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Matching statistic: St001175
St001175: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0
[2]
=> 0
[1,1]
=> 0
[3]
=> 0
[2,1]
=> 0
[1,1,1]
=> 0
[4]
=> 0
[3,1]
=> 0
[2,2]
=> 1
[2,1,1]
=> 0
[1,1,1,1]
=> 0
[5]
=> 0
[4,1]
=> 0
[3,2]
=> 1
[3,1,1]
=> 0
[2,2,1]
=> 1
[2,1,1,1]
=> 0
[1,1,1,1,1]
=> 0
[6]
=> 0
[5,1]
=> 0
[4,2]
=> 1
[4,1,1]
=> 0
[3,3]
=> 2
[3,2,1]
=> 1
[3,1,1,1]
=> 0
[2,2,2]
=> 2
[2,2,1,1]
=> 1
[2,1,1,1,1]
=> 0
[1,1,1,1,1,1]
=> 0
[7]
=> 0
[6,1]
=> 0
[5,2]
=> 1
[5,1,1]
=> 0
[4,3]
=> 2
[4,2,1]
=> 1
[4,1,1,1]
=> 0
[3,3,1]
=> 2
[3,2,2]
=> 2
[3,2,1,1]
=> 1
[3,1,1,1,1]
=> 0
[2,2,2,1]
=> 2
[2,2,1,1,1]
=> 1
[2,1,1,1,1,1]
=> 0
[1,1,1,1,1,1,1]
=> 0
[8]
=> 0
[7,1]
=> 0
[6,2]
=> 1
[6,1,1]
=> 0
[5,3]
=> 2
[5,2,1]
=> 1
Description
The size of a partition minus the hook length of the base cell.
This is, the number of boxes in the diagram of a partition that are neither in the first row nor in the first column.
Matching statistic: St000377
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000377: Integer partitions ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> 0
[2]
=> []
=> []
=> 0
[1,1]
=> [1]
=> [1]
=> 0
[3]
=> []
=> []
=> 0
[2,1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> 0
[4]
=> []
=> []
=> 0
[3,1]
=> [1]
=> [1]
=> 0
[2,2]
=> [2]
=> [1,1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[5]
=> []
=> []
=> 0
[4,1]
=> [1]
=> [1]
=> 0
[3,2]
=> [2]
=> [1,1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,1]
=> [2,1]
=> [3]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[6]
=> []
=> []
=> 0
[5,1]
=> [1]
=> [1]
=> 0
[4,2]
=> [2]
=> [1,1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> 2
[3,2,1]
=> [2,1]
=> [3]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[2,2,2]
=> [2,2]
=> [4]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[7]
=> []
=> []
=> 0
[6,1]
=> [1]
=> [1]
=> 0
[5,2]
=> [2]
=> [1,1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> 2
[4,2,1]
=> [2,1]
=> [3]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,2,2]
=> [2,2]
=> [4]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> 0
[8]
=> []
=> []
=> 0
[7,1]
=> [1]
=> [1]
=> 0
[6,2]
=> [2]
=> [1,1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> 2
[5,2,1]
=> [2,1]
=> [3]
=> 1
[2,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? ∊ {4,5,6}
[2,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? ∊ {4,5,6}
[2,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [6,4,3]
=> ? ∊ {4,5,6}
[3,3,3,3,3,1]
=> [3,3,3,3,1]
=> [7,2,1,1,1,1]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[3,3,3,3,2,2]
=> [3,3,3,2,2]
=> [7,6]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[3,3,3,3,2,1,1]
=> [3,3,3,2,1,1]
=> [6,3,1,1,1,1]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[3,3,3,2,2,1,1,1]
=> [3,3,2,2,1,1,1]
=> [6,5,2]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[3,2,2,2,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [6,2,2,2,1]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[3,2,2,2,2,2,1,1,1]
=> [2,2,2,2,2,1,1,1]
=> [6,4,1,1,1]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[3,2,2,2,2,1,1,1,1,1]
=> [2,2,2,2,1,1,1,1,1]
=> [6,4,3]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[2,2,2,2,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [6,2,2,2,2]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[2,2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,2,1,1]
=> [6,5,1,1,1]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
[2,2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,2,1,1,1,1]
=> [6,5,3]
=> ? ∊ {4,5,5,6,6,6,7,7,8,8}
Description
The dinv defect of an integer partition.
