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Your data matches 563 different statistics following compositions of up to 3 maps.
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Matching statistic: St000980
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000980: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [1,0,1,0]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,3,2]
=> [3,2]
=> [2]
=> [1,0,1,0]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [1,1,0,0]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,3,3]
=> [3,3]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 0
Description
The number of boxes weakly below the path and above the diagonal that lie below at least two peaks.
For example, the path 111011010000 has three peaks in positions 03,15,26. The boxes below 03 are 01,02,12, the boxes below 15 are 12,13,14,23,24,34, and the boxes below 26 are 23,24,25,34,35,45.
We thus obtain the four boxes in positions 12,23,24,34 that are below at least two peaks.
Matching statistic: St001435
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 44%●distinct values known / distinct values provided: 10%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001435: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 44%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[4,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[3,3,3]
=> [3,3]
=> [3]
=> [[3],[]]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[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
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[6,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[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
[2,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[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],[]]
=> ? = 0
[3,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[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],[]]
=> ? = 0
[4,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 3
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ? = 0
[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],[]]
=> ? = 0
[5,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[5,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[5,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0
[4,3,2,2,2]
=> [3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
Description
The number of missing boxes in the first row.
Matching statistic: St001438
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 44%●distinct values known / distinct values provided: 10%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001438: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 44%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[2,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[3,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[4,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[3,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[5,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[4,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[3,3,3]
=> [3,3]
=> [3]
=> [[3],[]]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 0
[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
[1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 0
[6,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 0
[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
[2,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[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],[]]
=> ? = 0
[3,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[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],[]]
=> ? = 0
[4,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 3
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ? = 0
[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],[]]
=> ? = 0
[5,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
[5,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0
[5,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0
[5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ? = 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0
[4,3,2,2,2]
=> [3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0
Description
The number of missing boxes of a skew partition.
Matching statistic: St001487
(load all 9 compositions to match this statistic)
(load all 9 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 44%●distinct values known / distinct values provided: 10%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001487: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 44%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[4,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[5,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[4,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[3,3,3]
=> [3,3]
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 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]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[6,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[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
[2,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[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],[]]
=> ? = 0 + 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[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],[]]
=> ? = 0 + 1
[4,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,3,3,3]
=> [3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 0 + 1
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0 + 1
[3,3,2,2,2]
=> [3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 0 + 1
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 3 + 1
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 0 + 1
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[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],[]]
=> ? = 0 + 1
[5,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[5,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[5,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[4,3,3,3]
=> [3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 0 + 1
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0 + 1
[4,3,2,2,2]
=> [3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
Description
The number of inner corners of a skew partition.
Matching statistic: St001490
(load all 11 compositions to match this statistic)
(load all 11 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 44%●distinct values known / distinct values provided: 10%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00179: Integer partitions —to skew partition⟶ Skew partitions
St001490: Skew partitions ⟶ ℤResult quality: 10% ●values known / values provided: 44%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[2,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[3,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[2,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[2,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[2,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[4,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[3,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[3,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[3,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[2,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[2,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[5,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[4,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[4,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[4,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[3,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[3,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[3,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[3,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[2,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
[2,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[6,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[5,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[5,2,1,1]
=> [2,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[5,1,1,1,1]
=> [1,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[4,3,2]
=> [3,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[4,3,1,1]
=> [3,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[4,2,2,1]
=> [2,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[4,2,1,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[3,3,3]
=> [3,3]
=> [3]
=> [[3],[]]
=> 1 = 0 + 1
[3,3,2,1]
=> [3,2,1]
=> [2,1]
=> [[2,1],[]]
=> 1 = 0 + 1
[3,3,1,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> [[1,1,1],[]]
=> 1 = 0 + 1
[3,2,2,2]
=> [2,2,2]
=> [2,2]
=> [[2,2],[]]
=> 1 = 0 + 1
[3,2,2,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [[2,1,1],[]]
=> 1 = 0 + 1
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1,1,1,1],[]]
=> 1 = 0 + 1
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[2,2,2,2,1]
=> [2,2,2,1]
=> [2,2,1]
=> [[2,2,1],[]]
=> 1 = 0 + 1
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [[2,1,1,1],[]]
=> 1 = 0 + 1
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1,1,1,1,1],[]]
=> 1 = 0 + 1
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 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]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[7,1,1,1]
=> [1,1,1]
=> [1,1]
=> [[1,1],[]]
=> 1 = 0 + 1
[6,2,2]
=> [2,2]
=> [2]
=> [[2],[]]
=> 1 = 0 + 1
[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
[2,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[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],[]]
=> ? = 0 + 1
[3,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[3,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[3,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[3,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[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],[]]
=> ? = 0 + 1
[4,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[4,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[4,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[4,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[4,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,3,3,3]
=> [3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 0 + 1
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0 + 1
[3,3,2,2,2]
=> [3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> [[2,2,2,1],[]]
=> ? = 0 + 1
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[2,2,1,1,1],[]]
=> ? = 0 + 1
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[2,1,1,1,1,1],[]]
=> ? = 0 + 1
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> [[2,2,2,2],[]]
=> ? = 3 + 1
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [2,2,2,1,1]
=> [[2,2,2,1,1],[]]
=> ? = 0 + 1
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [[2,2,1,1,1,1],[]]
=> ? = 0 + 1
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [[2,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[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],[]]
=> ? = 0 + 1
[5,2,2,2,2]
=> [2,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
[5,2,2,2,1,1]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [[2,2,1,1],[]]
=> ? = 0 + 1
[5,2,2,1,1,1,1]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[2,1,1,1,1],[]]
=> ? = 0 + 1
[5,2,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1,1,1,1,1,1],[]]
=> ? = 0 + 1
[5,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1,1,1,1,1,1,1],[]]
=> ? = 0 + 1
[4,3,3,3]
=> [3,3,3]
=> [3,3]
=> [[3,3],[]]
=> ? = 1 + 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [3,2,1]
=> [[3,2,1],[]]
=> ? = 0 + 1
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> [[3,1,1,1],[]]
=> ? = 0 + 1
[4,3,2,2,2]
=> [3,2,2,2]
=> [2,2,2]
=> [[2,2,2],[]]
=> ? = 0 + 1
Description
The number of connected components of a skew partition.
