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Matching statistic: St000507
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Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> [[1]]
=> 1
[2]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1]
=> [2]
=> [[1,2]]
=> 2
[3]
=> [2,1]
=> [[1,3],[2]]
=> 2
[2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[4]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[3,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[2,2]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 3
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[5]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 4
[4,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[3,2]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[3,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 2
[2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 3
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 4
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5
[6]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 5
[5,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4
[4,2]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 2
[4,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 3
[3,3]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 3
[3,2,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 4
[2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 4
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 5
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 5
[1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 6
[7]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 6
[6,1]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 5
[5,2]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 4
[5,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 5
[4,3]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 3
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1
[4,1,1,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 4
[3,3,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 2
[3,2,2]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 3
[3,2,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 5
[2,2,2,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 5
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 6
[2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 6
[1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 7
[8]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 7
[7,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 6
[6,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 5
[6,1,1]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 6
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> 4
[5,2,1]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> 4
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
Matching statistic: St001176
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St001176: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> [1]
=> 0 = 1 - 1
[2]
=> [1,1]
=> [2]
=> 0 = 1 - 1
[1,1]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[3]
=> [2,1]
=> [2,1]
=> 1 = 2 - 1
[2,1]
=> [1,1,1]
=> [3]
=> 0 = 1 - 1
[1,1,1]
=> [3]
=> [1,1,1]
=> 2 = 3 - 1
[4]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
[3,1]
=> [1,1,1,1]
=> [4]
=> 0 = 1 - 1
[2,2]
=> [2,1,1]
=> [3,1]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> 3 = 4 - 1
[5]
=> [3,2]
=> [2,2,1]
=> 3 = 4 - 1
[4,1]
=> [3,1,1]
=> [3,1,1]
=> 2 = 3 - 1
[3,2]
=> [1,1,1,1,1]
=> [5]
=> 0 = 1 - 1
[3,1,1]
=> [2,1,1,1]
=> [4,1]
=> 1 = 2 - 1
[2,2,1]
=> [2,2,1]
=> [3,2]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> 4 = 5 - 1
[6]
=> [3,3]
=> [2,2,2]
=> 4 = 5 - 1
[5,1]
=> [3,2,1]
=> [3,2,1]
=> 3 = 4 - 1
[4,2]
=> [2,1,1,1,1]
=> [5,1]
=> 1 = 2 - 1
[4,1,1]
=> [2,2,1,1]
=> [4,2]
=> 2 = 3 - 1
[3,3]
=> [3,1,1,1]
=> [4,1,1]
=> 2 = 3 - 1
[3,2,1]
=> [1,1,1,1,1,1]
=> [6]
=> 0 = 1 - 1
[3,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> 3 = 4 - 1
[2,2,2]
=> [2,2,2]
=> [3,3]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> 5 = 6 - 1
[7]
=> [4,3]
=> [2,2,2,1]
=> 5 = 6 - 1
[6,1]
=> [3,3,1]
=> [3,2,2]
=> 4 = 5 - 1
[5,2]
=> [3,2,1,1]
=> [4,2,1]
=> 3 = 4 - 1
[5,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> 4 = 5 - 1
[4,3]
=> [2,2,1,1,1]
=> [5,2]
=> 2 = 3 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> 0 = 1 - 1
[4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> 3 = 4 - 1
[3,3,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> 1 = 2 - 1
[3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> 4 = 5 - 1
[2,2,2,1]
=> [3,2,2]
=> [3,3,1]
=> 4 = 5 - 1
[2,2,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> 6 = 7 - 1
[8]
=> [4,4]
=> [2,2,2,2]
=> 6 = 7 - 1
[7,1]
=> [4,3,1]
=> [3,2,2,1]
=> 5 = 6 - 1
[6,2]
=> [3,3,1,1]
=> [4,2,2]
=> 4 = 5 - 1
[6,1,1]
=> [4,2,2]
=> [3,3,1,1]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [5,3]
=> 3 = 4 - 1
[5,2,1]
=> [4,1,1,1,1]
=> [5,1,1,1]
=> 3 = 4 - 1
Description
The size of a partition minus its first part.