This is the number of cells c in the diagram of an integer partition λ for which arm(c)−leg(c)∉{0,1}.
Matching statistic: St001176
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1]
=> 0
[3]
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> 0
[4]
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1]
=> 0
[2,2]
=> [2]
=> [1,1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[5]
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> 0
[3,2]
=> [2]
=> [1,1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[6]
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> 0
[4,2]
=> [2]
=> [1,1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[7]
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1]
=> 0
[5,2]
=> [2]
=> [1,1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0
[8]
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1]
=> 0
[6,2]
=> [2]
=> [1,1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> 3
[4,3,1]
=> [3,1]
=> [2,1,1]
=> 2
[4,2,2]
=> [2,2]
=> [2,2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 2
[9]
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> ? = 0
[12]
=> []
=> []
=> ? = 0
[13]
=> []
=> []
=> ? = 0
[14]
=> []
=> []
=> ? = 0
[15]
=> []
=> []
=> ? = 0
[16]
=> []
=> []
=> ? = 0
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000228
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> ?
=> ? = 0
[2]
=> []
=> []
=> ?
=> ? = 0
[1,1]
=> [1]
=> [1]
=> []
=> 0
[3]
=> []
=> []
=> ?
=> ? = 0
[2,1]
=> [1]
=> [1]
=> []
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[4]
=> []
=> []
=> ?
=> ? = 0
[3,1]
=> [1]
=> [1]
=> []
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[1,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[5]
=> []
=> []
=> ?
=> ? = 0
[4,1]
=> [1]
=> [1]
=> []
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[2,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[6]
=> []
=> []
=> ?
=> ? = 0
[5,1]
=> [1]
=> [1]
=> []
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[3,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[2,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[7]
=> []
=> []
=> ?
=> ? = 0
[6,1]
=> [1]
=> [1]
=> []
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[4,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[3,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[2,2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 0
[8]
=> []
=> []
=> ?
=> ? = 0
[7,1]
=> [1]
=> [1]
=> []
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> []
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2
[5,2,1]
=> [2,1]
=> [2,1]
=> [1]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [3]
=> []
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2
[4,2,2]
=> [2,2]
=> [2,2]
=> [2]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0
[3,3,2]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2
[9]
=> []
=> []
=> ?
=> ? = 0
[10]
=> []
=> []
=> ?
=> ? = 0
[11]
=> []
=> []
=> ?
=> ? = 0
[12]
=> []
=> []
=> ?
=> ? = 0
[13]
=> []
=> []
=> ?
=> ? = 0
[14]
=> []
=> []
=> ?
=> ? = 0
[15]
=> []
=> []
=> ?
=> ? = 0
[16]
=> []
=> []
=> ?
