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Your data matches 34 different statistics following compositions of up to 3 maps.
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Matching statistic: St001392
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
St001392: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 1
[1,1]
=> 0
[3]
=> 2
[2,1]
=> 0
[1,1,1]
=> 0
[4]
=> 3
[3,1]
=> 2
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 4
[4,1]
=> 3
[3,2]
=> 1
[3,1,1]
=> 2
[2,2,1]
=> 0
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 5
[5,1]
=> 4
[4,2]
=> 3
[4,1,1]
=> 3
[3,3]
=> 2
[3,2,1]
=> 0
[3,1,1,1]
=> 2
[2,2,2]
=> 1
[2,2,1,1]
=> 0
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 6
[6,1]
=> 5
[5,2]
=> 4
[5,1,1]
=> 4
[4,3]
=> 2
[4,2,1]
=> 3
[4,1,1,1]
=> 3
[3,3,1]
=> 2
[3,2,2]
=> 1
[3,2,1,1]
=> 0
[3,1,1,1,1]
=> 2
[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]
=> 7
[7,1]
=> 6
[6,2]
=> 5
[6,1,1]
=> 5
[5,3]
=> 4
[5,2,1]
=> 4
Description
The largest nonnegative integer which is not a part and is smaller than the largest part of the partition.
Matching statistic: St001232
(load all 19 compositions to match this statistic)
(load all 19 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 62%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
St001232: Dyck paths ⟶ ℤResult quality: 21% ●values known / values provided: 21%●distinct values known / distinct values provided: 62%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 1
[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]
=> [1,1,0,1,0,0]
=> 2
[4]
=> []
=> []
=> []
=> ? ∊ {0,3}
[3,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[2,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,3}
[5]
=> []
=> []
=> []
=> ? ∊ {0,3,4}
[4,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[3,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[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]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,3,4}
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,3,4}
[6]
=> []
=> []
=> []
=> ? ∊ {0,0,2,3,3,5}
[5,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[4,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,2,3,3,5}
[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]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,2,3,3,5}
[2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,2,3,3,5}
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,2,3,3,5}
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,2,3,3,5}
[7]
=> []
=> []
=> []
=> ? ∊ {0,0,0,2,3,4,5,6}
[6,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[5,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,2,3,4,5,6}
[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]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,2,3,4,5,6}
[3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,2,3,4,5,6}
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,2,3,4,5,6}
[2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,2,3,4,5,6}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,2,3,4,5,6}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,2,3,4,5,6}
[8]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[7,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[6,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[5,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[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]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[4,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[2,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[2,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,3,3,3,4,4,5,5,6,7}
[9]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[8,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[7,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[7,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[6,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[6,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[5,4]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[5,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[5,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[5,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[4,4,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[3,3,3]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[3,2,2,2]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[3,2,2,1,1]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[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]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? ∊ {0,0,0,0,0,2,3,3,3,3,4,4,4,5,5,5,6,6,7,8}
[9,1]
=> [1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0
[8,2]
=> [2]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> 1
[8,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> 2
[7,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0
[6,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> 2
[6,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[5,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 2
[5,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 3
Description
The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
Matching statistic: St000456
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 88%
Mp00100: Dyck paths —touch composition⟶ Integer compositions
Mp00184: Integer compositions —to threshold graph⟶ Graphs
St000456: Graphs ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 88%
Values
[1]
=> [1,0,1,0]
=> [1,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [2,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => ([(1,2)],3)
=> ? = 1 + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => ([(0,1),(0,2),(1,2)],3)
=> 3 = 2 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => ([(2,3)],4)
=> ? = 0 + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1] => ([(0,3),(1,3),(2,3)],4)
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => ([(1,3),(2,3)],4)
=> ? ∊ {1,2,3} + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3] => ([(2,3)],4)
=> ? ∊ {1,2,3} + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,3} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,2,4} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,2,4} + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,2,4} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,4} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 6 = 5 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,2,3,3,4} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {1,2,2,3,3,4} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {1,2,2,3,3,4} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {1,2,2,3,3,4} + 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {1,2,2,3,3,4} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,4] => ([(3,4)],5)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => ([(6,7)],8)
=> ? ∊ {0,2,2,2,3,3,4,5,6} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,3,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 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] => ([(6,7)],8)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 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]
=> [1,8] => ([(7,8)],9)
=> ? ∊ {0,1,2,2,3,3,3,3,4,4,5,5,6,7} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => ([(0,9),(1,9),(2,9),(3,9),(4,9),(5,9),(6,9),(7,9),(8,9)],10)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,8),(7,8)],9)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [7,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [4,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[4,3,2]
=> [1,1,0,0,1,0,1,0,1,0]
=> [2,1,1,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 8 = 7 + 1
[4,3,1,1]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,2,1,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 6 = 5 + 1
[4,2,2,1]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,2,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[3,3,3]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> [3,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[3,3,2,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[3,3,1,1,1]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> [2,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [1,5] => ([(4,5)],6)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,6] => ([(5,6)],7)
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,4,4,4,4,5,6,6,8} + 1
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [6,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[5,3,2]
=> [1,1,1,0,0,1,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,2,2,1]
=> [1,1,0,1,0,1,1,0,0,0,1,0]
=> [5,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,4,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 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: St001265
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001265: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 31%
Mp00102: Dyck paths —rise composition⟶ Integer compositions
Mp00231: Integer compositions —bounce path⟶ Dyck paths
St001265: Dyck paths ⟶ ℤResult quality: 14% ●values known / values provided: 14%●distinct values known / distinct values provided: 31%
Values
[1]
=> [1,0,1,0]
=> [1,1] => [1,0,1,0]
=> 0
[2]
=> [1,1,0,0,1,0]
=> [2,1] => [1,1,0,0,1,0]
=> 0
[1,1]
=> [1,0,1,1,0,0]
=> [1,2] => [1,0,1,1,0,0]
=> 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [3,1] => [1,1,1,0,0,0,1,0]
=> 0
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1] => [1,0,1,0,1,0]
=> 0
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,3] => [1,0,1,1,1,0,0,0]
=> 2
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4,1] => [1,1,1,1,0,0,0,0,1,0]
=> 0