This is the number of boxes in its diagram that are not in the first row.
Matching statistic: St000738
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000738: Standard tableaux ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
[1]
=> [1]
=> [1]
=> [[1]]
=> 1
[2]
=> [1,1]
=> [2]
=> [[1,2]]
=> 1
[1,1]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 2
[3]
=> [2,1]
=> [2,1]
=> [[1,3],[2]]
=> 2
[2,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[1,1,1]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[4]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 3
[3,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[2,2]
=> [2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 2
[2,1,1]
=> [3,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 3
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[5]
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 4
[4,1]
=> [3,1,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 3
[3,2]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[3,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 2
[2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 3
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 4
[1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
[6]
=> [3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 5
[5,1]
=> [3,2,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 4
[4,2]
=> [2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 2
[4,1,1]
=> [2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 3
[3,3]
=> [3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 3
[3,2,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 1
[3,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 4
[2,2,2]
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 4
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 5
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 5
[1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[7]
=> [4,3]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 6
[6,1]
=> [3,3,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 5
[5,2]
=> [3,2,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 4
[5,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 5
[4,3]
=> [2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 3
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 1
[4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 4
[3,3,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 2
[3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 3
[3,2,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 5
[2,2,2,1]
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 5
[2,2,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 6
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 6
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7
[8]
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 7
[7,1]
=> [4,3,1]
=> [3,2,2,1]
=> [[1,3,8],[2,5],[4,7],[6]]
=> 6
[6,2]
=> [3,3,1,1]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 5
[6,1,1]
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 6
[5,3]
=> [2,2,2,1,1]
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> 4
[5,2,1]
=> [4,1,1,1,1]
=> [5,1,1,1]
=> [[1,5,6,7,8],[2],[3],[4]]
=> 4
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12
Description
The first entry in the last row of a standard tableau.
For the last entry in the first row, see [[St000734]].
Matching statistic: St000228
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000228: Integer partitions ⟶ ℤResult quality: 92% ●values known / values provided: 99%●distinct values known / distinct values provided: 92%
Values
[1]
=> [1]
=> [1]
=> []
=> 0 = 1 - 1
[2]
=> [1,1]
=> [2]
=> []
=> 0 = 1 - 1
[1,1]
=> [2]
=> [1,1]
=> [1]
=> 1 = 2 - 1
[3]
=> [2,1]
=> [2,1]
=> [1]
=> 1 = 2 - 1
[2,1]
=> [1,1,1]
=> [3]
=> []
=> 0 = 1 - 1
[1,1,1]
=> [3]
=> [1,1,1]
=> [1,1]