=> ? = 0
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000369
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[1,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[2,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[3,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[4,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[3,3]
=> [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[2,2,2]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[5,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[4,3]
=> [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[3,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[3,2,2]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0
[8]
=> []
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1]
=> [1,0,1,0]
=> 0
[6,2]
=> [2]
=> [1,1]
=> [1,0,1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [2]
=> [1,1,0,0,1,0]
=> 0
[5,3]
=> [3]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [2,1]
=> [1,0,1,0,1,0]
=> 0
[4,4]
=> [4]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[4,3,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[4,2,2]
=> [2,2]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 0
[3,3,2]
=> [3,2]
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2
[9]
=> []
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> []
=> ? ∊ {0,5}
[4,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5}
[12]
=> []
=> []
=> []
=> ? ∊ {0,5,5,6}
[5,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,6}
[4,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,6}
[4,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,6}
[13]
=> []
=> []
=> []
=> ? ∊ {0,5,5,5,6,6,6}
[6,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,6,6,6}
[5,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,6,6,6}
[5,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,6,6,6}
[4,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,6,6,6}
[4,4,3,2]
=> [4,3,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,6,6,6}
[4,4,3,1,1]
=> [4,3,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,6,6,6}
[14]
=> []
=> []
=> []
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[7,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[6,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[6,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[5,5,4]
=> [5,4]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[5,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[5,4,3,2]
=> [4,3,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[5,4,3,1,1]
=> [4,3,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[4,4,4,2]
=> [4,4,2]
=> [8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[4,4,4,1,1]
=> [4,4,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[4,4,3,2,1]
=> [4,3,2,1]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[4,4,3,1,1,1]
=> [4,3,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,6,6,6,6,6,7,7}
[15]
=> []
=> []
=> []
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[8,4,3]
=> [4,3]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[7,4,4]
=> [4,4]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[7,4,3,1]
=> [4,3,1]
=> [7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[6,5,4]
=> [5,4]
=> [9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[6,4,4,1]
=> [4,4,1]
=> [8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[6,4,3,2]
=> [4,3,2]
=> [7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[6,4,3,1,1]
=> [4,3,1,1]
=> [7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[5,5,5]
=> [5,5]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[5,5,4,1]
=> [5,4,1]
=> [9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[5,5,3,2]
=> [5,3,2]
=> [7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[5,4,4,2]
=> [4,4,2]
=> [8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[5,4,4,1,1]
=> [4,4,1,1]
=> [8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[5,4,3,2,1]
=> [4,3,2,1]
=> [7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[5,4,3,1,1,1]
=> [4,3,1,1,1]
=> [7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? ∊ {0,5,5,5,5,5,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
Description
The dinv deficit of a Dyck path.
For a Dyck path D of semilength n, this is defined as
\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).
In other words, this is the number of boxes in the partition traced out by D for which the leg-length minus the arm-length is not in \{0,1\}.
See also [[St000376]] for the bounce deficit.
Matching statistic: St000394
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 89% ●values known / values provided: 89%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> []
=> []
=> ? = 0
[2]
=> []
=> []
=> []
=> ? = 0
[1,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3]
=> []
=> []
=> []
=> ? = 0
[2,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[1,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4]
=> []
=> []
=> []
=> ? = 0
[3,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[2,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[5]
=> []
=> []
=> []
=> ? = 0
[4,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[3,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[3,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[6]
=> []
=> []
=> []
=> ? = 0
[5,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[4,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[4,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[3,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[7]
=> []
=> []
=> []
=> ? = 0
[6,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[5,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[5,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[4,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[4,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[3,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[3,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0
[8]
=> []
=> []
=> []
=> ? = 0
[7,1]
=> [1]
=> [1,0]
=> [1,0]
=> 0
[6,2]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[6,1,1]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0
[5,3]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2
[5,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1
[5,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0
[4,4]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3
[4,3,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2
[4,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2
[4,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0
[3,3,2]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3
[3,3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2
[9]
=> []
=> []
=> []
=> ? = 0
[10]
=> []
=> []
=> []
=> ? = 0
[11]
=> []
=> []
=> []
=> ? = 0
[12]
=> []
=> []
=> []
=> ? = 0
[13]
=> []
=> []
=> []
=> ? ∊ {0,0}
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0}
[14]
=> []
=> []
=> []
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,1,2,2,3,4,5}
[15]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,3,3,2,2,1,1]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,3,2,2,1,1,1,1,1]
=> [3,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,3,2,1,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,3,1,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[3,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[2,2,2,2,1,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[2,2,2,1,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[2,2,1,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ?
