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Your data matches 148 different statistics following compositions of up to 3 maps.
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Matching statistic: St000685
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(load all 2 compositions to match this statistic)
Mp00199: Dyck paths —prime Dyck path⟶ Dyck paths
St000685: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000685: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,0]
=> [1,1,0,0]
=> 1
[1,0,1,0]
=> [1,1,0,1,0,0]
=> 1
[1,1,0,0]
=> [1,1,1,0,0,0]
=> 1
[1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0]
=> 2
[1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 1
[1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 3
[1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2
[1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,1,0,1,1,0,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,1,1,0,0,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,1,0,0,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 2
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,1,0,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,1,1,0,1,1,0,0,0,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,1,1,0,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1,0,1,1,1,0,0,0,0,0]
=> 1
Description
The dominant dimension of the LNakayama algebra associated to a Dyck path.
To every Dyck path there is an LNakayama algebra associated as described in [[St000684]].
Matching statistic: St000655
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%
Mp00312: Integer partitions —Glaisher-Franklin⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000655: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 93%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0,1,0]
=> [2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,0,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? = 2
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [6,1,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,2,2,2]
=> [1,1,0,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,2,2,1,1]
=> [1,0,1,1,0,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2,2,1,1]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,0,0,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,2,2,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 2
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? = 3
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [4,4,3,2,1]
=> [3,2,2,2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,3,4}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [5,4,2,2,1]
=> [5,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,3,4}
[1,0,1,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2,1]
=> [2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,3,4}
[1,0,1,0,1,1,0,1,0,1,0,0]
=> [4,3,2,2,1]
=> [3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,3,4}
[1,0,1,0,1,1,1,0,0,0,1,0]
=> [5,2,2,2,1]
=> [5,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,3,4}
[1,0,1,0,1,1,1,0,0,1,0,0]
=> [4,2,2,2,1]
=> [2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,3,4}
[1,1,0,0,1,1,0,0,1,1,0,0]
=> [4,4,2,2]
=> [2,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,2,3,4}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,1,2,3,4}
Description
The length of the minimal rise of a Dyck path.
For the length of a maximal rise, see [[St000444]].
Matching statistic: St000781
(load all 18 compositions to match this statistic)
(load all 18 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000781: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [3]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [3]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [5,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [5,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [4,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [5]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [5]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [4]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [4]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [4]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [3]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [3]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [3]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [2]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [7,3]
=> 1
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? ∊ {1,1,1,1,3,4}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? ∊ {1,1,1,1,3,4}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,1,3,4}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,1,3,4}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,1,3,4}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,1,3,4}
Description
The number of proper colouring schemes of a Ferrers diagram.
A colouring of a Ferrers diagram is proper if no two cells in a row or in a column have the same colour. The minimal number of colours needed is the maximum of the length and the first part of the partition, because we can restrict a latin square to the shape. We can associate to each colouring the integer partition recording how often each colour is used, see [1].
This statistic is the number of distinct such integer partitions that occur.
Matching statistic: St001256
(load all 30 compositions to match this statistic)
(load all 30 compositions to match this statistic)
Mp00027: Dyck paths —to partition⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001256: Dyck paths ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> []
=> ?
=> ?
=> ? = 1
[1,0,1,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,0,1,1,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,0,1,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2}
[1,1,0,1,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2}
[1,1,1,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2}
[1,0,1,0,1,0,1,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,0,0,1,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,0,1,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,1,3}
[1,1,1,1,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [3,3,2,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [4,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [3,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [2,2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [4,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [3,3,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [4,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [3,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [2,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [3,3,2]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [4,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [3,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,2,2]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [3,3,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,2,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [4,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [3,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,1,1]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,3]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [2,2]
=> [2]
=> [1,1,0,0,1,0]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,1]
=> [1]
=> [1,0,1,0]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [4]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,0,0,1,0,0]
=> [3]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,1,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,1,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> [4,3,2,1]
=> [1,0,1,0,1,0,1,0,1,0]
=> 2
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [5]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [4]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [3]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [2]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> []
=> []
=> ? ∊ {1,2,2,2,3,4}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ?