=> 2 = 3 - 1
[4]
=> [2,2]
=> [2,2]
=> [2]
=> 2 = 3 - 1
[3,1]
=> [1,1,1,1]
=> [4]
=> []
=> 0 = 1 - 1
[2,2]
=> [2,1,1]
=> [3,1]
=> [1]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [2,1,1]
=> [1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[5]
=> [3,2]
=> [2,2,1]
=> [2,1]
=> 3 = 4 - 1
[4,1]
=> [3,1,1]
=> [3,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,2]
=> [1,1,1,1,1]
=> [5]
=> []
=> 0 = 1 - 1
[3,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1]
=> 1 = 2 - 1
[2,2,1]
=> [2,2,1]
=> [3,2]
=> [2]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[6]
=> [3,3]
=> [2,2,2]
=> [2,2]
=> 4 = 5 - 1
[5,1]
=> [3,2,1]
=> [3,2,1]
=> [2,1]
=> 3 = 4 - 1
[4,2]
=> [2,1,1,1,1]
=> [5,1]
=> [1]
=> 1 = 2 - 1
[4,1,1]
=> [2,2,1,1]
=> [4,2]
=> [2]
=> 2 = 3 - 1
[3,3]
=> [3,1,1,1]
=> [4,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,2,1]
=> [1,1,1,1,1,1]
=> [6]
=> []
=> 0 = 1 - 1
[3,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[2,2,2]
=> [2,2,2]
=> [3,3]
=> [3]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> [2,1,1]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[7]
=> [4,3]
=> [2,2,2,1]
=> [2,2,1]
=> 5 = 6 - 1
[6,1]
=> [3,3,1]
=> [3,2,2]
=> [2,2]
=> 4 = 5 - 1
[5,2]
=> [3,2,1,1]
=> [4,2,1]
=> [2,1]
=> 3 = 4 - 1
[5,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> [2,1,1]
=> 4 = 5 - 1
[4,3]
=> [2,2,1,1,1]
=> [5,2]
=> [2]
=> 2 = 3 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> []
=> 0 = 1 - 1
[4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> [3]
=> 3 = 4 - 1
[3,3,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1]
=> 1 = 2 - 1
[3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> 4 = 5 - 1
[2,2,2,1]
=> [3,2,2]
=> [3,3,1]
=> [3,1]
=> 4 = 5 - 1
[2,2,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> 6 = 7 - 1
[8]
=> [4,4]
=> [2,2,2,2]
=> [2,2,2]
=> 6 = 7 - 1
[7,1]
=> [4,3,1]
=> [3,2,2,1]
=> [2,2,1]
=> 5 = 6 - 1
[6,2]
=> [3,3,1,1]
=> [4,2,2]
=> [2,2]
=> 4 = 5 - 1
[6,1,1]
=> [4,2,2]
=> [3,3,1,1]
=> [3,1,1]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [5,3]
=> [3]
=> 3 = 4 - 1
[5,2,1]
=> [4,1,1,1,1]
=> [5,1,1,1]
=> [1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> [1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,1,1,1,1,1,1,1,1,1,1]
=> ? = 12 - 1
Description
The size of a partition.
This statistic is the constant statistic of the level sets.
Matching statistic: St000377
St000377: Integer partitions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
[1]
=> 0 = 1 - 1
[2]
=> 0 = 1 - 1
[1,1]
=> 1 = 2 - 1
[3]
=> 1 = 2 - 1
[2,1]
=> 0 = 1 - 1
[1,1,1]
=> 2 = 3 - 1
[4]
=> 2 = 3 - 1
[3,1]
=> 0 = 1 - 1
[2,2]
=> 1 = 2 - 1
[2,1,1]
=> 2 = 3 - 1
[1,1,1,1]
=> 3 = 4 - 1
[5]
=> 3 = 4 - 1
[4,1]
=> 2 = 3 - 1
[3,2]
=> 0 = 1 - 1
[3,1,1]
=> 1 = 2 - 1
[2,2,1]
=> 2 = 3 - 1
[2,1,1,1]
=> 3 = 4 - 1
[1,1,1,1,1]
=> 4 = 5 - 1
[6]
=> 4 = 5 - 1
[5,1]
=> 3 = 4 - 1
[4,2]
=> 1 = 2 - 1
[4,1,1]
=> 2 = 3 - 1
[3,3]
=> 2 = 3 - 1
[3,2,1]
=> 0 = 1 - 1
[3,1,1,1]
=> 3 = 4 - 1
[2,2,2]
=> 3 = 4 - 1
[2,2,1,1]
=> 4 = 5 - 1
[2,1,1,1,1]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> 5 = 6 - 1
[7]
=> 5 = 6 - 1
[6,1]
=> 4 = 5 - 1
[5,2]
=> 3 = 4 - 1
[5,1,1]
=> 4 = 5 - 1
[4,3]
=> 2 = 3 - 1
[4,2,1]
=> 0 = 1 - 1
[4,1,1,1]
=> 3 = 4 - 1
[3,3,1]
=> 1 = 2 - 1
[3,2,2]
=> 2 = 3 - 1
[3,2,1,1]
=> 3 = 4 - 1
[3,1,1,1,1]
=> 4 = 5 - 1
[2,2,2,1]
=> 4 = 5 - 1
[2,2,1,1,1]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> 6 = 7 - 1
[8]
=> 6 = 7 - 1
[7,1]
=> 5 = 6 - 1
[6,2]
=> 4 = 5 - 1
[6,1,1]
=> 5 = 6 - 1
[5,3]
=> 3 = 4 - 1
[5,2,1]
=> 3 = 4 - 1
[9,4]
=> ? = 10 - 1
[9,2,2]
=> ? = 9 - 1
[9,2,1,1]
=> ? = 10 - 1
[8,4,1]
=> ? = 9 - 1
[7,2,2,2]
=> ? = 9 - 1
[6,6,1]
=> ? = 9 - 1
[6,2,2,1,1,1]
=> ? = 9 - 1
[4,2,2,2,2,1]
=> ? = 9 - 1
[3,3,2,2,2,1]
=> ? = 10 - 1
[9,4,1]
=> ? = 10 - 1
[9,2,2,1]
=> ? = 10 - 1
[5,2,2,2,2,1]
=> ? = 10 - 1
Description
The dinv defect of an integer partition.