=> ? ∊ {0,0,0,0,1,1,2,2,2,2,3,3,3,3,4,4,4,4,5,5,5,5,5,6,6,6}
[16]
=> []
=> []
=> []
=> ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,2,3,3,3,3,3,3,3,4,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7}
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000293
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00136: Binary words —rotate back-to-front⟶ Binary words
St000293: Binary words ⟶ ℤResult quality: 87% ●values known / values provided: 87%●distinct values known / distinct values provided: 100%
Values
[1]
=> []
=> => ? => ? = 0
[2]
=> []
=> => ? => ? = 0
[1,1]
=> [1]
=> 10 => 01 => 0
[3]
=> []
=> => ? => ? = 0
[2,1]
=> [1]
=> 10 => 01 => 0
[1,1,1]
=> [1,1]
=> 110 => 011 => 0
[4]
=> []
=> => ? => ? = 0
[3,1]
=> [1]
=> 10 => 01 => 0
[2,2]
=> [2]
=> 100 => 010 => 1
[2,1,1]
=> [1,1]
=> 110 => 011 => 0
[1,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[5]
=> []
=> => ? => ? = 0
[4,1]
=> [1]
=> 10 => 01 => 0
[3,2]
=> [2]
=> 100 => 010 => 1
[3,1,1]
=> [1,1]
=> 110 => 011 => 0
[2,2,1]
=> [2,1]
=> 1010 => 0101 => 1
[2,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 0
[6]
=> []
=> => ? => ? = 0
[5,1]
=> [1]
=> 10 => 01 => 0
[4,2]
=> [2]
=> 100 => 010 => 1
[4,1,1]
=> [1,1]
=> 110 => 011 => 0
[3,3]
=> [3]
=> 1000 => 0100 => 2
[3,2,1]
=> [2,1]
=> 1010 => 0101 => 1
[3,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[2,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[2,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011111 => 0
[7]
=> []
=> => ? => ? = 0
[6,1]
=> [1]
=> 10 => 01 => 0
[5,2]
=> [2]
=> 100 => 010 => 1
[5,1,1]
=> [1,1]
=> 110 => 011 => 0
[4,3]
=> [3]
=> 1000 => 0100 => 2
[4,2,1]
=> [2,1]
=> 1010 => 0101 => 1
[4,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[3,3,1]
=> [3,1]
=> 10010 => 01001 => 2
[3,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[3,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 0
[2,2,2,1]
=> [2,2,1]
=> 11010 => 01101 => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> 101110 => 010111 => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 011111 => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 1111110 => 0111111 => 0
[8]
=> []
=> => ? => ? = 0
[7,1]
=> [1]
=> 10 => 01 => 0
[6,2]
=> [2]
=> 100 => 010 => 1
[6,1,1]
=> [1,1]
=> 110 => 011 => 0
[5,3]
=> [3]
=> 1000 => 0100 => 2
[5,2,1]
=> [2,1]
=> 1010 => 0101 => 1
[5,1,1,1]
=> [1,1,1]
=> 1110 => 0111 => 0
[4,4]
=> [4]
=> 10000 => 01000 => 3
[4,3,1]
=> [3,1]
=> 10010 => 01001 => 2
[4,2,2]
=> [2,2]
=> 1100 => 0110 => 2
[4,2,1,1]
=> [2,1,1]
=> 10110 => 01011 => 1
[4,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 01111 => 0
[3,3,2]
=> [3,2]
=> 10100 => 01010 => 3
[3,3,1,1]
=> [3,1,1]
=> 100110 => 010011 => 2
[9]
=> []
=> => ? => ? = 0
[10]
=> []
=> => ? => ? = 0
[11]
=> []
=> => ? => ? = 0
[12]
=> []
=> => ? => ? ∊ {0,0}
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? ∊ {0,0}
[13]
=> []
=> => ? => ? ∊ {0,0,0,1,2,3}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? ∊ {0,0,0,1,2,3}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? ∊ {0,0,0,1,2,3}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? ∊ {0,0,0,1,2,3}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? ∊ {0,0,0,1,2,3}
[1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? => ? ∊ {0,0,0,1,2,3}
[14]
=> []
=> => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> 10101111110 => 01010111111 => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => 0111101111 => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> 11101111110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => 011011111111 => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> 1011111111110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[2,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> 1111111111110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 11111111111110 => ? => ? ∊ {0,0,0,0,1,1,2,2,2,3,3,3,4,4,4}
[15]
=> []
=> => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> 1010011110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> 1001101110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> 10010111110 => 01001011111 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> 100011111110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> 1011011110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> 10101111110 => 01010111111 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 0111011111 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> 11011111110 => 01101111111 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> 101111111110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[4,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> 111111111110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[3,3,3,2,1,1,1,1]
=> [3,3,2,1,1,1,1]
=> 1101011110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[3,3,3,1,1,1,1,1,1]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => 01100111111 => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
[3,3,2,2,2,1,1,1]
=> [3,2,2,2,1,1,1]
=> 1011101110 => ? => ? ∊ {0,0,0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,5,5,5,5,5}
Description
The number of inversions of a binary word.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 90%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 90%
Values
[1]
=> []
=> []
=> []
=> 0
[2]
=> []
=> []
=> []
=> 0
[1,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[3]
=> []
=> []
=> []
=> 0
[2,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[4]
=> []
=> []
=> []
=> 0
[3,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[2,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[2,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[5]
=> []
=> []
=> []
=> 0
[4,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[3,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[6]
=> []
=> []
=> []
=> 0
[5,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[4,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[4,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[3,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[7]
=> []
=> []
=> []
=> 0
[6,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[5,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[4,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[3,3,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 0
[8]
=> []
=> []
=> []
=> 0
[7,1]
=> [1]
=> [[1]]
=> [[1]]
=> 0
[6,2]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1
[6,1,1]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0
[5,3]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2
[5,2,1]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? = 0
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? ∊ {0,1,2,3,4,5}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ?