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [2,2] => [1,1,0,0,1,1,0,0]
=> 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,4] => [1,0,1,1,1,1,0,0,0,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [2,1,1] => [1,1,0,0,1,0,1,0]
=> 0
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,2,1] => [1,0,1,1,0,0,1,0]
=> 2
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,2] => [1,0,1,0,1,1,0,0]
=> 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,5] => [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [6,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5}
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [3,2] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1] => [1,0,1,0,1,0,1,0]
=> 0
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [2,3] => [1,1,0,0,1,1,1,0,0,0]
=> 2
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,6] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,5}
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [7,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,3,5,6}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,3,5,6}
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> 2
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> 0
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> 3
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,3] => [1,0,1,0,1,1,1,0,0,0]
=> 2
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,3,5,6}
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,7] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {0,3,5,6}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,5,5,6,7}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,5,5,6,7}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,2,3,4,5,5,6,7}
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,2,3,4,5,5,6,7}
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> 0
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> 0
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> 0
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> 2
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> 0
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> 4
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> 3
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> 2
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> 0
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,5,5,6,7}
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {0,2,3,4,5,5,6,7}
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,2,3,4,5,5,6,7}
[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]
=> [1,8] => [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {0,2,3,4,5,5,6,7}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [9,1] => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [5,2,1] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [3,3,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[3,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,1,0,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,3,3] => [1,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> [1,5,2] => [1,0,1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[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]
=> [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[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]
=> [1,9] => [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? ∊ {0,0,2,3,3,3,4,5,5,5,6,6,7,8}
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [10,1] => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [8,1,1] => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [7,1,1] => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [6,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> [6,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [5,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> [4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [5,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [4,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [4,2,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [3,2,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [2,4,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [5,2] => [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,5,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,4,1,1] => [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[4,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [1,6,1] => [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,4,2] => [1,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,3,2,1] => [1,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,5,1,1] => [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,7,1] => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [2,5] => [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,2,4] => [1,0,1,1,0,0,1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,2,2,3,3,3,4,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8,9}
Description
The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra.
Matching statistic: St001583
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 38%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St001583: Permutations ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 38%
Values
[1]
=> []
=> []
=> [] => ? = 0
[2]
=> []
=> []
=> [] => ? ∊ {0,1}
[1,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,1}
[3]
=> []
=> []
=> [] => ? ∊ {0,2}
[2,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,2}
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4]
=> []
=> []
=> [] => ? ∊ {2,3}
[3,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {2,3}
[2,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[5]
=> []
=> []
=> [] => ? ∊ {3,4}
[4,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {3,4}
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[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] => 0
[6]
=> []
=> []
=> [] => ? ∊ {0,4,5}
[5,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,4,5}
[4,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[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] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? ∊ {0,4,5}
[7]
=> []
=> []
=> [] => ? ∊ {0,0,2,4,4,6}
[6,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,0,2,4,4,6}
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 5
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ? ∊ {0,0,2,4,4,6}
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ? ∊ {0,0,2,4,4,6}
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? ∊ {0,0,2,4,4,6}
[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] => ? ∊ {0,0,2,4,4,6}
[8]
=> []
=> []
=> [] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[7,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[6,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[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] => 6
[4,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 5
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[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] => ? ∊ {0,0,0,1,2,3,3,4,4,4,5,7}
[9]
=> []
=> []
=> [] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[8,1]
=> [1]
=> [[1]]
=> [1] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[7,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[7,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[6,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[6,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[5,4]
=> [4]
=> [[1,2,3,4]]
=> [1,2,3,4] => 6
[5,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [2,1,3,4] => 5
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[5,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [3,2,1,4] => 3
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[4,4,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [2,1,3,4,5] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[4,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [3,4,1,2,5] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[4,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [3,2,1,4,5] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[4,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [4,2,5,1,3] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[4,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [4,3,2,1,5] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[3,3,3]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[3,3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [4,2,5,1,3,6] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[3,3,1,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [4,3,2,1,5,6] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[3,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[3,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [5,3,2,6,1,4] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [5,4,3,2,1,6] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [6,4,7,2,5,1,3] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [6,4,3,2,7,1,5] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1,7] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[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] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[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] => ? ∊ {0,0,0,0,0,1,2,2,3,3,3,4,4,4,4,5,5,6,7,8}
[8,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[8,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[7,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 3
[7,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [2,1,3] => 2
Description
The projective dimension of the simple module corresponding to the point in the poset of the symmetric group under bruhat order.