Matching statistic: St001695
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001695: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[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]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[3,3,3]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[4,4,4,1]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> ? = 0
[4,4,3,2]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> ? = 0
[4,4,3,1,1]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> ? = 0
[4,4,2,2,1]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 0
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> ? = 0
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[4,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 0
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> ? = 0
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,3,3,3,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> ? = 1
[3,3,3,2,2]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 0
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> ? = 0
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> ? = 0
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> ? = 0
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> ? = 0
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> ? = 0
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> ? = 0
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 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]]
=> ? = 3
[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
Description
The natural comajor index of a standard Young tableau.
A natural descent of a standard tableau T is an entry i such that i+1 appears in a higher row than i in English notation.
The natural comajor index of a tableau of size n with natural descent set D is then ∑d∈Dn−d.
Matching statistic: St001698
(load all 34 compositions to match this statistic)
(load all 34 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001698: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[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]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[3,3,3]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[4,4,4,1]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> ? = 0
[4,4,3,2]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> ? = 0
[4,4,3,1,1]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> ? = 0
[4,4,2,2,1]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 0
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> ? = 0
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[4,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 0
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> ? = 0
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,3,3,3,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> ? = 1
[3,3,3,2,2]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 0
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> ? = 0
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> ? = 0
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> ? = 0
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> ? = 0
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> ? = 0
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> ? = 0
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 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]]
=> ? = 3
[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
Description
The comajor index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001699
(load all 27 compositions to match this statistic)
(load all 27 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St001699: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[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]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,3,2]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[3,3,3]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4],[5],[7],[9]]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? = 0
[4,4,4,1]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> ? = 0
[4,4,3,2]
=> [4,3,2]
=> [[1,2,5,9],[3,4,8],[6,7]]
=> ? = 0
[4,4,3,1,1]
=> [4,3,1,1]
=> [[1,4,5,9],[2,7,8],[3],[6]]
=> ? = 0
[4,4,2,2,1]
=> [4,2,2,1]
=> [[1,3,8,9],[2,5],[4,7],[6]]
=> ? = 0
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [[1,5,8,9],[2,7],[3],[4],[6]]
=> ? = 0
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,7,8,9],[2],[3],[4],[5],[6]]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8],[7]]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> ? = 0
[4,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> ? = 0
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,4,9],[2,6],[3,8],[5],[7]]
=> ? = 0
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,6,9],[2,8],[3],[4],[5],[7]]
=> ? = 0
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,8,9],[2],[3],[4],[5],[6],[7]]
=> ? = 0
[4,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4],[6],[8]]
=> ? = 0
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> ? = 0
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,9],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[3,3,3,3,1]
=> [3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> ? = 1
[3,3,3,2,2]
=> [3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6],[8,9]]
=> ? = 0
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9],[5],[8]]
=> ? = 0
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> ? = 0
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [[1,3,10],[2,5],[4,7],[6,9],[8]]
=> ? = 0
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [[1,5,10],[2,7],[3,9],[4],[6],[8]]
=> ? = 0
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [[1,7,10],[2,9],[3],[4],[5],[6],[8]]
=> ? = 0
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,9,10],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 0
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5,10],[7],[9]]
=> ? = 0
[3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4],[5],[7],[9]]
=> ? = 0
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? = 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]]
=> ? = 3
[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
Description
The major index of a standard tableau minus the weighted size of its shape.
Matching statistic: St001712
(load all 35 compositions to match this statistic)
(load all 35 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St001712: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[3,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[2,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[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]]
=> 0
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 0
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 0
[4,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 0
[3,3,3]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[4,4,4,1]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> ? = 0
[4,4,3,2]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> ? = 0
[4,4,3,1,1]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> ? = 0
[4,4,2,2,1]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> ? = 0
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> ? = 0
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> ? = 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> ? = 0
[4,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> ? = 0
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> ? = 0
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> ? = 0
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> ? = 0
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> ? = 0
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> ? = 0
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 0
[3,3,3,3,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> ? = 1
[3,3,3,2,2]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> ? = 0
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> ? = 0
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> ? = 0
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> ? = 0
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> ? = 0
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> ? = 0
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> ? = 0
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> ? = 3
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> ? = 0
[3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> ? = 0
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> ? = 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]]
=> ? = 3
[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
Description
The number of natural descents of a standard Young tableau.
A natural descent of a standard tableau T is an entry i such that i+1 appears in a higher row than i in English notation.