This is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \not\in \{0,1\}$.
Matching statistic: St000394
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 92%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St000394: Dyck paths ⟶ ℤResult quality: 85% ●values known / values provided: 85%●distinct values known / distinct values provided: 92%
Values
[1]
=> [1]
=> [1,0]
=> [1,0]
=> 0 = 1 - 1
[2]
=> [1,1]
=> [1,1,0,0]
=> [1,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [2]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1 = 2 - 1
[3]
=> [2,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 1 = 2 - 1
[2,1]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 2 = 3 - 1
[4]
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[2,2]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> [1,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[5]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[4,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[3,2]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[2,2,1]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[6]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> [1,1,1,0,1,0,0,0]
=> 4 = 5 - 1
[5,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[4,2]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[4,1,1]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0]
=> 2 = 3 - 1
[3,3]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[3,2,1]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0]
=> 3 = 4 - 1
[2,2,2]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[7]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,0,0,1,0,0,0]
=> 5 = 6 - 1
[6,1]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0]
=> 4 = 5 - 1
[5,2]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[5,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 4 = 5 - 1
[4,3]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[4,1,1,1]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0]
=> 3 = 4 - 1
[3,3,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0]
=> 1 = 2 - 1
[3,2,2]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[2,2,2,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 4 = 5 - 1
[2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6 = 7 - 1
[8]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 6 = 7 - 1
[7,1]
=> [4,3,1]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 5 = 6 - 1
[6,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0,1,0,1,0]
=> 4 = 5 - 1
[6,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0]
=> 3 = 4 - 1
[5,2,1]
=> [4,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> 3 = 4 - 1
[8,2,1]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 7 - 1
[7,2,2]
=> [4,3,2,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[7,2,1,1]
=> [5,3,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0]
=> ? = 8 - 1
[6,2,2,1]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[5,5,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0,1,0,1,0,1,0]
=> ? = 7 - 1
[9,2,1]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0,1,0,1,0]
=> ? = 8 - 1
[8,2,2]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[8,2,1,1]
=> [5,3,3,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0]
=> ? = 9 - 1
[7,4,1]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,0,1,0,1,0]
=> ? = 8 - 1
[7,3,1,1]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0,1,0,1,0]
=> ? = 7 - 1
[7,2,2,1]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,1,0]
=> ? = 8 - 1
[6,6]
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 9 - 1
[6,2,2,1,1]
=> [4,3,2,2,1]
=> [1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 8 - 1
[4,2,2,2,2]
=> [4,3,3,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[9,4]
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 10 - 1
[9,2,2]
=> [5,4,2,1,1]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0,1,0,1,0]
=> ? = 9 - 1
[9,2,1,1]
=> [5,4,3,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0]
=> ? = 10 - 1
[8,5]
=> [5,3,3,2]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0]
=> ? = 10 - 1
[8,4,1]
=> [5,3,3,1,1]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0,1,0,1,0]
=> ? = 9 - 1
[7,2,2,2]
=> [4,3,3,2,1]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 9 - 1
[6,6,1]
=> [5,3,2,2,1]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0,1,0]
=> ? = 9 - 1
[6,2,2,1,1,1]
=> [4,4,2,2,1]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 9 - 1
[4,2,2,2,2,1]
=> [4,4,3,1,1]
=> [1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 9 - 1
[9,5]
=> [5,4,3,2]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0]
=> ? = 11 - 1
[9,4,1]
=> [5,4,3,1,1]
=> [1,0,1,1,1,0,1,1,1,0,0,0,0,1,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,0,1,0,1,0]
=> ? = 10 - 1
[9,2,2,1]
=> [5,4,2,2,1]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0,1,0]
=> ? = 10 - 1
[8,5,1]
=> [5,3,3,2,1]
=> [1,0,1,0,1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,1,0,0,1,0]
=> ? = 10 - 1
[9,5,1]
=> [5,4,3,2,1]
=> [1,0,1,1,1,0,1,1,1,0,0,1,0,0,0,1,0,0]
=> [1,1,1,1,1,0,0,1,0,0,1,0,0,1,0,0,1,0]
=> ? = 11 - 1
Description
The sum of the heights of the peaks of a Dyck path minus the number of peaks.