Matching statistic: St001498
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 31%
Mp00103: Dyck paths —peeling map⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001498: Dyck paths ⟶ ℤResult quality: 13% ●values known / values provided: 13%●distinct values known / distinct values provided: 31%
Values
[1]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> ? = 0
[2]
=> [1,1,0,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,1}
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,1}
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,2}
[2,1]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> ? ∊ {0,0,2}
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,2}
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1}
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1}
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1}
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 0
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,2,3,4}
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,2,3,4}
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,2,3,4}
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,1,2,3,4}
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,1,2,3,4}
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,1,0,0,0]
=> 0
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,1,0,0,0,0]
=> 0
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,0,0,0]
=> 2
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,2,3,4,5}
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,2,3,4,5}
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,2,3,4,5}
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> ? ∊ {0,0,1,2,3,4,5}
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,2,3,4,5}
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,2,3,4,5}
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,1,2,3,4,5}
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,0,0]
=> 3
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,1,0,0,0]
=> 0
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0]
=> 0
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,1,0,0,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,0,1,2,2,4,5,6}
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,1,1,0,0,0]
=> 3
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,1,1,0,0,0,0]
=> 2
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0,1,1,0,0]
=> 4
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,1,0,0,0]
=> 3
[2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[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]
=> [1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,0,0,1,1,5,5,6,7}
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,7,8}
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0,1,0,1,0]
=> [1,1,1,1,1,0,0,1,0,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,7,8}
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,1,1,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,7,8}
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,1,1,0,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,1,0,0,1,1,1,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,7,8}
[6,3]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 0
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,1,0,0,0]
=> 2
[6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 0
[5,4]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> 2
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,7,8}
[5,2,2]
=> [1,1,1,0,0,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,7,8}
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,7,8}
[5,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,0,0,0]
=> 3
[4,4,1]
=> [1,1,1,0,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {0,0,0,0,1,1,2,2,3,3,3,4,4,4,5,5,5,6,6,7,8}
[4,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 0
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,0]
=> 3
[2,2,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4
[2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 0
[6,4]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 0
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[6,2,2]
=> [1,1,1,1,0,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,1,0,0,0]
=> 2
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 3
[6,1,1,1,1]
=> [1,1,0,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 0
[5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 0
[5,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,1,1,0,0,0,0]
=> 0
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,1,1,0,0,0]
=> 3
[3,3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,1,0,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,0,1,1,0,0]
=> 3
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0,1,1,0,0]
=> 4
[2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 0
[2,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,1,1,0,0,0]
=> 0
[6,5]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,1,1,1,0,0,0,0,0]
=> 0
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,1,1,0,0,0,0]
=> 2
[6,3,2]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> 2
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> 3
[6,2,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> 4
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: St000772
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 25%
Mp00129: Dyck paths —to 321-avoiding permutation (Billey-Jockusch-Stanley)⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000772: Graphs ⟶ ℤResult quality: 11% ●values known / values provided: 11%●distinct values known / distinct values provided: 25%
Values
[1]
=> [1,0,1,0]
=> [2,1] => ([(0,1)],2)
=> 1 = 0 + 1
[2]
=> [1,1,0,0,1,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,1} + 1
[1,1]
=> [1,0,1,1,0,0]
=> [2,1,3] => ([(1,2)],3)
=> ? ∊ {0,1} + 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,2} + 1
[2,1]
=> [1,0,1,0,1,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 1 = 0 + 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [2,1,3,4] => ([(2,3)],4)
=> ? ∊ {0,2} + 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {1,2,3} + 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [3,1,4,2] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,3,2,4] => ([(2,3)],4)
=> ? ∊ {1,2,3} + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [2,4,1,3] => ([(0,3),(1,2),(2,3)],4)
=> 1 = 0 + 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [2,1,3,4,5] => ([(3,4)],5)
=> ? ∊ {1,2,3} + 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,1,2,3,4} + 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1,2,5,3] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,2,3,4} + 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [2,1,4,3] => ([(0,3),(1,2)],4)
=> ? ∊ {0,1,2,3,4} + 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [2,3,1,4] => ([(1,3),(2,3)],4)
=> ? ∊ {0,1,2,3,4} + 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [2,5,1,3,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [2,1,3,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,1,2,3,4} + 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ? ∊ {0,0,2,2,3,3,4,5} + 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1,2,3,6,4] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,4,2,5,3] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,2,2,3,3,4,5} + 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [3,1,2,5,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,2,2,3,3,4,5} + 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,2,4,3,5] => ([(3,4)],5)
=> ? ∊ {0,0,2,2,3,3,4,5} + 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 2 = 1 + 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [2,1,5,3,4] => ([(0,1),(2,4),(3,4)],5)
=> ? ∊ {0,0,2,2,3,3,4,5} + 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,3,2,4,5] => ([(3,4)],5)
=> ? ∊ {0,0,2,2,3,3,4,5} + 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [2,4,1,3,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,0,2,2,3,3,4,5} + 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [2,6,1,3,4,5] => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> 1 = 0 + 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] => ([(5,6)],7)
=> ? ∊ {0,0,2,2,3,3,4,5} + 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [6,1,2,3,4,7,5] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,5,2,3,6,4] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [4,1,2,3,6,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [3,4,1,5,2] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [2,1,3,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [3,1,4,2,5] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,3,5,2,4] => ([(1,4),(2,3),(3,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [2,4,5,1,3] => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 1 = 0 + 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [2,1,6,3,4,5] => ([(0,1),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [2,3,1,4,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [2,5,1,3,4,6] => ([(1,5),(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 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] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 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] => ([(6,7)],8)
=> ? ∊ {0,1,2,2,2,3,3,4,4,5,6} + 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [7,1,2,3,4,5,8,6] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,6,2,3,4,7,5] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [5,1,2,3,4,7,6] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,2,5,3,6,4] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [4,5,1,2,6,3] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [3,1,2,4,6,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,2,3,5,4,6] => ([(4,5)],6)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> [3,1,4,5,2] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[4,2,2]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,3,2,5,4] => ([(1,4),(2,3)],5)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[4,2,1,1]
=> [1,0,1,1,0,1,0,0,1,0]
=> [2,4,1,5,3] => ([(0,4),(1,3),(2,3),(2,4)],5)
=> 1 = 0 + 1
[4,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [2,1,3,6,4,5] => ([(1,2),(3,5),(4,5)],6)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[3,3,2]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,3,4,2,5] => ([(2,4),(3,4)],5)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[3,3,1,1]
=> [1,0,1,1,0,0,1,1,0,0]
=> [2,1,4,3,5] => ([(1,4),(2,3)],5)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [2,3,5,1,4] => ([(0,4),(1,4),(2,3),(3,4)],5)
=> 1 = 0 + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [2,5,6,1,3,4] => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 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] => ([(0,1),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[2,2,2,2]