=> ?
=> ? ∊ {1,2,2,2,3,4}
Description
Number of simple reflexive modules that are 2-stable reflexive.
See Definition 3.1. in the reference for the definition of 2-stable reflexive.
Matching statistic: St000706
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 89%●distinct values known / distinct values provided: 25%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000706: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 89%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,2}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,1,2}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,1,2}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 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]
=> []
=> ? ∊ {1,1,3}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,3}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,3}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {2,2,3}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {2,2,3}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {2,2,3}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
Description
The product of the factorials of the multiplicities of an integer partition.
Matching statistic: St000993
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 89%●distinct values known / distinct values provided: 25%
Mp00101: Dyck paths —decomposition reverse⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 89%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,1,0,0]
=> []
=> ? ∊ {1,1}
[1,1,0,0]
=> [1,1,0,0]
=> [1,0,1,0]
=> [1]
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,2}
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,1,2}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,1,2}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 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]
=> []
=> ? ∊ {1,1,3}
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,3}
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,3}
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {2,2,3}
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {2,2,3}
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {2,2,3}
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? ∊ {1,1,1,2,2,2,2,2,3,4}
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St001901
(load all 10 compositions to match this statistic)
(load all 10 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 89%●distinct values known / distinct values provided: 25%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001901: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 89%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? = 2
[1,1,1,0,0,0]
=> [3,2,1] => [2,1]
=> [1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,3}
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4]
=> []
=> ? ∊ {1,3}
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [2,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [4,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,4,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,3,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,4,6,3,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,2,5,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,4,2,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,3,5,2,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,3,6,5,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,5,3,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,2,2,2,2,2,3,4}
Description
The largest multiplicity of an irreducible representation contained in the higher Lie character for an integer partition.
Matching statistic: St001934
(load all 6 compositions to match this statistic)
(load all 6 compositions to match this statistic)
Mp00031: Dyck paths —to 312-avoiding permutation⟶ Permutations
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Mp00108: Permutations —cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001934: Integer partitions ⟶ ℤResult quality: 50% ●values known / values provided: 89%●distinct values known / distinct values provided: 50%
Values
[1,0]
=> [1] => [1]
=> []
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,1]
=> [1]
=> 1
[1,1,0,0]
=> [2,1] => [2]
=> []
=> ? = 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [2,1]
=> [1]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1]
=> [1]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [3]
=> []
=> ? = 2
[1,1,1,0,0,0]
=> [3,2,1] => [2,1]
=> [1]
=> 1
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [2,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [3,1]
=> [1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,2]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [3,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [4]
=> []
=> ? ∊ {1,3}
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [2,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,0]
=> [3,4,2,1] => [4]
=> []
=> ? ∊ {1,3}
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [2,2]
=> [2]
=> 1
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,1,1,1,1]
=> [1,1,1,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [4,1]
=> [1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [3,1,1]
=> [1,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,4,5,3,2] => [4,1]
=> [1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [3,2]
=> [2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,2,1]
=> [2,1]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [3,2]
=> [2]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [4,1]
=> [1]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [4,1]
=> [1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,4,5,3,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [3,2]
=> [2]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [2,1,1,1]
=> [1,1,1]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [2,2,1]
=> [2,1]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [4,1]
=> [1]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,1,1]
=> [1,1]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [3,4,2,1,5] => [4,1]
=> [1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [3,4,2,5,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,0,1,0,1,0,0,0]
=> [3,4,5,2,1] => [3,2]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [3,5,4,2,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [4,3,5,2,1] => [5]
=> []
=> ? ∊ {1,1,2,2,3}
[1,1,0,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,0,1,0,1,1,0,1,0,0,0]
=> [2,3,5,6,4,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,0,1,1,0,1,0,0,1,0,0]
=> [2,4,5,3,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,0,1,1,0,1,1,0,0,0,0]
=> [2,4,6,5,3,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,0,1,1,1,0,0,1,0,0,0]
=> [2,5,4,6,3,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,1,0,1,0,0,1,0,1,0,0]
=> [3,4,2,5,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [3,4,5,6,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,1,0,1,1,0,0,0,1,0,0]
=> [3,5,4,2,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,1,1,0,0,1,0,0,1,0,0]
=> [4,3,5,2,6,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [4,3,6,5,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [4,6,5,3,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [5,4,6,3,2,1] => [6]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,2,2,3,4}
Description
The number of monotone factorisations of genus zero of a permutation of given cycle type.