Matching statistic: St000376
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 78% ●values known / values provided: 78%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> [1,1,0,0,1,1,0,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> 3 = 4 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 3 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> 4 = 5 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6 = 7 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> 4 = 5 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> 5 = 6 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> [1,1,1,0,0,0,1,1,0,1,0,0]
=> 3 = 4 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8 - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 6 - 1
[3,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 7 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 10 - 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 11 - 1
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 10 - 1
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8 - 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 7 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ?
=> ? = 12 - 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 10 - 1
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 9 - 1
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 10 - 1
[8,5]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 10 - 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 9 - 1
[9,5]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 11 - 1
[9,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0]
=> ?
=> ? = 10 - 1
[9,2,2,1]
=> [1,1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 10 - 1
[8,5,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,1,0]
=> ?
=> ? = 10 - 1
[9,5,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,1,0,0,0,0,1,0]
=> ?
=> ? = 11 - 1
Description
The bounce deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{bounce}(D).$$
The zeta map [[Mp00032]] sends this statistic to the dinv deficit [[St000369]], both are thus equidistributed.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 92%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 92%
Values
[1]
=> [1]
=> [[1]]
=> [[1]]
=> 0 = 1 - 1
[2]
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 0 = 1 - 1
[1,1]
=> [2]
=> [[1,2]]
=> [[1],[2]]
=> 1 = 2 - 1
[3]
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1 = 2 - 1
[2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 0 = 1 - 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[4]
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2 = 3 - 1
[3,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 0 = 1 - 1
[2,2]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1 = 2 - 1
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 2 = 3 - 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[5]
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 3 = 4 - 1
[4,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 2 = 3 - 1
[3,2]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
[3,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1 = 2 - 1
[2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
[6]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 4 = 5 - 1
[5,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 3 = 4 - 1
[4,2]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1 = 2 - 1
[4,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 2 = 3 - 1
[3,3]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 2 = 3 - 1
[3,2,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 4 - 1
[2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 3 = 4 - 1