Matching statistic: St001803
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St001803: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 38%●distinct values known / distinct values provided: 10%
Values
[1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[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
[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]]
=> 0
[2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 0
[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
[4,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[3,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 0
[3,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 0
[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]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 0
[2,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 0
[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
[5,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[4,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 0
[4,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 0
[4,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[3,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 0
[3,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 0
[3,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 0
[3,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 0
[3,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[2,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 0
[2,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 0
[2,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 0
[2,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 0
[1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 0
[6,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0
[5,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 0
[5,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 0
[5,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0
[4,3,2]
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 0
[4,3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 0
[4,2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 0
[4,2,1,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 0
[4,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0
[3,3,3]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 0
[3,3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 0
[3,3,1,1,1]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 0
[3,2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 0
[3,2,2,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 0
[3,2,1,1,1,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 0
[3,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 0
[2,2,2,2,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> [[1,3,5,7],[2,4,6]]
=> 0
[2,2,2,1,1,1]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> [[1,3,5,6,7],[2,4]]
=> 0
[2,2,1,1,1,1,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> [[1,3,4,5,6,7],[2]]
=> 0
[2,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 0
[2,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> ? = 0
[2,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> ? = 0
[2,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,3,5,6,7,8,9],[2,4]]
=> ? = 0
[2,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 0
[3,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> ? = 1
[3,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> ? = 0
[3,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> ? = 0
[3,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> ? = 0
[3,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> ? = 0
[3,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> ? = 0
[3,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 0
[3,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> ? = 0
[3,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> ? = 0
[3,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,3,5,6,7,8,9],[2,4]]
=> ? = 0
[3,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 0
[2,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 3
[2,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [[1,3,5,7,9,10],[2,4,6,8]]
=> ? = 0
[2,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> [[1,3,5,7,8,9,10],[2,4,6]]
=> ? = 0
[2,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [[1,3,5,6,7,8,9,10],[2,4]]
=> ? = 0
[4,4,4,1]
=> [4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9]]
=> [[1,5,9],[2,6],[3,7],[4,8]]
=> ? = 0
[4,4,3,2]
=> [4,3,2]
=> [[1,2,3,4],[5,6,7],[8,9]]
=> [[1,5,8],[2,6,9],[3,7],[4]]
=> ? = 0
[4,4,3,1,1]
=> [4,3,1,1]
=> [[1,2,3,4],[5,6,7],[8],[9]]
=> [[1,5,8,9],[2,6],[3,7],[4]]
=> ? = 0
[4,4,2,2,1]
=> [4,2,2,1]
=> [[1,2,3,4],[5,6],[7,8],[9]]
=> [[1,5,7,9],[2,6,8],[3],[4]]
=> ? = 0
[4,4,2,1,1,1]
=> [4,2,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9]]
=> [[1,5,7,8,9],[2,6],[3],[4]]
=> ? = 0
[4,4,1,1,1,1,1]
=> [4,1,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8],[9]]
=> [[1,5,6,7,8,9],[2],[3],[4]]
=> ? = 0
[4,3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> ? = 1
[4,3,3,2,1]
=> [3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9]]
=> [[1,4,7,9],[2,5,8],[3,6]]
=> ? = 0
[4,3,3,1,1,1]
=> [3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9]]
=> [[1,4,7,8,9],[2,5],[3,6]]
=> ? = 0
[4,3,2,2,2]
=> [3,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9]]
=> [[1,4,6,8],[2,5,7,9],[3]]
=> ? = 0
[4,3,2,2,1,1]
=> [3,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9]]
=> [[1,4,6,8,9],[2,5,7],[3]]
=> ? = 0
[4,3,2,1,1,1,1]
=> [3,2,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9]]
=> [[1,4,6,7,8,9],[2,5],[3]]
=> ? = 0
[4,3,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9]]
=> [[1,4,5,6,7,8,9],[2],[3]]
=> ? = 0
[4,2,2,2,2,1]
=> [2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9]]
=> [[1,3,5,7,9],[2,4,6,8]]
=> ? = 0
[4,2,2,2,1,1,1]
=> [2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9]]
=> [[1,3,5,7,8,9],[2,4,6]]
=> ? = 0
[4,2,2,1,1,1,1,1]
=> [2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9]]
=> [[1,3,5,6,7,8,9],[2,4]]
=> ? = 0
[4,2,1,1,1,1,1,1,1]
=> [2,1,1,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7],[8],[9]]
=> [[1,3,4,5,6,7,8,9],[2]]
=> ? = 0
[3,3,3,3,1]
=> [3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10]]
=> [[1,4,7,10],[2,5,8],[3,6,9]]
=> ? = 1
[3,3,3,2,2]
=> [3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10]]
=> [[1,4,7,9],[2,5,8,10],[3,6]]
=> ? = 0
[3,3,3,2,1,1]
=> [3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10]]
=> [[1,4,7,9,10],[2,5,8],[3,6]]
=> ? = 0
[3,3,3,1,1,1,1]
=> [3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7],[8],[9],[10]]
=> [[1,4,7,8,9,10],[2,5],[3,6]]
=> ? = 0
[3,3,2,2,2,1]
=> [3,2,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10]]
=> [[1,4,6,8,10],[2,5,7,9],[3]]
=> ? = 0
[3,3,2,2,1,1,1]
=> [3,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8],[9],[10]]
=> [[1,4,6,8,9,10],[2,5,7],[3]]
=> ? = 0
[3,3,2,1,1,1,1,1]
=> [3,2,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10]]
=> [[1,4,6,7,8,9,10],[2,5],[3]]
=> ? = 0
[3,3,1,1,1,1,1,1,1]
=> [3,1,1,1,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7],[8],[9],[10]]
=> [[1,4,5,6,7,8,9,10],[2],[3]]
=> ? = 0
[3,2,2,2,2,2]
=> [2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10]]
=> ? = 3
[3,2,2,2,2,1,1]
=> [2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10]]
=> [[1,3,5,7,9,10],[2,4,6,8]]
=> ? = 0
[3,2,2,2,1,1,1,1]
=> [2,2,2,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10]]
=> [[1,3,5,7,8,9,10],[2,4,6]]
=> ? = 0
[3,2,2,1,1,1,1,1,1]
=> [2,2,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10]]
=> [[1,3,5,6,7,8,9,10],[2,4]]
=> ? = 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]]
=> [[1,3,5,7,9,11],[2,4,6,8,10]]
=> ? = 3
[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
Description
The maximal overlap of the cylindrical tableau associated with a tableau.
A cylindrical tableau associated with a standard Young tableau T is the skew row-strict tableau obtained by gluing two copies of T such that the inner shape is a rectangle.
The overlap, recorded in this statistic, equals maxC(2ℓ(T)−ℓ(C)), where ℓ denotes the number of rows of a tableau and the maximum is taken over all cylindrical tableaux.
In particular, the statistic equals 0, if and only if the last entry of the first row is larger than or equal to the first entry of the last row. Moreover, the statistic attains its maximal value, the number of rows of the tableau minus 1, if and only if the tableau consists of a single column.
