inversions of all the entries define the [[http://www.findstat.org/Permutations#The_Lehmer_code_and_the_major_code_of_a_permutation|Lehmer code]] of a permutation.
Matching statistic: St001801
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001801: Permutations ⟶ ℤResult quality: 61% ●values known / values provided: 61%●distinct values known / distinct values provided: 80%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? = 0
[1,1]
=> [1]
=> [[1]]
=> [1] => 0
[3]
=> []
=> []
=> [] => ? = 0
[2,1]
=> [1]
=> [[1]]
=> [1] => 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[4]
=> []
=> []
=> [] => ? = 2
[3,1]
=> [1]
=> [[1]]
=> [1] => 0
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[5]
=> []
=> []
=> [] => ? = 0
[4,1]
=> [1]
=> [[1]]
=> [1] => 0
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[6]
=> []
=> []
=> [] => ? = 2
[5,1]
=> [1]
=> [[1]]
=> [1] => 0
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[7]
=> []
=> []
=> [] => ? = 0
[6,1]
=> [1]
=> [[1]]
=> [1] => 0
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 3
[8]
=> []
=> []
=> [] => ? ∊ {0,3}
[7,1]
=> [1]
=> [[1]]
=> [1] => 0
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 0
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[4,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 0
[4,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 2
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 1
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? ∊ {0,3}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,0,1,1,1}
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? ∊ {0,0,0,1,1,1}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? ∊ {0,0,0,1,1,1}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? ∊ {0,0,0,1,1,1}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? ∊ {0,0,0,1,1,1}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,0,0,1,1,1}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[3,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[3,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[3,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[3,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[2,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,1,2] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9]]
=> [9,8,7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,1,1,1,1,2,2,2,3,3,3}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,4,3]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> [5,6,7,1,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,4,2,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> [7,5,6,1,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> [7,6,5,1,2,3,4] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,3,3,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> [7,4,5,6,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,3,2,2]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> [6,7,4,5,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,3,2,1,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> [7,6,4,5,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,3,1,1,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> [7,6,5,4,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [7,5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [7,6,5,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [7,6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,3,3,2]
=> [3,3,2]
=> [[1,2,3],[4,5,6],[7,8]]
=> [7,8,4,5,6,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,3,3,1,1]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> [8,7,4,5,6,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,3,2,2,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> [8,6,7,4,5,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,3,2,1,1,1]
=> [3,2,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8]]
=> [8,7,6,4,5,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,3,1,1,1,1,1]
=> [3,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8]]
=> [8,7,6,5,4,1,2,3] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,2,2,2,2]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [7,8,5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> [8,7,5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8]]
=> [8,7,6,5,3,4,1,2] => ? ∊ {0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
Description
Half the number of preimage-image pairs of different parity in a permutation.
Matching statistic: St001948
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001948: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? ∊ {0,0}
[1,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0}
[3]
=> []
=> []
=> [] => ? ∊ {0,0}
[2,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[4]
=> []
=> []
=> [] => ? ∊ {0,2}
[3,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[5]
=> []
=> []
=> [] => ? ∊ {0,2}
[4,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,2}
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[6]
=> []
=> []
=> [] => ? ∊ {1,2}
[5,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2}
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[7]
=> []
=> []
=> [] => ? ∊ {2,2,3}
[6,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {2,2,3}
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {2,2,3}
[8]
=> []
=> []
=> [] => ? ∊ {0,1,2,2,3}
[7,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1,2,2,3}
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 0
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 2
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 0
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 0
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 3
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {0,1,2,2,3}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {0,1,2,2,3}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? ∊ {0,1,2,2,3}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,0,1,2,2,2,3,3}
[8,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,0,1,2,2,2,3,3}
[7,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 1
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ? ∊ {0,0,0,1,2,2,2,3,3}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {0,0,0,1,2,2,2,3,3}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,1,2,2,2,3,3}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => ? ∊ {0,0,0,1,2,2,2,3,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? ∊ {0,0,0,1,2,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,0,1,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1,8] => ? ∊ {0,0,0,1,2,2,2,3,3}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[9,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1,8] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1,9] => ? ∊ {0,0,0,0,0,0,1,1,2,2,2,2,3,3,3,3}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[10,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[5,5,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[5,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[5,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {0,0,0,0,0,0,0,0,0,0,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3}
Description
The number of augmented double ascents of a permutation.