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> [[1],[2],[3],[4],[5],[6]]
=> 5 = 6 - 1
[7]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7]]
=> 5 = 6 - 1
[6,1]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7]]
=> 4 = 5 - 1
[5,2]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6],[7]]
=> 3 = 4 - 1
[5,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> [[1,2,4],[3,5],[6],[7]]
=> 4 = 5 - 1
[4,3]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> [[1,2,3,4,6],[5,7]]
=> 2 = 3 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
[4,1,1,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> [[1,2,4,6],[3,5,7]]
=> 3 = 4 - 1
[3,3,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6],[7]]
=> 1 = 2 - 1
[3,2,2]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7]]
=> 2 = 3 - 1
[3,2,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> [[1,2,3,4],[5],[6],[7]]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> [[1,2,3],[4],[5],[6],[7]]
=> 4 = 5 - 1
[2,2,2,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 4 = 5 - 1
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 6 = 7 - 1
[8]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> 6 = 7 - 1
[7,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> [[1,2,5],[3,6],[4,7],[8]]
=> 5 = 6 - 1
[6,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> [[1,2,3,6],[4,7],[5,8]]
=> 4 = 5 - 1
[6,1,1]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 5 = 6 - 1
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> [[1,2,3,5,7],[4,6,8]]
=> 3 = 4 - 1
[5,2,1]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6],[7],[8]]
=> 3 = 4 - 1
[9,2]
=> [5,4,1,1]
=> [[1,4,5,6,11],[2,8,9,10],[3],[7]]
=> [[1,2,3,7],[4,8],[5,9],[6,10],[11]]
=> ? = 8 - 1
[9,1,1]
=> [5,4,2]
=> [[1,2,5,6,11],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11]]
=> ? = 9 - 1
[8,3]
=> [4,4,2,1]
=> [[1,3,6,7],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11]]
=> ? = 8 - 1
[8,2,1]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> [[1,2,3,4,8],[5,9],[6,10],[7,11]]
=> ? = 7 - 1
[8,1,1,1]
=> [5,3,3]
=> [[1,2,3,10,11],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10],[11]]
=> ? = 9 - 1
[7,4]
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 8 - 1
[7,2,2]
=> [4,3,2,1,1]
=> [[1,4,7,11],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11]]
=> ? = 7 - 1
[7,2,1,1]
=> [5,3,2,1]
=> [[1,3,6,10,11],[2,5,9],[4,8],[7]]
=> [[1,2,4,7],[3,5,8],[6,9],[10],[11]]
=> ? = 8 - 1
[6,2,2,1]
=> [5,2,2,1,1]
=> [[1,4,9,10,11],[2,6],[3,8],[5],[7]]
=> [[1,2,3,5,7],[4,6,8],[9],[10],[11]]
=> ? = 7 - 1
[6,2,1,1,1]
=> [3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11]]
=> ? = 7 - 1
[6,1,1,1,1,1]
=> [3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11]]
=> ? = 8 - 1
[5,5,1]
=> [5,3,1,1,1]
=> [[1,5,6,10,11],[2,8,9],[3],[4],[7]]
=> [[1,2,3,4,7],[5,8],[6,9],[10],[11]]
=> ? = 7 - 1
[4,2,2,2,1]
=> [3,3,3,1,1]
=> [[1,4,5],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11]]
=> ? = 7 - 1
[3,3,3,1,1]
=> [4,2,2,2,1]
=> [[1,3,10,11],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9],[10],[11]]
=> ? = 7 - 1
[3,3,2,2,1]
=> [4,3,2,2]
=> [[1,2,7,11],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11]]
=> ? = 8 - 1
[3,2,2,2,2]
=> [4,3,3,1]
=> [[1,3,4,11],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11]]
=> ? = 8 - 1
[2,2,2,2,2,1]
=> [4,4,3]
=> [[1,2,3,7],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11]]
=> ? = 9 - 1
[9,3]
=> [5,4,2,1]
=> [[1,3,6,7,12],[2,5,10,11],[4,9],[8]]
=> [[1,2,4,8],[3,5,9],[6,10],[7,11],[12]]
=> ? = 9 - 1
[9,2,1]
=> [5,4,1,1,1]
=> [[1,5,6,7,12],[2,9,10,11],[3],[4],[8]]
=> [[1,2,3,4,8],[5,9],[6,10],[7,11],[12]]
=> ? = 8 - 1
[9,1,1,1]
=> [5,4,3]