The following 553 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000491The number of inversions of a set partition. St000497The lcb statistic of a set partition. St000555The number of occurrences of the pattern {{1,3},{2}} in a set partition. St000559The number of occurrences of the pattern {{1,3},{2,4}} in a set partition. St000562The number of internal points of a set partition. St000563The number of overlapping pairs of blocks of a set partition. St000565The major index of a set partition. St000572The dimension exponent of a set partition. St000581The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 2 is maximal. St000582The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000585The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal, (1,3) are consecutive in a block. St000594The number of occurrences of the pattern {{1,3},{2}} such that 1,2 are minimal, (1,3) are consecutive in a block. St000600The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal, (1,3) are consecutive in a block. St000602The number of occurrences of the pattern {{1,3},{2}} such that 1 is minimal. St000610The number of occurrences of the pattern {{1,3},{2}} such that 2 is maximal. St000613The number of occurrences of the pattern {{1,3},{2}} such that 2 is minimal, 3 is maximal, (1,3) are consecutive in a block. St000748The major index of the permutation obtained by flattening the set partition. St000695The number of blocks in the first part of the atomic decomposition of a set partition. St000058The order of a permutation. St000091The descent variation of a composition. St000234The number of global ascents of a permutation. St001781The interlacing number of a set partition. St001839The number of excedances of a set partition. St001840The number of descents of a set partition. St001841The number of inversions of a set partition. St001842The major index of a set partition. St001843The Z-index of a set partition. St000221The number of strong fixed points of a permutation. St000279The size of the preimage of the map 'cycle-as-one-line notation' from Permutations to Permutations. St000375The number of non weak exceedences of a permutation that are mid-points of a decreasing subsequence of length 3. St000623The number of occurrences of the pattern 52341 in a permutation. St000689The maximal n such that the minimal generator-cogenerator module in the LNakayama algebra of a Dyck path is n-rigid. St001204Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001314The number of tilting modules of arbitrary projective dimension that have no simple modules as a direct summand in the corresponding Nakayama algebra. St001381The fertility of a permutation. St001444The rank of the skew-symmetric form which is non-zero on crossing arcs of a perfect matching. St001466The number of transpositions swapping cyclically adjacent numbers in a permutation. St001549The number of restricted non-inversions between exceedances. St001551The number of restricted non-inversions between exceedances where the rightmost exceedance is linked. St001552The number of inversions between excedances and fixed points of a permutation. St001663The number of occurrences of the Hertzsprung pattern 132 in a permutation. St001810The number of fixed points of a permutation smaller than its largest moved point. St001811The Castelnuovo-Mumford regularity of a permutation. St001837The number of occurrences of a 312 pattern in the restricted growth word of a perfect matching. St001850The number of Hecke atoms of a permutation. St000056The decomposition (or block) number of a permutation. St000486The number of cycles of length at least 3 of a permutation. St000694The number of affine bounded permutations that project to a given permutation. St001174The Gorenstein dimension of the algebra A/I when I is the tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St001195The global dimension of the algebra A/AfA of the corresponding Nakayama algebra A with minimal left faithful projective-injective module Af. 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]/(xn). St001256Number of simple reflexive modules that are 2-stable reflexive. St001461The number of topologically connected components of the chord diagram of a permutation. 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. St001359The number of permutations in the equivalence class of a permutation obtained by taking inverses of cycles. St000255The number of reduced Kogan faces with the permutation as type. St000317The cycle descent number of a permutation. St000355The number of occurrences of the pattern 21-3. St000406The number of occurrences of the pattern 3241 in a permutation. St000407The number of occurrences of the pattern 2143 in a permutation. St000425The number of occurrences of the pattern 132 or of the pattern 213 in a permutation. St000516The number of stretching pairs of a permutation. St000646The number of big ascents of a permutation. St000650The number of 3-rises of a permutation. St000663The number of right floats of a permutation. St000664The number of right ropes of a permutation. St000666The number of right tethers of a permutation. St000709The number of occurrences of 14-2-3 or 14-3-2. St000732The number of double deficiencies of a permutation. St000750The number of occurrences of the pattern 4213 in a permutation. St000751The number of occurrences of either of the pattern 2143 or 2143 in a permutation. St000799The number of occurrences of the vincular pattern |213 in a permutation. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001059Number of occurrences of the patterns 41352,42351,51342,52341 in a permutation. St001301The first Betti number of the order complex associated with the poset. St001550The number of inversions between exceedances where the greater exceedance is linked. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001705The number of occurrences of the pattern 2413 in a permutation. St001715The number of non-records in a permutation. St001766The number of cells which are not occupied by the same tile in all reduced pipe dreams corresponding to a permutation. St001847The number of occurrences of the pattern 1432 in a permutation. St000078The number of alternating sign matrices whose left key is the permutation. St000570The Edelman-Greene number of a permutation. St000908The length of the shortest maximal antichain in a poset. St000914The sum of the values of the Möbius function of a poset. St001162The minimum jump of a permutation. St001236The dominant dimension of the corresponding Comp-Nakayama algebra. St001344The neighbouring number of a permutation. St001634The trace of the Coxeter matrix of the incidence algebra of a poset. St001735The number of permutations with the same set of runs. St001737The number of descents of type 2 in a permutation. St000485The length of the longest cycle of a permutation. St000360The number of occurrences of the pattern 32-1. St000367The number of simsun double descents of a permutation. St000802The number of occurrences of the vincular pattern |321 in a permutation. St001513The number of nested exceedences of a permutation. St001728The number of invisible descents of a permutation. St000358The number of occurrences of the pattern 31-2. St000622The number of occurrences of the patterns 2143 or 4231 in a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. St001559The number of transpositions that are smaller or equal to a permutation in Bruhat order while not being inversions. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St001537The number of cyclic crossings of a permutation. St001906Half of the difference between the total displacement and the number of inversions and the reflection length of a permutation. St001696The natural major index of a standard Young tableau. St000745The index of the last row whose first entry is the row number in a standard Young tableau. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001139The number of occurrences of hills of size 2 in a Dyck path. St001141The number of occurrences of hills of size 3 in a Dyck path. St001741The largest integer such that all patterns of this size are contained in the permutation. St000042The number of crossings of a perfect matching. St000296The length of the symmetric border of a binary word. St000877The depth of the binary word interpreted as a path. St000878The number of ones minus the number of zeros of a binary word. St000885The number of critical steps in the Catalan decomposition of a binary word. St001047The maximal number of arcs crossing a given arc of a perfect matching. St000326The position of the first one in a binary word after appending a 1 at the end. St000876The number of factors in the Catalan decomposition of a binary word. St000733The row containing the largest entry of a standard tableau. St001371The length of the longest Yamanouchi prefix of a binary word. St000210Minimum over maximum difference of elements in cycles. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000800The number of occurrences of the vincular pattern |231 in a permutation. St000487The length of the shortest cycle of a permutation. St000501The size of the first part in the decomposition of a permutation. St000542The number of left-to-right-minima of a permutation. St000990The first ascent of a permutation. St001468The smallest fixpoint of a permutation. St000232The number of crossings of a set partition. St001051The depth of the label 1 in the decreasing labelled unordered tree associated with the set partition. St000807The sum of the heights of the valleys of the associated bargraph. St000181The number of connected components of the Hasse diagram for the poset. St000805The number of peaks of the associated bargraph. St001890The maximum magnitude of the Möbius function of a poset. St000253The crossing number of a set partition. St001434The number of negative sum pairs of a signed permutation. St000447The number of pairs of vertices of a graph with distance 3. St000449The number of pairs of vertices of a graph with distance 4. St000629The defect of a binary word. St000217The number of occurrences of the pattern 312 in a permutation. St000674The number of hills of a Dyck path. St000879The number of long braid edges in the graph of braid moves of a permutation. 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. 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. St001125The number of simple modules that satisfy the 2-regular condition 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. St001221The number of simple modules in the corresponding LNakayama algebra that have 2 dimensional second Extension group with the regular module. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. 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. St001498The normalised height of a Nakayama algebra with magnitude 1. 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. St000889The number of alternating sign matrices with the same antidiagonal sums. St001113Number of indecomposable projective non-injective modules with reflexive Auslander-Reiten sequences in the corresponding Nakayama algebra. St001159Number of simple modules with dominant dimension equal to the global dimension in the corresponding Nakayama algebra. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of Ext2A(S,A) for a simple module S over the corresponding Nakayama algebra A. 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. St001481The minimal height of a peak of a Dyck path. 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. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000022The number of fixed points of a permutation. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000039The number of crossings of a permutation. St000065The number of entries equal to -1 in an alternating sign matrix. St000095The number of triangles of a graph. St000096The number of spanning trees of a graph. St000119The number of occurrences of the pattern 321 in a permutation. St000123The difference in Coxeter length of a permutation and its image under the Simion-Schmidt map. St000126The number of occurrences of the contiguous pattern [.,[.,[.,[.,[.,.]]]]] in a binary tree. St000127The number of occurrences of the contiguous pattern [.,[.,[.,[[.,.],.]]]] in a binary tree. St000128The number of occurrences of the contiguous pattern [.,[.,[[.,[.,.]],.]]] in a binary tree. St000131The number of occurrences of the contiguous pattern [.,[[[[.,.],.],.],. St000133The "bounce" of a permutation. St000153The number of adjacent cycles of a permutation. St000214The number of adjacencies of a permutation. St000215The number of adjacencies of a permutation, zero appended. St000219The number of occurrences of the pattern 231 in a permutation. St000223The number of nestings in the permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000268The number of strongly connected orientations of a graph. St000276The size of the preimage of the map 'to graph' from Ordered trees to Graphs. St000297The number of leading ones in a binary word. St000303The determinant of the product of the incidence matrix and its transpose of a graph divided by 4. St000315The number of isolated vertices of a graph. St000322The skewness of a graph. St000323The minimal crossing number of a graph. St000344The number of strongly connected outdegree sequences of a graph. St000357The number of occurrences of the pattern 12-3. St000365The number of double ascents of a permutation. St000366The number of double descents of a permutation. St000368The Altshuler-Steinberg determinant of a graph. St000370The genus of a graph. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000372The number of mid points of increasing subsequences of length 3 in a permutation. St000379The number of Hamiltonian cycles in a graph. St000403The Szeged index minus the Wiener index of a graph. St000441The number of successions of a permutation. St000461The rix statistic of a permutation. St000478Another weight of a partition according to Alladi. St000534The number of 2-rises of a permutation. St000546The number of global descents of a permutation. St000637The length of the longest cycle in a graph. St000648The number of 2-excedences of a permutation. St000665The number of rafts of a permutation. St000713The dimension of the irreducible representation of Sp(4) labelled by an integer partition. St000714The number of semistandard Young tableau of given shape, with entries at most 2. St000731The number of double exceedences of a permutation. St000779The tier of a permutation. St000804The number of occurrences of the vincular pattern |123 in a permutation. St000873The aix statistic of a permutation. St000929The constant term of the character polynomial of an integer partition. St000951The dimension of Ext1(D(A),A) of the corresponding LNakayama algebra. St000954Number of times the corresponding LNakayama algebra has Exti(D(A),A)=0 for i>0. St000974The length of the trunk of an ordered tree. St000989The number of final rises of a permutation. St001008Number of indecomposable injective modules with projective dimension 1 in the Nakayama algebra corresponding to the Dyck path. St001022Number of simple modules with projective dimension 3 in the Nakayama algebra corresponding to the Dyck path. St001025Number of simple modules with projective dimension 4 in the Nakayama algebra corresponding to the Dyck path. St001046The maximal number of arcs nesting a given arc of a perfect matching. St001069The coefficient of the monomial xy of the Tutte polynomial of the graph. St001070The absolute value of the derivative of the chromatic polynomial of the graph at 1. St001071The beta invariant of the graph. St001082The number of boxed occurrences of 123 in a permutation. St001086The number of occurrences of the consecutive pattern 132 in a permutation. St001130The number of two successive successions in a permutation. St001140Number of indecomposable modules with projective and injective dimension at least two in the corresponding Nakayama algebra. St001163The number of simple modules with dominant dimension at least three in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive 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. St001193The dimension of Ext1A(A/AeA,A) in the corresponding Nakayama algebra A such that eA is a minimal faithful projective-injective module. 