An augmented double ascent of a permutation π is a double ascent of the augmented permutation ˜π obtained from π by adding an initial 0.
A double ascent of ˜π then is a position i such that ˜π(i)<˜π(i+1)<˜π(i+2).
Matching statistic: St001960
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 80%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00025: Dyck paths —to 132-avoiding permutation⟶ Permutations
St001960: Permutations ⟶ ℤResult quality: 56% ●values known / values provided: 56%●distinct values known / distinct values provided: 80%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? ∊ {0,0}
[1,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0}
[3]
=> []
=> []
=> [] => ? ∊ {0,1}
[2,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,1}
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[4]
=> []
=> []
=> [] => ? ∊ {1,2}
[3,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2}
[2,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[5]
=> []
=> []
=> [] => ? ∊ {1,2}
[4,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {1,2}
[3,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[6]
=> []
=> []
=> [] => ? ∊ {0,2}
[5,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,2}
[4,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2
[7]
=> []
=> []
=> [] => ? ∊ {0,2,3}
[6,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,2,3}
[5,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {0,2,3}
[8]
=> []
=> []
=> [] => ? ∊ {0,0,1,3,3}
[7,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,1,3,3}
[6,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [2,3,1] => 0
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [2,1,3] => 0
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [4,3,2,1] => 2
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [3,4,2,1] => 1
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,2,3] => 0
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [3,2,4,1] => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [3,2,1,4] => 1
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [2,3,4,1] => 0
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [4,3,5,2,1] => 2
[3,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,1,2,4] => 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,5,1] => 2
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,5] => 2
[2,2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,2,3,4] => 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,2,5] => 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {0,0,1,3,3}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {0,0,1,3,3}
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,1,3,3}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,1,1,1,2,3,3,3}
[8,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,1,1,1,2,3,3,3}
[7,2]
=> [2]
=> [1,0,1,0]
=> [2,1] => 0
[7,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,2] => 0
[6,3]
=> [3]
=> [1,0,1,0,1,0]
=> [3,2,1] => 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ? ∊ {0,0,1,1,1,2,3,3,3}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {0,0,1,1,1,2,3,3,3}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {0,0,1,1,1,2,3,3,3}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => ? ∊ {0,0,1,1,1,2,3,3,3}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? ∊ {0,0,1,1,1,2,3,3,3}
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,1,1,1,2,3,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1,8] => ? ∊ {0,0,1,1,1,2,3,3,3}
[10]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[9,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[4,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[4,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[4,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[4,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[3,3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,3,6,1] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[3,3,1,1,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,7,2,1] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[3,2,2,1,1,1]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,2,6] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[3,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,7,1] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [5,4,1,2,3,6] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,2,7] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,8,1] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1,8] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,3,2,1,9] => ? ∊ {0,0,0,0,1,1,1,2,2,2,2,3,3,3,3,3}
[11]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[10,1]
=> [1]
=> [1,0]
=> [1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[5,5,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,6,4,3,2,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[5,4,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,6,3,2,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[5,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,6,2,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
[5,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,6,1] => ? ∊ {0,0,0,0,0,0,0,1,1,1,1,1,1,1,2,2,2,2,2,2,2,3,3,3,3,3,3,3}
Description
The number of descents of a permutation minus one if its first entry is not one.
This statistic appears in [1, Theorem 2.3] in a gamma-positivity result, see also [2].