=> [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,3,2,4,5,6] => ([(4,5)],6)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> [2,4,1,3,5,6] => ([(2,5),(3,4),(4,5)],6)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 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] => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 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] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,6),(6,7)],8)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 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] => ([(7,8)],9)
=> ? ∊ {0,1,1,2,2,2,3,3,3,3,4,4,4,5,5,6,7} + 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,2,3,4,5,6,7,8,10,9] => ([(8,9)],10)
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8} + 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [8,1,2,3,4,5,6,9,7] => ([(0,8),(1,8),(2,8),(3,8),(4,8),(5,8),(6,7),(7,8)],9)
=> ? ∊ {0,0,1,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8} + 1
[6,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,0,1,0]
=> [5,6,1,2,3,7,4] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1 = 0 + 1
[5,3,1]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [4,1,5,2,6,3] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,2,1,1]
=> [1,1,0,1,1,0,1,0,0,0,1,0]
=> [3,5,1,2,6,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [2,5,1,6,3,4] => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> [2,4,6,1,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[3,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [2,6,7,1,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1 = 0 + 1
[6,3,1]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0]
=> [5,1,6,2,3,7,4] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1 = 0 + 1
[6,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,0,1,0]
=> [4,6,1,2,3,7,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> 1 = 0 + 1
[5,4,1]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> [4,1,2,5,6,3] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[5,3,1,1]
=> [1,1,0,1,1,0,0,1,0,0,1,0]
=> [3,1,5,2,6,4] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[5,2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0,1,0]
=> [2,5,1,3,6,4] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> [2,4,1,6,3,5] => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
=> 1 = 0 + 1
[4,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,1,0,0,0]
=> [2,6,1,7,3,4,5] => ([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)
=> 1 = 0 + 1
[3,2,2,2,1]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [2,3,6,1,4,5] => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 1 = 0 + 1
[3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [2,5,7,1,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> 1 = 0 + 1
[6,4,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0]
=> [5,1,2,6,3,7,4] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 1
[6,3,1,1]
=> [1,1,1,0,1,1,0,0,1,0,0,0,1,0]
=> [4,1,6,2,3,7,5] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1 = 0 + 1
[6,2,1,1,1]
=> [1,1,0,1,1,1,0,1,0,0,0,0,1,0]
=> [3,6,1,2,4,7,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[5,3,2,1]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [3,4,5,1,6,2] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 0 + 1
[5,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [2,6,1,3,7,4,5] => ([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[4,3,3,1]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [3,1,4,6,2,5] => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 1 = 0 + 1
[4,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [2,4,5,6,1,3] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 0 + 1
[4,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,1,0,0,0]
=> [2,5,1,7,3,4,6] => ([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)
=> 1 = 0 + 1
[3,2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,1,0,0,0,0]
=> [2,4,7,1,3,5,6] => ([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 1
[6,5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0]
=> [5,1,2,3,6,7,4] => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
=> 1 = 0 + 1
[6,4,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0,1,0]
=> [4,1,2,6,3,7,5] => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
=> 1 = 0 + 1
[6,3,2,1]
=> [1,1,1,0,1,0,1,0,1,0,0,0,1,0]
=> [4,5,6,1,2,7,3] => ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> 1 = 0 + 1
[6,3,1,1,1]
=> [1,1,0,1,1,1,0,0,1,0,0,0,1,0]
=> [3,1,6,2,4,7,5] => ([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)
=> 1 = 0 + 1
[6,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0,1,0]
=> [2,6,1,3,4,7,5] => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
=> 1 = 0 + 1
[5,4,2,1]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [3,4,1,5,6,2] => ([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> 1 = 0 + 1
[5,3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [2,4,5,1,6,3] => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 1 = 0 + 1
Description
The multiplicity of the largest distance Laplacian eigenvalue in a connected graph.
The distance Laplacian of a graph is the (symmetric) matrix with row and column sums $0$, which has the negative distances between two vertices as its off-diagonal entries. This statistic is the largest multiplicity of an eigenvalue.
For example, the cycle on four vertices has distance Laplacian
$$
\left(\begin{array}{rrrr}
4 & -1 & -2 & -1 \\
-1 & 4 & -1 & -2 \\
-2 & -1 & 4 & -1 \\
-1 & -2 & -1 & 4
\end{array}\right).
$$
Its eigenvalues are $0,4,4,6$, so the statistic is $1$.
The path on four vertices has eigenvalues $0, 4.7\dots, 6, 9.2\dots$ and therefore also statistic $1$.
The graphs with statistic $n-1$, $n-2$ and $n-3$ have been characterised, see [1].
Matching statistic: St000205
(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
St000205: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 44%
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000205: Integer partitions ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1,0]
=> [1,0]
=> []
=> ? = 0
[2]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? = 1
[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]
=> []
=> ? = 2
[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]
=> 0
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? = 3
[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]
=> 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 2
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,2,4}
[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]
=> 3
[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]
=> ? ∊ {1,2,4}
[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]
=> ? ∊ {1,2,4}
[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]
=> []
=> ? ∊ {2,2,3,3,4}
[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]
=> ? ∊ {2,2,3,3,4}
[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]
=> 1
[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]
=> ? ∊ {2,2,3,3,4}
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 0
[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]
=> 5
[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]
=> ? ∊ {2,2,3,3,4}
[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]
=> ? ∊ {2,2,3,3,4}
[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]
=> []
=> ? ∊ {0,2,2,2,3,4,5,6}
[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]
=> ? ∊ {0,2,2,2,3,4,5,6}
[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]
=> 3
[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]
=> ? ∊ {0,2,2,2,3,4,5,6}
[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]
=> 0
[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]
=> 1
[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]
=> ? ∊ {0,2,2,2,3,4,5,6}
[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]
=> ? ∊ {0,2,2,2,3,4,5,6}
[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]
=> 4
[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]
=> ? ∊ {0,2,2,2,3,4,5,6}
[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]
=> ? ∊ {0,2,2,2,3,4,5,6}
[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]
=> ? ∊ {0,2,2,2,3,4,5,6}
[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]
=> []
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> 1
[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]
=> 3
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> 2
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> ? ∊ {0,1,2,2,3,3,3,4,4,4,5,5,6,7}
[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]
=> []
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[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
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> [6]
=> 0
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0]
=> [8,7]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0]
=> [4]
=> 0
[6,2,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0,1,0]
=> [7,5]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[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]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> 0
[5,3,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0]
=> [6,3]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[5,2,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0,1,0,1,0]
=> [7,6,4]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[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]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [5,1]
=> 0
[4,3,2]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,2]
=> 1
[4,3,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0]
=> [6,5,2]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[4,2,2,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0]
=> [6,4,3]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[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]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[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]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 0
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0]
=> [5,3,1]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0]
=> [6,5,4,1]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [4,3,2]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[3,2,2,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0]
=> [6,5,3,2]
=> ? ∊ {0,0,1,2,2,2,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7,8}
[8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> [7]
=> 0
[7,3]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0]
=> [5]
=> 0
[6,4]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0]
=> [3]
=> 0
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> 0
[5,4,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [6,2]
=> 1
[5,3,2]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0]
=> [5,3]
=> 3
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [4,1]
=> 0
[4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> [3,2]
=> 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [5,2,1]
=> 7
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,3,1]
=> 5
[9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0,0]
=> [8]
=> 0
Description
Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight.