=> ? ∊ {0,1,2,3,4,5}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> [[1,2,3,4,5,6,8,10],[7,9,11]]
=> ? ∊ {0,1,2,3,4,5}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ?
=> ? ∊ {0,1,2,3,4,5}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? ∊ {0,1,2,3,4,5}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? ∊ {0,1,2,3,4,5}
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> [[1,2,3,4,5,6,9],[7,10],[8,11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> [[1,2,4,6,8,10],[3,5,7,9,11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> [[1,2,3,4,5,6,8,10],[7,9,11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,11],[2],[3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,2,3,4,5,6,7,8,9,10],[11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> [[1,2,3,4,5,6,7,8,9,10,11]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7,12],[9],[11]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5],[7],[9],[11]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3,12],[4],[5],[6],[7],[9],[11]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> [[1,2,3,4,5,6,7,8,9,11],[10,12]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,12],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ?
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13]]
=> ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,7}
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [[1,2,3,4,8],[5,9],[6,10],[7,11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,6,7,11],[2,9,10],[3],[4],[5],[8]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9],[10],[11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,5,10,11],[2,7],[3,9],[4],[6],[8]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,5,8],[2,7,11],[3,10],[4],[6],[9]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> [[1,2,3,4,5,6,9],[7,10],[8,11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,4,11],[2,6],[3,8],[5,10],[7],[9]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,6,11],[2,8],[3,10],[4],[5],[7],[9]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ?
=> ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
Description
The number of descents of a standard tableau.
Entry i of a standard Young tableau is a descent if i+1 appears in a row below the row of i.
Matching statistic: St000245
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 90%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000245: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 90%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [[1]]
=> [1] => 0
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [[1]]
=> [1] => 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [[1]]
=> [1] => 0
[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]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [[1]]
=> [1] => 0
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [[1]]
=> [1] => 0
[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] => 2
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [[1]]
=> [1] => 0
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [[1]]
=> [1] => 0
[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] => 2
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 0
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [11,9,10,5,6,7,8,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [11,10,9,5,6,7,8,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [10,11,8,9,5,6,7,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,5,6,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
Description
The number of ascents of a permutation.