Matching statistic: St001586
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001586: Integer partitions ⟶ ℤResult quality: 53% ●values known / values provided: 53%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? = 0
[1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> 0
[3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? = 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> [2]
=> 0
[1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 2
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 0
[2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> 0
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? = 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 0
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 2
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? = 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> 0
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 0
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [5,4]
=> 0
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 0
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 2
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 2
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> []
=> ? ∊ {0,2,2,3}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6]
=> 0
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> 0
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [6,5]
=> 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,3]
=> 0
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> [6,5,4]
=> 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 0
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> 0
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3]
=> ? ∊ {0,2,2,3}
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> 2
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2]
=> ? ∊ {0,2,2,3}
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? ∊ {0,2,2,3}
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> []
=> ? ∊ {0,1,2,2,2,3,3}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7]
=> 0
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> [5]
=> 0
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [7,6]
=> 0
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [6,4]
=> 0
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> [7,6,5]
=> ? ∊ {0,1,2,2,2,3,3}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,2]
=> 0
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0]
=> [6,5,3]
=> 2
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4]
=> ? ∊ {0,1,2,2,2,3,3}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 0
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [5,4,1]
=> 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,3,2]
=> 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2]
=> 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3]
=> ? ∊ {0,1,2,2,2,3,3}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,1]
=> ? ∊ {0,1,2,2,2,3,3}
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2]
=> ? ∊ {0,1,2,2,2,3,3}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> ? ∊ {0,1,2,2,2,3,3}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8]
=> 0
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [8,7,6]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> [6,5,4,2,1]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,1]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1]
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> [9,8,7]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0,1,0]
=> [8,7,5]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0,1,0]
=> [8,7,6,4]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0]
=> [7,6,5,2]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0]
=> [7,6,4,3]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,3]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,2,2,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> [7,6,5,3,2]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,2]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,2,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1]
=> ? ∊ {0,0,0,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3}
[11]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0]
=> []
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[9,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,1,0]
=> [10,9]
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[8,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
[7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,1,0]
=> ?
=> ? ∊ {0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,2,2,2,3,3,3,3,3,3,3,3}
Description
The number of odd parts smaller than the largest even part in an integer partition.
Matching statistic: St001413
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St001413: Binary words ⟶ ℤResult quality: 43% ●values known / values provided: 43%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1,0]
=> []
=> ? => ? = 0
[2]
=> [1,0,1,0]
=> [1]
=> 1 => 0
[1,1]
=> [1,1,0,0]
=> []
=> ? => ? = 0
[3]
=> [1,0,1,0,1,0]
=> [2,1]
=> 01 => 0
[2,1]
=> [1,0,1,1,0,0]
=> [1,1]
=> 11 => 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [1]
=> 1 => 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 101 => 0
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [2,2,1]
=> 001 => 1
[2,2]
=> [1,1,1,0,0,0]
=> []
=> ? => ? = 2
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,1,1]
=> 011 => 0
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 01 => 0
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 0101 => 0
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> 1101 => 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1]
=> 111 => 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> 1001 => 2
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 0 => 0
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> 1011 => 0
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 101 => 0
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> 10101 => 0
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> 00101 => 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> 0001 => 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [4,3,3,2,1]
=> 01101 => 2
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> 1 => 0
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> 1111 => 2
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> 01001 => 0
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? => ? = 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 10 => 0
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1,1]
=> 01011 => 0
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 0101 => 0
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [6,5,4,3,2,1]
=> ? => ? ∊ {1,1,2,3}
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [5,5,4,3,2,1]
=> ? => ? ∊ {1,1,2,3}
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [3,3,3,2,1]
=> 11101 => 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [5,4,4,3,2,1]
=> ? => ? ∊ {1,1,2,3}
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> 0111 => 0
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> 00001 => 2
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [5,4,3,3,2,1]
=> ? => ? ∊ {1,1,2,3}
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 11 => 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> 1111 => 2
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [4,3,1,1,1]
=> 01111 => 0
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,2,1]
=> 101001 => 0
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1 => 0
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> 010 => 0
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1,1]
=> 101011 => 0