Given $\lambda$ count how many ''integer partitions'' $w$ (weight) there are, such that
$P_{\lambda,w}$ is non-integral, i.e., $w$ such that the Gelfand-Tsetlin polytope $P_{\lambda,w}$ has at least one non-integral vertex.
Matching statistic: St000733
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Mp00226: Standard tableaux —row-to-column-descents⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Values
[1]
=> [[1]]
=> [[1]]
=> [[1]]
=> 1 = 0 + 1
[2]
=> [[1,2]]
=> [[1,2]]
=> [[1],[2]]
=> 2 = 1 + 1
[1,1]
=> [[1],[2]]
=> [[1],[2]]
=> [[1,2]]
=> 1 = 0 + 1
[3]
=> [[1,2,3]]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1 = 0 + 1
[1,1,1]
=> [[1],[2],[3]]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1 = 0 + 1
[4]
=> [[1,2,3,4]]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[3,1]
=> [[1,3,4],[2]]
=> [[1,2,4],[3]]
=> [[1,3],[2],[4]]
=> 3 = 2 + 1
[2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[2,1,1]
=> [[1,4],[2],[3]]
=> [[1,3],[2],[4]]
=> [[1,2,4],[3]]
=> 1 = 0 + 1
[1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[5]
=> [[1,2,3,4,5]]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 5 = 4 + 1
[4,1]
=> [[1,3,4,5],[2]]
=> [[1,2,4,5],[3]]
=> [[1,3],[2],[4],[5]]
=> 4 = 3 + 1
[3,2]
=> [[1,2,5],[3,4]]
=> [[1,3,4],[2,5]]
=> [[1,2],[3,5],[4]]
=> 2 = 1 + 1
[3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,3,5],[2],[4]]
=> [[1,2,4],[3],[5]]
=> 3 = 2 + 1
[2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1 = 0 + 1
[2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,4],[2],[3],[5]]
=> [[1,2,3,5],[4]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1 = 0 + 1
[6]
=> [[1,2,3,4,5,6]]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 6 = 5 + 1
[5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2,4,5,6],[3]]
=> [[1,3],[2],[4],[5],[6]]
=> 5 = 4 + 1
[4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3,4,6],[2,5]]
=> [[1,2],[3,5],[4],[6]]
=> 4 = 3 + 1
[4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,3,5,6],[2],[4]]
=> [[1,2,4],[3],[5],[6]]
=> 4 = 3 + 1
[3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,3,5],[2,4,6]]
=> [[1,2],[3,4],[5,6]]
=> 3 = 2 + 1
[3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> [[1,3,6],[2,5],[4]]
=> 1 = 0 + 1
[3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,4,6],[2],[3],[5]]
=> [[1,2,3,5],[4],[6]]
=> 3 = 2 + 1
[2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,4],[2,5],[3,6]]
=> [[1,2,3],[4,5,6]]
=> 2 = 1 + 1
[2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,3],[2,5],[4],[6]]
=> [[1,2,4,6],[3,5]]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,5],[2],[3],[4],[6]]
=> [[1,2,3,4,6],[5]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1 = 0 + 1
[7]
=> [[1,2,3,4,5,6,7]]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7 = 6 + 1
[6,1]
=> [[1,3,4,5,6,7],[2]]
=> [[1,2,4,5,6,7],[3]]
=> [[1,3],[2],[4],[5],[6],[7]]
=> 6 = 5 + 1
[5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3,4,6,7],[2,5]]
=> [[1,2],[3,5],[4],[6],[7]]
=> 5 = 4 + 1
[5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [[1,3,5,6,7],[2],[4]]
=> [[1,2,4],[3],[5],[6],[7]]
=> 5 = 4 + 1
[4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[1,3,5,6],[2,4,7]]
=> [[1,2],[3,4],[5,7],[6]]
=> 3 = 2 + 1
[4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[1,2,4,7],[3,5],[6]]
=> [[1,3,6],[2,5],[4],[7]]
=> 4 = 3 + 1
[4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[1,4,6,7],[2],[3],[5]]
=> [[1,2,3,5],[4],[6],[7]]
=> 4 = 3 + 1
[3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6,7],[4]]
=> [[1,3,4],[2,6],[5,7]]
=> 3 = 2 + 1
[3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,4,6],[2,5],[3,7]]
=> [[1,2,3],[4,5,7],[6]]
=> 2 = 1 + 1
[3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,3,5],[2,6],[4],[7]]
=> [[1,2,4,7],[3,6],[5]]
=> 1 = 0 + 1
[3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,5,7],[2],[3],[4],[6]]
=> [[1,2,3,4,6],[5],[7]]
=> 3 = 2 + 1
[2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,4],[2,6],[3],[5],[7]]
=> [[1,2,3,5,7],[4,6]]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,6],[2],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 1 = 0 + 1
[8]
=> [[1,2,3,4,5,6,7,8]]
=> [[1,2,3,4,5,6,7,8]]
=> [[1],[2],[3],[4],[5],[6],[7],[8]]
=> 8 = 7 + 1
[7,1]
=> [[1,3,4,5,6,7,8],[2]]
=> [[1,2,4,5,6,7,8],[3]]
=> [[1,3],[2],[4],[5],[6],[7],[8]]
=> 7 = 6 + 1
[6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3,4,6,7,8],[2,5]]
=> [[1,2],[3,5],[4],[6],[7],[8]]
=> 6 = 5 + 1
[6,1,1]
=> [[1,4,5,6,7,8],[2],[3]]
=> [[1,3,5,6,7,8],[2],[4]]
=> [[1,2,4],[3],[5],[6],[7],[8]]
=> 6 = 5 + 1
[5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,3,5,6,8],[2,4,7]]
=> [[1,2],[3,4],[5,7],[6],[8]]
=> 5 = 4 + 1
[5,2,1]
=> [[1,3,6,7,8],[2,5],[4]]
=> [[1,2,4,7,8],[3,5],[6]]
=> [[1,3,6],[2,5],[4],[7],[8]]
=> 5 = 4 + 1
[8,1]
=> [[1,3,4,5,6,7,8,9],[2]]
=> [[1,2,4,5,6,7,8,9],[3]]
=> [[1,3],[2],[4],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3,4,6,7,8,9],[2,5]]
=> [[1,2],[3,5],[4],[6],[7],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[7,1,1]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> [[1,3,5,6,7,8,9],[2],[4]]
=> [[1,2,4],[3],[5],[6],[7],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[6,3]
=> [[1,2,3,7,8,9],[4,5,6]]
=> [[1,3,5,6,8,9],[2,4,7]]
=> [[1,2],[3,4],[5,7],[6],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[6,2,1]
=> [[1,3,6,7,8,9],[2,5],[4]]
=> [[1,2,4,7,8,9],[3,5],[6]]
=> [[1,3,6],[2,5],[4],[7],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[6,1,1,1]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> [[1,4,6,7,8,9],[2],[3],[5]]
=> [[1,2,3,5],[4],[6],[7],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,3,5,7,8],[2,4,6,9]]
=> [[1,2],[3,4],[5,6],[7,9],[8]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[5,3,1]