Matching statistic: St000441
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 90%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000441: Permutations ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 90%
Values
[1]
=> []
=> []
=> [] => 0
[2]
=> []
=> []
=> [] => 0
[1,1]
=> [1]
=> [[1]]
=> [1] => 0
[3]
=> []
=> []
=> [] => 0
[2,1]
=> [1]
=> [[1]]
=> [1] => 0
[1,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4]
=> []
=> []
=> [] => 0
[3,1]
=> [1]
=> [[1]]
=> [1] => 0
[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]
=> []
=> []
=> [] => 0
[4,1]
=> [1]
=> [[1]]
=> [1] => 0
[3,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[3,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[6]
=> []
=> []
=> [] => 0
[5,1]
=> [1]
=> [[1]]
=> [1] => 0
[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] => 2
[3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[7]
=> []
=> []
=> [] => 0
[6,1]
=> [1]
=> [[1]]
=> [1] => 0
[5,2]
=> [2]
=> [[1,2]]
=> [1,2] => 1
[5,1,1]
=> [1,1]
=> [[1],[2]]
=> [2,1] => 0
[4,3]
=> [3]
=> [[1,2,3]]
=> [1,2,3] => 2
[4,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 0
[3,3,1]
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 2
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 2
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 0
[8]
=> []
=> []
=> [] => 0
[7,1]
=> [1]
=> [[1]]
=> [1] => 0
[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] => 2
[5,2,1]
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? = 0
[2,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[2,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,1,2,3,4,5}
[3,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,3,1,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,2,2,2,2,1]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,2,2,2,1,1,1]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,2,2,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,2,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,2,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[3,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> [11,12,9,10,7,8,5,6,3,4,1,2] => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,2,2,2,1,1]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,2,2,1,1,1,1]
=> [2,2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,2,1,1,1,1,1,1]
=> [2,2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,2,1,1,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[2,2,1,1,1,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12],[13]]
=> ? => ? ∊ {0,0,1,1,2,2,2,3,3,3,4,4,4,4,5,5,5,5,6,6,6,6,7}
[4,4,4,3]
=> [4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11]]
=> [9,10,11,5,6,7,8,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,4,2,1]
=> [4,4,2,1]
=> [[1,2,3,4],[5,6,7,8],[9,10],[11]]
=> [11,9,10,5,6,7,8,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,4,1,1,1]
=> [4,4,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9],[10],[11]]
=> [11,10,9,5,6,7,8,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,3,1]
=> [4,3,3,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11]]
=> [11,8,9,10,5,6,7,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,2,2]
=> [4,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11]]
=> [10,11,8,9,5,6,7,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,2,1,1]
=> [4,3,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10],[11]]
=> [11,10,8,9,5,6,7,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,3,1,1,1,1]
=> [4,3,1,1,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,2,2,1]
=> [4,2,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,5,6,1,2,3,4] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,2,1,1,1]
=> [4,2,2,1,1,1]
=> [[1,2,3,4],[5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,2,1,1,1,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,4,1,1,1,1,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,3,2]
=> [3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11]]
=> [10,11,7,8,9,4,5,6,1,2,3] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,3,1,1]
=> [3,3,3,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11]]
=> [11,10,7,8,9,4,5,6,1,2,3] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,2,2,1]
=> [3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11]]
=> [11,9,10,7,8,4,5,6,1,2,3] => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,2,1,1,1]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,3,1,1,1,1,1]
=> [3,3,1,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,2,2]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,2,1,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,2,1,1,1,1]
=> [3,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
[4,3,2,1,1,1,1,1,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? => ? ∊ {0,0,0,1,1,1,2,2,2,2,2,3,3,3,3,3,3,4,4,4,4,4,4,4,4,5,5,5,5,5,5,5,5,5,6,6,6,6,6,6,6,6,6,6,6,7,7,7,7,7,7,8,8}
Description
The number of successions of a permutation.
A succession of a permutation \pi is an index i such that \pi(i)+1 = \pi(i+1). Successions are also known as ''small ascents'' or ''1-rises''.