=> [[1,2,3,7,12],[4,5,6,11],[8,9,10]]
=> [[1,4,8],[2,5,9],[3,6,10],[7,11],[12]]
=> ? = 10 - 1
[8,4]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 9 - 1
[8,2,2]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> [[1,2,3,5,9],[4,6,10],[7,11],[8,12]]
=> ? = 8 - 1
[8,2,1,1]
=> [5,3,3,1]
=> [[1,3,4,11,12],[2,6,7],[5,9,10],[8]]
=> [[1,2,5,8],[3,6,9],[4,7,10],[11],[12]]
=> ? = 9 - 1
[7,4,1]
=> [5,3,2,1,1]
=> [[1,4,7,11,12],[2,6,10],[3,9],[5],[8]]
=> [[1,2,3,5,8],[4,6,9],[7,10],[11],[12]]
=> ? = 8 - 1
[7,3,1,1]
=> [3,3,2,2,1,1]
=> [[1,4,9],[2,6,12],[3,8],[5,11],[7],[10]]
=> [[1,2,3,5,7,10],[4,6,8,11],[9,12]]
=> ? = 7 - 1
[7,2,2,1]
=> [5,2,2,2,1]
=> [[1,3,10,11,12],[2,5],[4,7],[6,9],[8]]
=> [[1,2,4,6,8],[3,5,7,9],[10],[11],[12]]
=> ? = 8 - 1
[6,6]
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 9 - 1
[6,2,2,1,1]
=> [4,3,2,2,1]
=> [[1,3,8,12],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11],[12]]
=> ? = 8 - 1
[6,2,1,1,1,1]
=> [3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12]]
=> ? = 8 - 1
[4,2,2,2,2]
=> [4,3,3,1,1]
=> [[1,4,5,12],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11],[12]]
=> ? = 8 - 1
[3,3,2,2,2]
=> [4,3,3,2]
=> [[1,2,5,12],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12]]
=> ? = 9 - 1
[3,3,2,2,1,1]
=> [6,4,2]
=> [[1,2,5,6,11,12],[3,4,9,10],[7,8]]
=> [[1,3,7],[2,4,8],[5,9],[6,10],[11],[12]]
=> ? = 10 - 1
[3,2,2,2,2,1]
=> [4,4,3,1]
=> [[1,3,4,8],[2,6,7,12],[5,10,11],[9]]
=> [[1,2,5,9],[3,6,10],[4,7,11],[8,12]]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [12]
=> [[1,2,3,4,5,6,7,8,9,10,11,12]]
=> [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 12 - 1
[9,4]
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 10 - 1
[9,2,2]
=> [5,4,2,1,1]
=> [[1,4,7,8,13],[2,6,11,12],[3,10],[5],[9]]
=> [[1,2,3,5,9],[4,6,10],[7,11],[8,12],[13]]
=> ? = 9 - 1
[9,2,1,1]
=> [5,4,3,1]
=> [[1,3,4,8,13],[2,6,7,12],[5,10,11],[9]]
=> [[1,2,5,9],[3,6,10],[4,7,11],[8,12],[13]]
=> ? = 10 - 1
[8,5]
=> [5,3,3,2]
=> [[1,2,5,12,13],[3,4,8],[6,7,11],[9,10]]
=> [[1,3,6,9],[2,4,7,10],[5,8,11],[12],[13]]
=> ? = 10 - 1
[8,4,1]
=> [5,3,3,1,1]
=> [[1,4,5,12,13],[2,7,8],[3,10,11],[6],[9]]
=> [[1,2,3,6,9],[4,7,10],[5,8,11],[12],[13]]
=> ? = 9 - 1
[7,2,2,2]
=> [4,3,3,2,1]
=> [[1,3,6,13],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13]]
=> ? = 9 - 1
[6,6,1]
=> [5,3,2,2,1]
=> [[1,3,8,12,13],[2,5,11],[4,7],[6,10],[9]]
=> [[1,2,4,6,9],[3,5,7,10],[8,11],[12],[13]]
=> ? = 9 - 1
[6,2,2,1,1,1]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> [[1,2,4,6,10],[3,5,7,11],[8,12],[9,13]]
=> ? = 9 - 1
[4,2,2,2,2,1]
=> [4,4,3,1,1]
=> [[1,4,5,9],[2,7,8,13],[3,11,12],[6],[10]]
=> [[1,2,3,6,10],[4,7,11],[5,8,12],[9,13]]
=> ? = 9 - 1
[3,3,2,2,2,1]
=> [4,4,3,2]
=> [[1,2,5,9],[3,4,8,13],[6,7,12],[10,11]]
=> [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13]]
=> ? = 10 - 1
[9,5]
=> [5,4,3,2]
=> [[1,2,5,9,14],[3,4,8,13],[6,7,12],[10,11]]
=> [[1,3,6,10],[2,4,7,11],[5,8,12],[9,13],[14]]
=> ? = 11 - 1
[9,4,1]
=> [5,4,3,1,1]
=> [[1,4,5,9,14],[2,7,8,13],[3,11,12],[6],[10]]
=> [[1,2,3,6,10],[4,7,11],[5,8,12],[9,13],[14]]
=> ? = 10 - 1
[9,2,2,1]
=> [5,4,2,2,1]
=> [[1,3,8,9,14],[2,5,12,13],[4,7],[6,11],[10]]
=> [[1,2,4,6,10],[3,5,7,11],[8,12],[9,13],[14]]
=> ? = 10 - 1
[8,5,1]
=> [5,3,3,2,1]
=> [[1,3,6,13,14],[2,5,9],[4,8,12],[7,11],[10]]
=> [[1,2,4,7,10],[3,5,8,11],[6,9,12],[13],[14]]
=> ? = 10 - 1
[5,2,2,2,2,1]
=> [4,4,3,2,1]
=> [[1,3,6,10],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14]]
=> ? = 10 - 1
[9,5,1]
=> [5,4,3,2,1]
=> [[1,3,6,10,15],[2,5,9,14],[4,8,13],[7,12],[11]]
=> [[1,2,4,7,11],[3,5,8,12],[6,9,13],[10,14],[15]]
=> ? = 11 - 1
Description
The number of descents of a standard tableau.