=> [[1,3,4,8,9],[2,6,7],[5]]
=> [[1,2,5,7,9],[3,6,8],[4]]
=> [[1,3,4],[2,6],[5,8],[7],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,4,6,8,9],[2,5],[3,7]]
=> [[1,2,3],[4,5,7],[6],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[5,2,1,1]
=> [[1,4,7,8,9],[2,6],[3],[5]]
=> [[1,3,5,8,9],[2,6],[4],[7]]
=> [[1,2,4,7],[3,6],[5],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[5,1,1,1,1]
=> [[1,6,7,8,9],[2],[3],[4],[5]]
=> [[1,5,7,8,9],[2],[3],[4],[6]]
=> [[1,2,3,4,6],[5],[7],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> [[1,2,5,6],[3,7,8,9],[4]]
=> [[1,3,4],[2,7],[5,8],[6,9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> [[1,3,4,7],[2,6,8],[5,9]]
=> [[1,2,5],[3,6,9],[4,8],[7]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> [[1,2,6,9],[3,4],[5,7],[8]]
=> [[1,3,5,8],[2,4,7],[6],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> [[1,4,6,9],[2,7],[3],[5],[8]]
=> [[1,2,3,5,8],[4,7],[6],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> [[1,6,8,9],[2],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6],[8],[9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> [[1,2,4],[3,5,7],[6,8],[9]]
=> [[1,3,6,9],[2,5,8],[4,7]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> [[1,4,7],[2,8,9],[3],[5],[6]]
=> [[1,2,3,5,6],[4,8],[7,9]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,5,8],[2,6],[3,7],[4,9]]
=> [[1,2,3,4],[5,6,7,9],[8]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> [[1,3,7],[2,5],[4,8],[6],[9]]
=> [[1,2,4,6,9],[3,5,8],[7]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> [[1,5,7],[2,8],[3],[4],[6],[9]]
=> [[1,2,3,4,6,9],[5,8],[7]]
=> ? ∊ {0,0,0,1,1,2,3,3,3,3,3,4,4,4,4,5,5,5,6,6,7} + 1
[9,1]
=> [[1,3,4,5,6,7,8,9,10],[2]]
=> [[1,2,4,5,6,7,8,9,10],[3]]
=> [[1,3],[2],[4],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3,4,6,7,8,9,10],[2,5]]
=> [[1,2],[3,5],[4],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[8,1,1]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> [[1,3,5,6,7,8,9,10],[2],[4]]
=> [[1,2,4],[3],[5],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[7,3]
=> [[1,2,3,7,8,9,10],[4,5,6]]
=> [[1,3,5,6,8,9,10],[2,4,7]]
=> [[1,2],[3,4],[5,7],[6],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[7,2,1]
=> [[1,3,6,7,8,9,10],[2,5],[4]]
=> [[1,2,4,7,8,9,10],[3,5],[6]]
=> [[1,3,6],[2,5],[4],[7],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[7,1,1,1]
=> [[1,5,6,7,8,9,10],[2],[3],[4]]
=> [[1,4,6,7,8,9,10],[2],[3],[5]]
=> [[1,2,3,5],[4],[6],[7],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [[1,3,5,7,8,10],[2,4,6,9]]
=> [[1,2],[3,4],[5,6],[7,9],[8],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[6,3,1]
=> [[1,3,4,8,9,10],[2,6,7],[5]]
=> [[1,2,5,7,9,10],[3,6,8],[4]]
=> [[1,3,4],[2,6],[5,8],[7],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,4,6,8,9,10],[2,5],[3,7]]
=> [[1,2,3],[4,5,7],[6],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[6,2,1,1]
=> [[1,4,7,8,9,10],[2,6],[3],[5]]
=> [[1,3,5,8,9,10],[2,6],[4],[7]]
=> [[1,2,4,7],[3,6],[5],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[6,1,1,1,1]
=> [[1,6,7,8,9,10],[2],[3],[4],[5]]
=> [[1,5,7,8,9,10],[2],[3],[4],[6]]
=> [[1,2,3,4,6],[5],[7],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[5,4,1]
=> [[1,3,4,5,10],[2,7,8,9],[6]]
=> [[1,2,5,6,9],[3,7,8,10],[4]]
=> [[1,3,4],[2,7],[5,8],[6,10],[9]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[5,3,2]
=> [[1,2,5,9,10],[3,4,8],[6,7]]
=> [[1,3,4,7,10],[2,6,8],[5,9]]
=> [[1,2,5],[3,6,9],[4,8],[7],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[5,2,2,1]
=> [[1,3,8,9,10],[2,5],[4,7],[6]]
=> [[1,2,6,9,10],[3,4],[5,7],[8]]
=> [[1,3,5,8],[2,4,7],[6],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[5,2,1,1,1]
=> [[1,5,8,9,10],[2,7],[3],[4],[6]]
=> [[1,4,6,9,10],[2,7],[3],[5],[8]]
=> [[1,2,3,5,8],[4,7],[6],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[5,1,1,1,1,1]
=> [[1,7,8,9,10],[2],[3],[4],[5],[6]]
=> [[1,6,8,9,10],[2],[3],[4],[5],[7]]
=> [[1,2,3,4,5,7],[6],[8],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8]]
=> [[1,3,4,8],[2,6,9,10],[5,7]]
=> [[1,2,5],[3,6,7],[4,9],[8,10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> [[1,3,6,7],[2,8,9,10],[4],[5]]
=> [[1,2,4,5],[3,8],[6,9],[7,10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7,9],[2,5,8],[3,6,10]]
=> [[1,2,3],[4,5,6],[7,8,10],[9]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[4,3,1,1,1]
=> [[1,5,6,10],[2,8,9],[3],[4],[7]]
=> [[1,4,7,9],[2,8,10],[3],[5],[6]]
=> [[1,2,3,5,6],[4,8],[7,10],[9]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,5,8,10],[2,6],[3,7],[4,9]]
=> [[1,2,3,4],[5,6,7,9],[8],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[4,2,2,1,1]
=> [[1,4,9,10],[2,6],[3,8],[5],[7]]
=> [[1,3,7,10],[2,5],[4,8],[6],[9]]
=> [[1,2,4,6,9],[3,5,8],[7],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[4,2,1,1,1,1]
=> [[1,6,9,10],[2,8],[3],[4],[5],[7]]
=> [[1,5,7,10],[2,8],[3],[4],[6],[9]]
=> [[1,2,3,4,6,9],[5,8],[7],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[4,1,1,1,1,1,1]
=> [[1,8,9,10],[2],[3],[4],[5],[6],[7]]
=> [[1,7,9,10],[2],[3],[4],[5],[6],[8]]
=> [[1,2,3,4,5,6,8],[7],[9],[10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> [[1,2,5],[3,7,8],[4,9,10],[6]]
=> [[1,3,4,6],[2,7,9],[5,8,10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> [[1,4,6],[2,5,9],[3,8],[7,10]]
=> [[1,2,3,7],[4,5,8,10],[6,9]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> [[1,3,5],[2,6,8],[4,9],[7],[10]]
=> [[1,2,4,7,10],[3,6,9],[5,8]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> [[1,5,8],[2,9,10],[3],[4],[6],[7]]
=> [[1,2,3,4,6,7],[5,9],[8,10]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
[3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> [[1,2,8],[3,4],[5,6],[7,9],[10]]
=> [[1,3,5,7,10],[2,4,6,9],[8]]