The following 249 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000672The number of minimal elements in Bruhat order not less than the permutation. St000074The number of special entries. St000211The rank of the set partition. St000502The number of successions of a set partitions. St000728The dimension of a set partition. St001298The number of repeated entries in the Lehmer code of a permutation. St001438The number of missing boxes of a skew partition. St001033The normalized area of the parallelogram polyomino associated with the Dyck path. St001520The number of strict 3-descents. St000473The number of parts of a partition that are strictly bigger than the number of ones. St000456The monochromatic index of a connected graph. St000358The number of occurrences of the pattern 31-2. St000365The number of double ascents of a permutation. St001423The number of distinct cubes in a binary word. St001556The number of inversions of the third entry of a permutation. St000356The number of occurrences of the pattern 13-2. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St000359The number of occurrences of the pattern 23-1. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000497The lcb statistic of a set partition. St000572The dimension exponent of a set partition. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St000516The number of stretching pairs of a permutation. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St001021Sum of the differences between projective and codominant dimension of the non-projective indecomposable injective modules in the Nakayama algebra corresponding to the Dyck path. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St001841The number of inversions of a set partition. St001843The Z-index of a set partition. St000223The number of nestings in the permutation. St000664The number of right ropes of a permutation. St001394The genus of a permutation. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St000833The comajor index of a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000617The number of global maxima of a Dyck path. St001535The number of cyclic alignments of a permutation. St000237The number of small exceedances. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001330The hat guessing number of a graph. St001727The number of invisible inversions of a permutation. St000463The number of admissible inversions of a permutation. St001052The length of the exterior of a permutation. St001096The size of the overlap set of a permutation. St000842The breadth of a permutation. St000035The number of left outer peaks of a permutation. St001301The first Betti number of the order complex associated with the poset. St001396Number of triples of incomparable elements in a finite poset. St001695The natural comajor index of a standard Young tableau. St001698The comajor index of a standard tableau minus the weighted size of its shape. St001699The major index of a standard tableau minus the weighted size of its shape. St001712The number of natural descents of a standard Young tableau. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001181Number of indecomposable injective modules with grade at least 3 in the corresponding Nakayama algebra. St001186Number of simple modules with grade at least 3 in the corresponding Nakayama algebra. St001231The number of simple modules that are non-projective and non-injective with the property that they have projective dimension equal to one and that also the Auslander-Reiten translates of the module and the inverse Auslander-Reiten translate of the module have the same projective dimension. St001234The number of indecomposable three dimensional modules with projective dimension one. St000355The number of occurrences of the pattern 21-3. St001498The normalised height of a Nakayama algebra with magnitude 1. St000039The number of crossings of a permutation. St000360The number of occurrences of the pattern 32-1. St000800The number of occurrences of the vincular pattern |231 in a permutation. St001089Number of indecomposable projective non-injective modules minus the number of indecomposable projective non-injective modules with dominant dimension equal to the injective dimension in the corresponding Nakayama algebra. St001728The number of invisible descents of a permutation. St000454The largest eigenvalue of a graph if it is integral. St001162The minimum jump of a permutation. St001665The number of pure excedances of a permutation. St000353The number of inner valleys of a permutation. St000354The number of recoils of a permutation. St001469The holeyness of a permutation. St001729The number of visible descents of a permutation. St001737The number of descents of type 2 in a permutation. St001928The number of non-overlapping descents in a permutation. St000470The number of runs in a permutation. St000366The number of double descents of a permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000406The number of occurrences of the pattern 3241 in a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St000834The number of right outer peaks of a permutation. St000031The number of cycles in the cycle decomposition of a permutation. St000629The defect of a binary word. St001715The number of non-records in a permutation. St001816Eigenvalues of the top-to-random operator acting on a simple module. St000352The Elizalde-Pak rank of a permutation. St000007The number of saliances of the permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St000233The number of nestings of a set partition. St001846The number of elements which do not have a complement in the lattice. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001720The minimal length of a chain of small intervals in a lattice. St000565The major index of a set partition. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St000402Half the size of the symmetry class of a permutation. St000742The number of big ascents of a permutation after prepending zero. St000308The height of the tree associated to a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000562The number of internal points of a set partition. St000731The number of double exceedences of a permutation. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St001596The number of two-by-two squares inside a skew partition. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St000204The number of internal nodes of a binary tree. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St000871The number of very big ascents of a permutation. St001305The number of induced cycles on four vertices in a graph. St001311The cyclomatic number of a graph. St001317The minimal number of occurrences of the forest-pattern in a linear ordering of the vertices of the graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001323The independence gap of a graph. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. 