A monotone factorisation of genus zero of a permutation $\pi\in\mathfrak S_n$ with $\ell$ cycles, including fixed points, is a tuple of $r = n - \ell$ transpositions
$$
(a_1, b_1),\dots,(a_r, b_r)
$$
with $b_1 \leq \dots \leq b_r$ and $a_i < b_i$ for all $i$, whose product, in this order, is $\pi$.
For example, the cycle $(2,3,1)$ has the two factorizations $(2,3)(1,3)$ and $(1,2)(2,3)$.
Matching statistic: St001199
(load all 46 compositions to match this statistic)
(load all 46 compositions to match this statistic)
Mp00023: Dyck paths —to non-crossing permutation⟶ Permutations
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 88%●distinct values known / distinct values provided: 25%
Mp00068: Permutations —Simion-Schmidt map⟶ Permutations
Mp00127: Permutations —left-to-right-maxima to Dyck path⟶ Dyck paths
St001199: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 88%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1] => [1] => [1,0]
=> ? = 1
[1,0,1,0]
=> [1,2] => [1,2] => [1,0,1,0]
=> 1
[1,1,0,0]
=> [2,1] => [2,1] => [1,1,0,0]
=> ? = 1
[1,0,1,0,1,0]
=> [1,2,3] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,0,1,1,0,0]
=> [1,3,2] => [1,3,2] => [1,0,1,1,0,0]
=> 1
[1,1,0,0,1,0]
=> [2,1,3] => [2,1,3] => [1,1,0,0,1,0]
=> 1
[1,1,0,1,0,0]
=> [2,3,1] => [2,3,1] => [1,1,0,1,0,0]
=> 1
[1,1,1,0,0,0]
=> [3,2,1] => [3,2,1] => [1,1,1,0,0,0]
=> ? = 2
[1,0,1,0,1,0,1,0]
=> [1,2,3,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,2,4,3] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,3,2,4] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,3,4,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,4,3,2] => [1,4,3,2] => [1,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,0]
=> [2,1,3,4] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,0,1,1,0,0]
=> [2,1,4,3] => [2,1,4,3] => [1,1,0,0,1,1,0,0]
=> 1
[1,1,0,1,0,0,1,0]
=> [2,3,1,4] => [2,4,1,3] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,0,1,0,0]
=> [2,3,4,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,0,1,1,0,0,0]
=> [2,4,3,1] => [2,4,3,1] => [1,1,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,0,1,0]
=> [3,2,1,4] => [3,2,1,4] => [1,1,1,0,0,0,1,0]
=> 1
[1,1,1,0,0,1,0,0]
=> [3,2,4,1] => [3,2,4,1] => [1,1,1,0,0,1,0,0]
=> 1
[1,1,1,0,1,0,0,0]
=> [4,2,3,1] => [4,2,3,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,3}
[1,1,1,1,0,0,0,0]
=> [4,3,2,1] => [4,3,2,1] => [1,1,1,1,0,0,0,0]
=> ? ∊ {1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [1,2,3,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,2,3,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,2,4,3,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,2,4,5,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,2,5,4,3] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,3,2,4,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,3,2,5,4] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,3,4,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,3,4,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,3,5,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,4,3,2,5] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,4,3,5,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,5,3,4,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,5,4,3,2] => [1,5,4,3,2] => [1,0,1,1,1,1,0,0,0,0]
=> 1
[1,1,0,0,1,0,1,0,1,0]
=> [2,1,3,4,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [2,1,3,5,4] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [2,1,4,3,5] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [2,1,4,5,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [2,1,5,4,3] => [2,1,5,4,3] => [1,1,0,0,1,1,1,0,0,0]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [2,3,1,4,5] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [2,3,1,5,4] => [2,5,1,4,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [2,3,4,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [2,3,4,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [2,3,5,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [2,4,3,1,5] => [2,5,4,1,3] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [2,4,3,5,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [2,5,3,4,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [2,5,4,3,1] => [2,5,4,3,1] => [1,1,0,1,1,1,0,0,0,0]
=> 1
[1,1,1,0,0,0,1,0,1,0]
=> [3,2,1,4,5] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [3,2,1,5,4] => [3,2,1,5,4] => [1,1,1,0,0,0,1,1,0,0]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [3,2,4,1,5] => [3,2,5,1,4] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [3,2,4,5,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [3,2,5,4,1] => [3,2,5,4,1] => [1,1,1,0,0,1,1,0,0,0]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [5,2,3,4,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3}
[1,1,1,0,1,1,0,0,0,0]
=> [5,2,4,3,1] => [5,2,4,3,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,0,1,0,0,0]
=> [5,3,2,4,1] => [5,3,2,4,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,1,0,0,0,0]
=> [5,3,4,2,1] => [5,3,4,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [5,4,3,2,1] => [5,4,3,2,1] => [1,1,1,1,1,0,0,0,0,0]