=> ? ∊ {0,0,0,0,1,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6,7,7,8} + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000777
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 44%
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00160: Permutations —graph of inversions⟶ Graphs
St000777: Graphs ⟶ ℤResult quality: 7% ●values known / values provided: 7%●distinct values known / distinct values provided: 44%
Values
[1]
=> [1,0]
=> [1] => ([],1)
=> 1 = 0 + 1
[2]
=> [1,0,1,0]
=> [1,2] => ([],2)
=> ? = 0 + 1
[1,1]
=> [1,1,0,0]
=> [2,1] => ([(0,1)],2)
=> 2 = 1 + 1
[3]
=> [1,0,1,0,1,0]
=> [1,2,3] => ([],3)
=> ? ∊ {0,0} + 1
[2,1]
=> [1,0,1,1,0,0]
=> [1,3,2] => ([(1,2)],3)
=> ? ∊ {0,0} + 1
[1,1,1]
=> [1,1,0,1,0,0]
=> [2,3,1] => ([(0,2),(1,2)],3)
=> 3 = 2 + 1
[4]
=> [1,0,1,0,1,0,1,0]
=> [1,2,3,4] => ([],4)
=> ? ∊ {0,0,3} + 1
[3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,2,4,3] => ([(2,3)],4)
=> ? ∊ {0,0,3} + 1
[2,2]
=> [1,1,1,0,0,0]
=> [3,2,1] => ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,3,4,2] => ([(1,3),(2,3)],4)
=> ? ∊ {0,0,3} + 1
[1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [2,3,4,1] => ([(0,3),(1,3),(2,3)],4)
=> 3 = 2 + 1
[5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => ([],5)
=> ? ∊ {0,0,0,1,4} + 1
[4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => ([(3,4)],5)
=> ? ∊ {0,0,0,1,4} + 1
[3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,4,3,2] => ([(1,2),(1,3),(2,3)],4)
=> ? ∊ {0,0,0,1,4} + 1
[3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => ([(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,4} + 1
[2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [3,2,4,1] => ([(0,3),(1,2),(1,3),(2,3)],4)
=> 4 = 3 + 1
[2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => ([(1,4),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,1,4} + 1
[1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => ([(0,4),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
[6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6] => ([],6)
=> ? ∊ {0,0,0,0,3,4,5} + 1
[5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,6,5] => ([(4,5)],6)
=> ? ∊ {0,0,0,0,3,4,5} + 1
[4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => ([(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,3,4,5} + 1
[4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,5,6,4] => ([(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,3,4,5} + 1
[3,3]
=> [1,1,1,0,1,0,0,0]
=> [4,2,3,1] => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
[3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => ([(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,3,4,5} + 1
[3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,4,5,6,3] => ([(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,3,4,5} + 1
[2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [4,3,2,1] => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
[2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,2] => ([(1,5),(2,5),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,3,4,5} + 1
[1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
[7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7] => ([],7)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,7,6] => ([(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,6,5,4] => ([(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,6,7,5] => ([(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,5,4,6,3] => ([(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,4] => ([(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [4,2,3,5,1] => ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 5 = 4 + 1
[3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,4,3,5,6,2] => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,3] => ([(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [4,3,2,5,1] => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,1] => ([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,1,2,2,4,5,6} + 1
[1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
[8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8] => ([],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[7,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,8,7] => ([(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,7,6,5] => ([(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[6,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,7,8,6] => ([(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,6,4,5,3] => ([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[5,2,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,2,3,6,5,7,4] => ([(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,2,3,4,6,7,8,5] => ([(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,5,3,4,6,2] => ([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,2,6,5,4,3] => ([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[4,2,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,2,5,4,6,7,3] => ([(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,2,3,5,6,7,8,4] => ([(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[3,3,2]
=> [1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 4 = 3 + 1
[3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [4,2,3,5,6,1] => ([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> [1,5,4,3,6,2] => ([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[3,2,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,4,3,5,6,7,2] => ([(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[3,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,2,4,5,6,7,8,3] => ([(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