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). St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St000542The number of left-to-right-minima of a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000317The cycle descent number of a permutation. St000461The rix statistic of a permutation. St000488The number of cycles of a permutation of length at most 2. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000951The dimension of Ext^{1}(D(A),A) of the corresponding LNakayama algebra. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001217The projective dimension of the indecomposable injective module I[n-2] in the corresponding Nakayama algebra with simples enumerated from 0 to n-1. St001219Number of simple modules S in the corresponding Nakayama algebra such that the Auslander-Reiten sequence ending at S has the property that all modules in the exact sequence are reflexive. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. St001265The maximal i such that the i-th simple module has projective dimension equal to the global dimension in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001513The number of nested exceedences of a permutation. St001549The number of restricted non-inversions between exceedances. St001550The number of inversions between exceedances where the greater exceedance is linked. St001553The number of indecomposable summands of the square of the Jacobson radical as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001557The number of inversions of the second entry of a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001811The Castelnuovo-Mumford regularity of a permutation. St001856The number of edges in the reduced word graph of a permutation. St001960The number of descents of a permutation minus one if its first entry is not one. St000021The number of descents of a permutation. St000023The number of inner peaks of a permutation. St000162The number of nontrivial cycles in the cycle decomposition of a permutation. St000333The dez statistic, the number of descents of a permutation after replacing fixed points by zeros. St000486The number of cycles of length at least 3 of a permutation. St000654The first descent of a permutation. St000711The number of big exceedences of a permutation. St000779The tier of a permutation. St000864The number of circled entries of the shifted recording tableau of a permutation. St000882The number of connected components of short braid edges in the graph of braid moves of a permutation. St001000Number of indecomposable modules with projective 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. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c_0,c_1,...,c_{n−1}] such that n=c_0 < c_i for all i > 0 a special CNakayama algebra. St001205The number of non-simple indecomposable projective-injective modules of the algebra eAe in the corresponding Nakayama algebra A with minimal faithful projective-injective module eA. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. St001256Number of simple reflexive modules that are 2-stable reflexive. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. 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. St001431Half of the Loewy length minus one of a modified stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. 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. St001661Half the permanent of the Identity matrix plus the permutation matrix associated to the permutation. St001859The number of factors of the Stanley symmetric function associated with a permutation. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000061The number of nodes on the left branch of a binary tree. St000092The number of outer peaks of a permutation. St000099The number of valleys of a permutation, including the boundary. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000991The number of right-to-left minima of a permutation. St001165Number of simple modules with even projective dimension in the corresponding Nakayama algebra. 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. St001201The grade of the simple module S_0 in the special CNakayama algebra corresponding to the Dyck path. 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. St001235The global dimension of the corresponding Comp-Nakayama algebra. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001526The Loewy length of the Auslander-Reiten translate of the regular module as a bimodule of the Nakayama algebra corresponding to the Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St001964The interval resolution global dimension of a poset. St000058The order of a permutation. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St000632The jump number of the poset. St000298The order dimension or Dushnik-Miller dimension of a poset. St001435The number of missing boxes in the first row. St000307The number of rowmotion orbits of a poset. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. 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. St001613The binary logarithm of the size of the center of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001616The number of neutral elements in a lattice. St000252The number of nodes of degree 3 of a binary tree. St000370The genus of a graph. St000646The number of big ascents of a permutation. St000649The number of 3-excedences of a permutation. St000650The number of 3-rises of a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St000570The Edelman-Greene number of a permutation. 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}. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001490The number of connected components of a skew partition. St000624The normalized sum of the minimal distances to a greater element. St000848The balance constant multiplied with the number of linear extensions of a poset. St000849The number of 1/3-balanced pairs in a poset. St000910The number of maximal chains of minimal length in a poset. St001105The number of greedy linear extensions of a poset. St001106The number of supergreedy linear extensions of a poset. St000872The number of very big descents of a permutation. St000450The number of edges minus the number of vertices plus 2 of a graph. St000640The rank of the largest boolean interval in a poset. St000181The number of connected components of the Hasse diagram for the poset. St001890The maximum magnitude of the Möbius function of a poset. St000243The number of cyclic valleys and cyclic peaks of a permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000753The Grundy value for the game of Kayles on a binary word. St001771The number of occurrences of the signed pattern 1-2 in a signed permutation. St001866The nesting alignments of a signed permutation. St001870The number of positive entries followed by a negative entry in a signed permutation. St001895The oddness of a signed permutation. St000627The exponent of a binary word. St001880The number of 2-Gorenstein indecomposable injective modules in the incidence algebra of the lattice. St001896The number of right descents of a signed permutations. St000455The second largest eigenvalue of a graph if it is integral.
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