=> ? ∊ {1,1,2,2,3}
[1,1,1,0,1,0,1,0,1,0,0,0]
=> [6,2,3,4,5,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,1,0,1,1,0,0,0,0]
=> [6,2,3,5,4,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,1,1,0,0,1,0,0,0]
=> [6,2,4,3,5,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,1,1,0,1,0,0,0,0]
=> [6,2,4,5,3,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [6,2,5,4,3,1] => [6,2,5,4,3,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,1,0,1,0,0,0]
=> [6,3,2,4,5,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,0,1,1,0,0,0,0]
=> [6,3,2,5,4,1] => [6,3,2,5,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,1,0,0,1,0,0,0]
=> [6,3,4,2,5,1] => [6,3,5,2,4,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,1,0,1,0,0,0,0]
=> [6,3,4,5,2,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [6,3,5,4,2,1] => [6,3,5,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,0,0,1,0,0,0]
=> [6,4,3,2,5,1] => [6,4,3,2,5,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,0,1,0,0,0,0]
=> [6,4,3,5,2,1] => [6,4,3,5,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [6,5,3,4,2,1] => [6,5,3,4,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [6,5,4,3,2,1] => [6,5,4,3,2,1] => [1,1,1,1,1,1,0,0,0,0,0,0]
=> ? ∊ {1,1,1,1,1,1,1,2,2,2,2,2,3,4}
Description
The dominant dimension of $eAe$ for the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$.
Matching statistic: St001568
(load all 8 compositions to match this statistic)
(load all 8 compositions to match this statistic)
Mp00099: Dyck paths —bounce path⟶ Dyck paths
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 83%●distinct values known / distinct values provided: 25%
Mp00132: Dyck paths —switch returns and last double rise⟶ Dyck paths
Mp00027: Dyck paths —to partition⟶ Integer partitions
St001568: Integer partitions ⟶ ℤResult quality: 25% ●values known / values provided: 83%●distinct values known / distinct values provided: 25%
Values
[1,0]
=> [1,0]
=> [1,0]
=> []
=> ? = 1
[1,0,1,0]
=> [1,0,1,0]
=> [1,0,1,0]
=> [1]
=> ? ∊ {1,1}
[1,1,0,0]
=> [1,1,0,0]
=> [1,1,0,0]
=> []
=> ? ∊ {1,1}
[1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [1,0,1,0,1,0]
=> [2,1]
=> 1
[1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,1,2}
[1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [1,1,0,0,1,0]
=> [2]
=> 1
[1,1,0,1,0,0]
=> [1,0,1,1,0,0]
=> [1,1,0,1,0,0]
=> [1]
=> ? ∊ {1,1,2}
[1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> [1,1,1,0,0,0]
=> []
=> ? ∊ {1,1,2}
[1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> [2,1]
=> 1
[1,0,1,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,3}
[1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,0,1,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,3}
[1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> [1]
=> ? ∊ {1,1,1,3}
[1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0]
=> []
=> ? ∊ {1,1,1,3}
[1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> [4,3,2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,0,1,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> [3,2,1]
=> 1
[1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> [4,2,1]
=> 1
[1,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,0,1,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,0,1,1,1,0,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> [2,1]
=> 1
[1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {1,1,2,2,3}
[1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> [4,3,2]
=> 1
[1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> [3,2]
=> 1
[1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> [4,3,1]
=> 1
[1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,0,1,1,0,0,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> [3,1]
=> 1
[1,1,0,1,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,0,1,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {1,1,2,2,3}
[1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> [4,3]
=> 1
[1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> [4,2]
=> 1
[1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,1,0,0,1,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> [4,1]
=> 1
[1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,1,0,1,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> [4]
=> 1
[1,1,1,1,0,0,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> [3]
=> 1
[1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> [2]
=> 1
[1,1,1,1,0,1,0,0,0,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1]
=> ? ∊ {1,1,2,2,3}
[1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> []
=> ? ∊ {1,1,2,2,3}
[1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [4,3,2,1]
=> 1
[1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,0]
=> [5,3,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> [5,4,2,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> [5,4,3,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> [5,4,3,1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,0,1,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> [5,4,3]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,0,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> [5,4,2]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,0,1,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,0,1,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> [1]