[2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [4,3,2,5,6,1] => ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[2,2,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> [3,2,4,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[2,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,3,4,5,6,7,8,2] => ([(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 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]
=> [2,3,4,5,6,7,8,1] => ([(0,7),(1,7),(2,7),(3,7),(4,7),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,1,1,2,3,4,4,5,5,6,7} + 1
[9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5,6,7,8,9] => ([],9)
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,4,5,5,6,6,7,8} + 1
[8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,4,5,6,7,9,8] => ([(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,4,5,5,6,6,7,8} + 1
[7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,2,3,4,5,8,7,6] => ([(5,6),(5,7),(6,7)],8)
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,4,5,5,6,6,7,8} + 1
[7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,2,3,4,5,6,8,9,7] => ([(6,8),(7,8)],9)
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,4,5,5,6,6,7,8} + 1
[6,3]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,2,3,7,5,6,4] => ([(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? ∊ {0,0,0,0,0,0,0,0,1,2,2,2,2,3,3,3,3,4,5,5,6,6,7,8} + 1
[4,4,1]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> [5,2,3,4,6,1] => ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
[3,3,2,1]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> [5,2,4,3,6,1] => ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 6 = 5 + 1
[3,3,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [4,2,3,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[2,2,2,2,1]
=> [1,1,1,1,0,1,0,0,0,1,0,0]
=> [5,3,4,2,6,1] => ([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[2,2,2,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [4,3,2,5,6,7,1] => ([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[4,4,2]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[4,4,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> [5,2,3,4,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[3,3,3,1]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> [5,4,3,2,6,1] => ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[3,3,2,2]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => ([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 5 = 4 + 1
[3,3,2,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [5,2,4,3,6,7,1] => ([(0,6),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => ([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> [5,3,4,2,6,7,1] => ([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[5,5,1]
=> [1,1,1,0,1,0,1,0,1,0,0,1,0,0]
=> [6,2,3,4,5,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[4,4,3]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => ([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[4,4,2,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,0]
=> [6,2,3,5,4,7,1] => ([(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[3,3,3,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4 = 3 + 1
[3,3,3,1,1]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [5,4,3,2,6,7,1] => ([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[3,3,2,2,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [6,2,4,5,3,7,1] => ([(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 7 = 6 + 1
[2,2,2,2,2,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,0]
=> [6,3,4,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [7,2,3,4,5,6,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
[5,5,2]
=> [1,1,1,0,1,0,1,0,1,1,0,0,0,0]
=> [7,2,3,4,6,5,1] => ([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 4 = 3 + 1
[4,4,4]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
[4,4,3,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [6,2,5,4,3,7,1] => ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [7,2,3,5,6,4,1] => ([(0,5),(0,6),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
[3,3,3,3]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
[3,3,3,2,1]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [6,4,3,5,2,7,1] => ([(0,6),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 6 = 5 + 1
[3,3,2,2,2]
=> [1,1,1,0,1,1,0,1,0,1,0,0,0,0]
=> [7,2,4,5,6,3,1] => ([(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 5 = 4 + 1
Description
The number of distinct eigenvalues of the distance Laplacian of a connected graph.
The following 24 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000462The major index minus the number of excedences of a permutation. St000054The first entry of the permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000801The number of occurrences of the vincular pattern |312 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001438The number of missing boxes of a skew partition. St001557The number of inversions of the second entry of a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St000060The greater neighbor of the maximum. St000154The sum of the descent bottoms of a permutation. St000199The column of the unique '1' in the last row of the alternating sign matrix. St000200The row of the unique '1' in the last column of the alternating sign matrix. St001000Number of indecomposable modules with projective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001879The number of indecomposable summands of the top of the first syzygy of the dual of the regular module in the incidence algebra of the lattice. St001875The number of simple modules with projective dimension at most 1. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St001463The number of distinct columns in the nullspace of a graph. St000259The diameter of a connected graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St000422The energy of a graph, if it is integral.
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