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2,1]
=> 10101 => 0
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [7,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [6,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [4,4,4,3,2,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [6,5,5,4,3,2,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [3,2,2,2,1]
=> 10001 => 0
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [5,3,3,3,2,1]
=> 111101 => 2
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [6,5,4,4,3,2,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 01 => 0
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [4,2,1,1,1]
=> 00111 => 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [2,2,2,2,1]
=> 00001 => 2
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [5,4,2,2,2,1]
=> 100001 => 3
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,3,2,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> 11 => 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> 011 => 0
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [4,1,1,1,1]
=> 01111 => 0
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [5,4,3,1,1,1]
=> 101111 => 0
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,2,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> 1 => 0
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3]
=> 01 => 0
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [5,4,3,2]
=> 1010 => 0
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2,1]
=> ? => ? ∊ {0,0,1,1,1,1,2,2,3}
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [8,7,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [7,7,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [5,5,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [7,6,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [4,3,3,3,2,1]
=> 011101 => 0
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [6,4,4,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [7,6,5,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [6,5,3,3,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[4,2,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,2,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[3,2,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,1,1,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [6,5,4,3,2]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [7,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,1,1,1,1,1,1,2,2,2,2,3,3}
[10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [9,8,7,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [8,8,7,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [6,6,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [8,7,7,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [5,4,4,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [7,5,5,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [8,7,6,6,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[6,3,1]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [6,4,3,3,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [4,4,4,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [7,6,4,4,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,5,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[5,3,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [6,5,3,2,2,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[5,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [6,3,3,3,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [7,6,5,3,3,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [8,7,6,5,4,4,3,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[4,3,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [6,5,4,2,1,1,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
[4,2,2,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [6,5,2,2,2,2,1]
=> ? => ? ∊ {0,0,0,0,0,0,0,0,1,1,1,1,1,1,1,1,2,2,2,2,2,2,3,3,3,3,3}
Description
Half the length of the longest even length palindromic prefix of a binary word.
The following 84 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001435The number of missing boxes in the first row. St001438The number of missing boxes of a skew partition. St001552The number of inversions between excedances and fixed points of a permutation. St001520The number of strict 3-descents. St000534The number of 2-rises of a permutation. St000512The number of invariant subsets of size 3 when acting with a permutation of given cycle type. St001423The number of distinct cubes in a binary word. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St000718The largest Laplacian eigenvalue of a graph if it is integral. St001498The normalised height of a Nakayama algebra with magnitude 1. St001651The Frankl number of a lattice. St000260The radius of a connected graph. St000058The order of a permutation. St000454The largest eigenvalue of a graph if it is integral. St000259The diameter of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St001629The coefficient of the integer composition in the quasisymmetric expansion of the relabelling action of the symmetric group on cycles. St000850The number of 1/2-balanced pairs in a poset. St001095The number of non-isomorphic posets with precisely one further covering relation. St000633The size of the automorphism group of a poset. St001399The distinguishing number of a poset. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001199The dominant dimension of eAe for the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000741The Colin de Verdière graph invariant. St000516The number of stretching pairs of a permutation. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St001964The interval resolution global dimension of a poset. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000365The number of double ascents of a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000650The number of 3-rises of a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001130The number of two successive successions in a permutation. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001550The number of inversions between exceedances where the greater exceedance is linked. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001715The number of non-records in a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000358The number of occurrences of the pattern 31-2. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001514The dimension of the top of the Auslander-Reiten translate of the regular modules as a bimodule. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St001816Eigenvalues of the top-to-random operator acting on a simple module. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St000461The rix statistic of a permutation. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c0,c1,...,cn−1] by adding c0 to cn−1. St001510The number of self-evacuating linear extensions of a finite poset. St001820The size of the image of the pop stack sorting operator. St001632The number of indecomposable injective modules I with dimExt1(I,A)=1 for the incidence algebra A of a poset. St000181The number of connected components of the Hasse diagram for the poset. St001490The number of connected components of a skew partition. St001890The maximum magnitude of the Möbius function of a poset. St001330The hat guessing number of a graph. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001862The number of crossings of a signed permutation. St001882The number of occurrences of a type-B 231 pattern in a signed permutation. St001487The number of inner corners of a skew partition. St000455The second largest eigenvalue of a graph if it is integral. St001846The number of elements which do not have a complement in the lattice.
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