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
[1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> []
=> ? ∊ {1,1,1,1,1,1,1,1,1,1,1,2,2,2,2,2,3,4}
Description
The smallest positive integer that does not appear twice in the partition.
The following 138 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St000284The Plancherel distribution on integer partitions. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd. St000668The least common multiple of the parts of the partition. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000707The product of the factorials of the parts. St000708The product of the parts of an integer partition. St000770The major index of an integer partition when read from bottom to top. St000815The number of semistandard Young tableaux of partition weight of given shape. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000929The constant term of the character polynomial of an integer partition. St000933The number of multipartitions of sizes given by an integer partition. St001128The exponens consonantiae of a partition. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000934The 2-degree of an integer partition. St000003The number of standard Young tableaux of the partition. St000278The size of the preimage of the map 'to partition' from Integer compositions to Integer partitions. St000326The position of the first one in a binary word after appending a 1 at the end. St000913The number of ways to refine the partition into singletons. St000999Number of indecomposable projective module with injective dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001006Number of simple modules with projective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001009Number of indecomposable injective modules with projective dimension g when g is the global dimension of the Nakayama algebra corresponding to the Dyck path. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001013Number of indecomposable injective modules with codominant dimension equal to the global dimension in the Nakayama algebra corresponding to the Dyck path. St001063Numbers of 3-torsionfree simple modules in the corresponding Nakayama algebra. St001064Number of simple modules in the corresponding Nakayama algebra that are 3-syzygy modules. St001188The number of simple modules $S$ with grade $\inf \{ i \geq 0 | Ext^i(S,A) \neq 0 \}$ at least two in the Nakayama algebra $A$ corresponding to the Dyck path. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001196The global dimension of $A$ minus the global dimension of $eAe$ for the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001244The number of simple modules of projective dimension one that are not 1-regular for the Nakayama algebra associated to a Dyck path. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001507The sum of projective dimension of simple modules with even projective dimension divided by 2 in the LNakayama algebra corresponding to Dyck paths. St001722The number of minimal chains with small intervals between a binary word and the top element. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St000260The radius of a connected graph. St001487The number of inner corners of a skew partition. St001490The number of connected components of a skew partition. St001933The largest multiplicity of a part in an integer partition. St000764The number of strong records in an integer composition. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000805The number of peaks of the associated bargraph. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St001964The interval resolution global dimension of a poset. St001162The minimum jump of a permutation. St001344The neighbouring number of a permutation. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St000667The greatest common divisor of the parts of the partition. St000255The number of reduced Kogan faces with the permutation as type. St000876The number of factors in the Catalan decomposition of a binary word. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St000889The number of alternating sign matrices with the same antidiagonal sums. St001289The vector space dimension of the n-fold tensor product of D(A), where n is maximal such that this n-fold tensor product is nonzero. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St001481The minimal height of a peak of a Dyck path. St000570The Edelman-Greene number of a permutation. St000441The number of successions of a permutation. St000451The length of the longest pattern of the form k 1 2. St000665The number of rafts of a permutation. St000914The sum of the values of the Möbius function of a poset. St001890The maximum magnitude of the Möbius function of a poset. St000788The number of nesting-similar perfect matchings of a perfect matching. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000694The number of affine bounded permutations that project to a given permutation. St001208The number of connected components of the quiver of $A/T$ when $T$ is the 1-tilting module corresponding to the permutation in the Auslander algebra $A$ of $K[x]/(x^n)$. St001590The crossing number of a perfect matching. St001830The chord expansion number of a perfect matching. St001832The number of non-crossing perfect matchings in the chord expansion of a perfect matching. St001859The number of factors of the Stanley symmetric function associated with a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000022The number of fixed points of a permutation. St000534The number of 2-rises of a permutation. St000908The length of the shortest maximal antichain in a poset. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St001735The number of permutations with the same set of runs. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000028The number of stack-sorts needed to sort a permutation. St000456The monochromatic index of a connected graph. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) $[c_0,c_1,...,c_{n−1}]$ by adding $c_0$ to $c_{n−1}$. St000731The number of double exceedences of a permutation. St001330The hat guessing number of a graph. St001432The order dimension of the partition. St001780The order of promotion on the set of standard tableaux of given shape. St001899The total number of irreducible representations contained in the higher Lie character for an integer partition. St001900The number of distinct irreducible representations contained in the higher Lie character for an integer partition. St001908The number of semistandard tableaux of distinct weight whose maximal entry is the length of the partition. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001875The number of simple modules with projective dimension at most 1. St001198The number of simple modules in the algebra $eAe$ with projective dimension at most 1 in the corresponding Nakayama algebra $A$ with minimal faithful projective-injective module $eA$. St001206The maximal dimension of an indecomposable projective $eAe$-module (that is the height of the corresponding Dyck path) of the corresponding Nakayama algebra with minimal faithful projective-injective module $eA$. St000181The number of connected components of the Hasse diagram for the poset. St000618The number of self-evacuating tableaux of given shape. St001280The number of parts of an integer partition that are at least two. St001442The number of standard Young tableaux whose major index is divisible by the size of a given integer partition. St001562The value of the complete homogeneous symmetric function evaluated at 1. St001563The value of the power-sum symmetric function evaluated at 1. St001564The value of the forgotten symmetric functions when all variables set to 1. St001593This is the number of standard Young tableaux of the given shifted shape. St001601The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on trees. St001609The number of coloured trees such that the multiplicities of colours are given by a partition. St001767The largest minimal number of arrows pointing to a cell in the Ferrers diagram in any assignment. St001913The number of preimages of an integer partition in Bulgarian solitaire. St001924The number of cells in an integer partition whose arm and leg length coincide. St001936The number of transitive factorisations of a permutation of given cycle type into star transpositions. St001967The coefficient of the monomial corresponding to the integer partition in a certain power series. St001968The coefficient of the monomial corresponding to the integer partition in a certain power series. St001846The number of elements which do not have a complement in the lattice. St001195The global dimension of the algebra $A/AfA$ of the corresponding Nakayama algebra $A$ with minimal left faithful projective-injective module $Af$. St001876The number of 2-regular simple modules in the incidence algebra of the lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001498The normalised height of a Nakayama algebra with magnitude 1. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St000058The order of a permutation. St001820The size of the image of the pop stack sorting operator. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001095The number of non-isomorphic posets with precisely one further covering relation. St001130The number of two successive successions in a permutation. St000741The Colin de Verdière graph invariant. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000669The number of permutations obtained by switching ascents or descents of size 2. St000237The number of small exceedances. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001556The number of inversions of the third entry of a permutation. St000883The number of longest increasing subsequences of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St001465The number of adjacent transpositions in the cycle decomposition of a permutation.
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