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. St001222Number of simple modules in the corresponding LNakayama algebra that have a unique 2-extension with the regular module. St001230The number of simple modules with injective dimension equal to the dominant dimension equal to one and the dual property. St001241The number of non-zero radicals of the indecomposable projective modules that have injective dimension and projective dimension at most one. St001253The number of non-projective indecomposable reflexive modules in the corresponding Nakayama algebra. 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. St001266The largest vector space dimension of an indecomposable non-projective module that is reflexive in the corresponding Nakayama algebra. St001283The number of finite solvable groups that are realised by the given partition over the complex numbers. St001284The number of finite groups that are realised by the given partition over the complex numbers. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St001305The number of induced cycles on four vertices in a graph. St001309The number of four-cliques in a graph. St001310The number of induced diamond graphs 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. St001325The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph. St001328The minimal number of occurrences of the bipartite-pattern in a linear ordering of the vertices of the graph. St001329The minimal number of occurrences of the outerplanar pattern in a linear ordering of the vertices of the graph. St001331The size of the minimal feedback vertex set. St001334The minimal number of occurrences of the 3-colorable pattern in a linear ordering of the vertices of the graph. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001367The smallest number which does not occur as degree of a vertex in a graph. St001394The genus of a permutation. St001465The number of adjacent transpositions in the cycle decomposition of a permutation. St001477The number of nowhere zero 5-flows of a graph. St001478The number of nowhere zero 4-flows of a graph. St001572The minimal number of edges to remove to make a graph bipartite. St001573The minimal number of edges to remove to make a graph triangle-free. St001577The minimal number of edges to add or remove to make a graph a cograph. St001594The number of indecomposable projective modules in the Nakayama algebra corresponding to the Dyck path such that the UC-condition is satisfied. St001633The number of simple modules with projective dimension two in the incidence algebra of the poset. St001640The number of ascent tops in the permutation such that all smaller elements appear before. St001682The number of distinct positions of the pattern letter 1 in occurrences of 123 in a permutation. St001690The length of a longest path in a graph such that after removing the paths edges, every vertex of the path has distance two from some other vertex of the path. St001736The total number of cycles in a graph. St001793The difference between the clique number and the chromatic number of a graph. St001795The binary logarithm of the evaluation of the Tutte polynomial of the graph at (x,y) equal to (-1,-1). St001797The number of overfull subgraphs of a graph. St001871The number of triconnected components of a graph. St001947The number of ties in a parking function. St001970The signature of a graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000035The number of left outer peaks of a permutation. St000115The single entry in the last row. St000266The number of spanning subgraphs of a graph with the same connected components. St000267The number of maximal spanning forests contained in a graph. St000272The treewidth of a graph. St000286The number of connected components of the complement of a graph. St000287The number of connected components of a graph. St000310The minimal degree of a vertex of a graph. St000314The number of left-to-right-maxima of a permutation. St000335The difference of lower and upper interactions. St000382The first part of an integer composition. St000392The length of the longest run of ones in a binary word. St000450The number of edges minus the number of vertices plus 2 of a graph. St000456The monochromatic index of a connected graph. St000535The rank-width of a graph. St000544The cop number of a graph. St000627The exponent of a binary word. St000654The first descent of a permutation. St000669The number of permutations obtained by switching ascents or descents of size 2. St000685The dominant dimension of the LNakayama algebra associated to a Dyck path. St000688The global dimension minus the dominant dimension of the LNakayama algebra associated to a Dyck path. St000696The number of cycles in the breakpoint graph of a permutation. St000740The last entry of a permutation. St000756The sum of the positions of the left to right maxima of a permutation. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000775The multiplicity of the largest eigenvalue in a graph. St000785The number of distinct colouring schemes of a graph. St000884The number of isolated descents of a permutation. St000930The k-Gorenstein degree of the corresponding Nakayama algebra with linear quiver. St000948The chromatic discriminant of a graph. St000964Gives the dimension of Ext^g(D(A),A) of the corresponding LNakayama algebra, when g denotes the global dimension of that algebra. St000968We make a CNakayama algebra out of the LNakayama algebra (corresponding to the Dyck path) [c0,c1,...,cn−1] by adding c0 to cn−1. St000991The number of right-to-left minima of a permutation. 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. St001011Number of simple modules of projective dimension 2 in the Nakayama algebra corresponding to the Dyck path. St001026The maximum of the projective dimensions of the indecomposable non-projective injective modules minus the minimum 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. St001066The number of simple reflexive modules in the corresponding Nakayama algebra. St001085The number of occurrences of the vincular pattern |21-3 in a permutation. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001191Number of simple modules S with ExtiA(S,A)=0 for all i=0,1,...,g−1 in the corresponding Nakayama algebra A with global dimension g. St001194The injective dimension of A/AfA in the corresponding Nakayama algebra A when Af is the minimal faithful projective-injective left A-module St001201The grade of the simple module S0 in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series L=[c0,c1,...,cn−1] such that n=c0<ci for all i>0 a special CNakayama algebra. St001212The number of simple modules in the corresponding Nakayama algebra that have non-zero second Ext-group with the regular module. St001233The number of indecomposable 2-dimensional modules with projective dimension one. 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. St001257The dominant dimension of the double dual of A/J when A is the corresponding Nakayama algebra with Jacobson radical J. St001271The competition number of a graph. St001273The projective dimension of the first term in an injective coresolution of the regular module. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001363The Euler characteristic of a graph according to Knill. St001418Half of the global dimension of the stable Auslander algebra of the Nakayama algebra corresponding to the Dyck path. St001469The holeyness of a permutation. St001475The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,0). St001476The evaluation of the Tutte polynomial of the graph at (x,y) equal to (1,-1). St001483The number of simple module modules that appear in the socle of the regular module but have no nontrivial selfextensions with the regular module. St001493The number of simple modules with maximal even projective dimension in the corresponding Nakayama algebra. St001496The number of graphs with the same Laplacian spectrum as the given graph. St001501The dominant dimension of magnitude 1 Nakayama algebras. St001503The largest distance of a vertex to a vertex in a cycle in the resolution quiver of the corresponding Nakayama algebra. St001518The number of graphs with the same ordinary spectrum as the given graph. St001546The number of monomials in the Tutte polynomial of a graph. St001592The maximal number of simple paths between any two different vertices of a graph. St001652The length of a longest interval of consecutive numbers. St001662The length of the longest factor of consecutive numbers in a permutation. St001665The number of pure excedances of a permutation. St001743The discrepancy of a graph. St001792The arboricity of a graph. St001796The absolute value of the quotient of the Tutte polynomial of the graph at (1,1) and (-1,-1). St001828The Euler characteristic of a graph. St001941The evaluation at 1 of the modified Kazhdan--Lusztig R polynomial (as in [1, Section 5. St000007The number of saliances of the permutation. St000061The number of nodes on the left branch of a binary tree. St000062The length of the longest increasing subsequence of the permutation. St000308The height of the tree associated to a permutation. St000483The number of times a permutation switches from increasing to decreasing or decreasing to increasing. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St000686The finitistic dominant dimension of a Dyck path. St000822The Hadwiger number of the graph. St000834The number of right outer peaks of a permutation. St000842The breadth of a permutation. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001014Number of indecomposable injective modules with codominant dimension equal to the dominant dimension of the Nakayama algebra corresponding to the Dyck path. St001029The size of the core of a graph. St001111The weak 2-dynamic chromatic number of a graph. St001210Gives the maximal vector space dimension of the first Ext-group between an indecomposable module X and the regular module A, when A is the Nakayama algebra corresponding to the Dyck path. St001226The number of integers i such that the radical of the i-th indecomposable projective module has vanishing first extension group with the Jacobson radical J in the corresponding Nakayama algebra. 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. St001316The domatic number of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001716The 1-improper chromatic number of a graph. St001108The 2-dynamic chromatic number of a graph. St001117The game chromatic index of a graph. St001023Number of simple modules with projective dimension at most 3 in the Nakayama algebra corresponding to the Dyck path. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length 3. St000404The number of occurrences of the pattern 3241 or of the pattern 4231 in a permutation. St000408The number of occurrences of the pattern 4231 in a permutation. St000440The number of occurrences of the pattern 4132 or of the pattern 4231 in a permutation. St000647The number of big descents of a permutation. St000862The number of parts of the shifted shape of a permutation. St000475The number of parts equal to 1 in a partition. St001037The number of inner corners of the upper path of the parallelogram polyomino associated with the Dyck path. St001785The number of ways to obtain a partition as the multiset of antidiagonal lengths of the Ferrers diagram of a partition. St000759The smallest missing part in an integer partition. St000687The dimension of Hom(I,P) for the LNakayama algebra of a Dyck path. St000658The number of rises of length 2 of a Dyck path. St000659The number of rises of length at least 2 of a Dyck path. St001732The number of peaks visible from the left. St000295The length of the border of a binary word. St001730The number of times the path corresponding to a binary word crosses the base line. St001884The number of borders of a binary word. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St000057The Shynar inversion number of a standard tableau. St000068The number of minimal elements in a poset. St000218The number of occurrences of the pattern 213 in a permutation. St000220The number of occurrences of the pattern 132 in a permutation. St000356The number of occurrences of the pattern 13-2. St000405The number of occurrences of the pattern 1324 in a permutation. St000871The number of very big ascents of a permutation. St001083The number of boxed occurrences of 132 in a permutation. St000651The maximal size of a rise in a permutation. St000742The number of big ascents of a permutation after prepending zero. St000288The number of ones in a binary word. St000289The decimal representation of a binary word. St000290The major index of a binary word. St000291The number of descents of a binary word. St000292The number of ascents of a binary word. St000293The number of inversions of a binary word. St000347The inversion sum of a binary word. St000348The non-inversion sum of a binary word. St000389The number of runs of ones of odd length in a binary word. St000390The number of runs of ones in a binary word. St000391The sum of the positions of the ones in a binary word. St000628The balance of a binary word. St000682The Grundy value of Welter's game on a binary word. St000691The number of changes of a binary word. St000753The Grundy value for the game of Kayles on a binary word. St000792The Grundy value for the game of ruler on a binary word. St000875The semilength of the longest Dyck word in the Catalan factorisation of a binary word. St000921The number of internal inversions of a binary word. St001355Number of non-empty prefixes of a binary word that contain equally many 0's and 1's. St001372The length of a longest cyclic run of ones of a binary word. St001419The length of the longest palindromic factor beginning with a one of a binary word. St001420Half the length of a longest factor which is its own reverse-complement of a binary word. St001421Half the length of a longest factor which is its own reverse-complement and begins with a one of a binary word. St001436The index of a given binary word in the lex-order among all its cyclic shifts. St001485The modular major index of a binary word. St000256The number of parts from which one can substract 2 and still get an integer partition. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000543The size of the conjugacy class of a binary word. St000626The minimal period of a binary word. St000630The length of the shortest palindromic decomposition of a binary word. St000781The number of proper colouring schemes of a Ferrers diagram. St000847The number of standard Young tableaux whose descent set is the binary word. St000883The number of longest increasing subsequences of a permutation. St000983The length of the longest alternating subword. St001313The number of Dyck paths above the lattice path given by a binary word. St001385The number of conjugacy classes of subgroups with connected subgroups of sizes prescribed by an integer partition. St001568The smallest positive integer that does not appear twice in the partition. St001838The number of nonempty primitive factors of a binary word. St001885The number of binary words with the same proper border set. St000143The largest repeated part of a partition. St000402Half the size of the symmetry class of a permutation. St000011The number of touch points (or returns) of a Dyck path. St000386The number of factors DDU in a Dyck path. St000660The number of rises of length at least 3 of a Dyck path. St001398Number of subsets of size 3 of elements in a poset that form a "v". St000640The rank of the largest boolean interval in a poset. St001942The number of loops of the quiver corresponding to the reduced incidence algebra of a poset. St000717The number of ordinal summands of a poset. St001001The number of indecomposable modules with projective and injective dimension equal to the global dimension of the Nakayama algebra corresponding to the Dyck path. St001260The permanent of an alternating sign matrix. St001804The minimal height of the rectangular inner shape in a cylindrical tableau associated to a tableau. St001207The Lowey length of the algebra A/T when T is the 1-tilting module corresponding to the permutation in the Auslander algebra of K[x]/(xn). St000744The length of the path to the largest entry in a standard Young tableau. St001515The vector space dimension of the socle of the first syzygy module of the regular module (as a bimodule). St000044The number of vertices of the unicellular map given by a perfect matching. St000017The number of inversions of a standard tableau. St001721The degree of a binary word. St000016The number of attacking pairs of a standard tableau. St000246The number of non-inversions of a permutation. St000463The number of admissible inversions of a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St001004The number of indices that are either left-to-right maxima or right-to-left minima. St000993The multiplicity of the largest part of an integer partition. St000130The number of occurrences of the contiguous pattern [.,[[.,.],[[.,.],.]]] in a binary tree. St000132The number of occurrences of the contiguous pattern [[.,.],[.,[[.,.],.]]] in a binary tree. St000657The smallest part of an integer composition. St001667The maximal size of a pair of weak twins for a permutation. St000769The major index of a composition regarded as a word. St000766The number of inversions of an integer composition. St000768The number of peaks in an integer composition. St000760The length of the longest strictly decreasing subsequence of parts of an integer composition. St000764The number of strong records in an integer composition. St000036The evaluation at 1 of the Kazhdan-Lusztig polynomial with parameters given by the identity and the permutation. St001095The number of non-isomorphic posets with precisely one further covering relation. St001396Number of triples of incomparable elements in a finite poset. St001532The leading coefficient of the Poincare polynomial of the poset cone. St001533The largest coefficient of the Poincare polynomial of the poset cone. St000741The Colin de Verdière graph invariant. St000761The number of ascents in an integer composition. St000763The sum of the positions of the strong records of an integer composition. St000124The cardinality of the preimage of the Simion-Schmidt map. St000762The sum of the positions of the weak records of an integer composition. St000765The number of weak records in an integer composition. St000900The minimal number of repetitions of a part in an integer composition. St000902 The minimal number of repetitions of an integer composition. St000455The second largest eigenvalue of a graph if it is integral. St000383The last part of an integer composition. St001964The interval resolution global dimension of a poset. St000754The Grundy value for the game of removing nestings in a perfect matching. St000788The number of nesting-similar perfect matchings of a perfect matching. St001132The number of leaves in the subtree whose sister has label 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000787The number of flips required to make a perfect matching noncrossing. St000735The last entry on the main diagonal of a standard tableau. St001035The convexity degree of the parallelogram polyomino associated with the Dyck path. St000002The number of occurrences of the pattern 123 in a permutation. St000508Eigenvalues of the random-to-random operator acting on a simple module. St000281The size of the preimage of the map 'to poset' from Binary trees to Posets. St000282The size of the preimage of the map 'to poset' from Ordered trees to Posets. St000298The order dimension or Dushnik-Miller dimension of a poset. St000396The register function (or Horton-Strahler number) of a binary tree. St001462The number of factors of a standard tableaux under concatenation. St000205Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and partition weight. St000206Number of non-integral Gelfand-Tsetlin polytopes with prescribed top row and integer composition weight. St001137Number of simple modules that are 3-regular in the corresponding Nakayama algebra. St000529The number of permutations whose descent word is the given binary word. St000352The Elizalde-Pak rank of a permutation. St000670The reversal length of a permutation. St001335The cardinality of a minimal cycle-isolating set of a graph. St000098The chromatic number of a graph. St000312The number of leaves in a graph. St000313The number of degree 2 vertices of a graph. St001133The smallest label in the subtree rooted at the sister of 1 in the decreasing labelled binary unordered tree associated with the perfect matching. St000237The number of small exceedances. St000054The first entry of the permutation. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001845The number of join irreducibles minus the rank of a lattice. St000782The indicator function of whether a given perfect matching is an L & P matching. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001613The binary logarithm of the size of the center of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001881The number of factors of a lattice as a Cartesian product of lattices. St001043The depth of the leaf closest to the root in the binary unordered tree associated with the perfect matching. St000233The number of nestings of a set partition. St000454The largest eigenvalue of a graph if it is integral. St000264The girth of a graph, which is not a tree. St000583The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1, 2 are maximal. St000374The number of exclusive right-to-left minima of a permutation. St000451The length of the longest pattern of the form k 1 2. St000359The number of occurrences of the pattern 23-1. St000496The rcs statistic of a set partition. St000996The number of exclusive left-to-right maxima of a permutation. St000052The number of valleys of a Dyck path not on the x-axis. St000590The number of occurrences of the pattern {{1},{2,3}} such that 2 is minimal, 1 is maximal, (2,3) are consecutive in a block. St000596The number of occurrences of the pattern {{1},{2},{3}} such that 3 is minimal, 1 is maximal. St000254The nesting number of a set partition. St000729The minimal arc length of a set partition.
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