Entry $i$ of a standard Young tableau is a descent if $i+1$ appears in a row below the row of $i$.
Matching statistic: St000369
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 83%
St000369: Dyck paths ⟶ ℤResult quality: 72% ●values known / values provided: 72%●distinct values known / distinct values provided: 83%
Values
[1]
=> [1,0,1,0]
=> 0 = 1 - 1
[2]
=> [1,1,0,0,1,0]
=> 0 = 1 - 1
[1,1]
=> [1,0,1,1,0,0]
=> 1 = 2 - 1
[3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
[2,1]
=> [1,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 2 = 3 - 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 2 = 3 - 1
[3,1]
=> [1,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[2,2]
=> [1,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3 = 4 - 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 3 = 4 - 1
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,2]
=> [1,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 1 = 2 - 1
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 2 = 3 - 1
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3 = 4 - 1
[1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 4 = 5 - 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 3 = 4 - 1
[4,2]
=> [1,1,1,0,0,1,0,0,1,0]
=> 1 = 2 - 1
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> 2 = 3 - 1
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> 0 = 1 - 1
[3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3 = 4 - 1
[2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 4 - 1
[2,2,1,1]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4 = 5 - 1
[2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 4 = 5 - 1
[1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> 5 = 6 - 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
=> 4 = 5 - 1
[5,2]
=> [1,1,1,1,0,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[5,1,1]
=> [1,1,1,0,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> 2 = 3 - 1
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> 0 = 1 - 1
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> 3 = 4 - 1
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> 1 = 2 - 1
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> 2 = 3 - 1
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [1,0,1,1,1,1,0,0,1,0,0,0]
=> 4 = 5 - 1
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4 = 5 - 1
[2,2,1,1,1]
=> [1,0,1,1,1,0,1,1,0,0,0,0]
=> 5 = 6 - 1
[2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> 5 = 6 - 1
[1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 6 = 7 - 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[7,1]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 6 - 1
[6,2]
=> [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[6,1,1]
=> [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
=> 5 = 6 - 1
[5,3]
=> [1,1,1,1,0,0,0,1,0,0,1,0]
=> 3 = 4 - 1
[5,2,1]
=> [1,1,1,0,1,0,1,0,0,0,1,0]
=> 3 = 4 - 1
[5,1,1,1]
=> [1,1,0,1,1,1,0,0,0,0,1,0]
=> 4 = 5 - 1
[4,4]
=> [1,1,1,1,0,0,0,0,1,1,0,0]
=> 4 = 5 - 1
[4,3,1]
=> [1,1,0,1,0,0,1,0,1,0]
=> 0 = 1 - 1
[1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 8 - 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[8,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[7,2]
=> [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[7,1,1]
=> [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 8 - 1
[1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 - 1
[9,1]
=> [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[8,2]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[8,1,1]
=> [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[7,3]
=> [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 7 - 1
[7,2,1]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
=> ? = 6 - 1
[7,1,1,1]
=> [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[3,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 8 - 1
[2,2,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0]
=> ? = 9 - 1
[2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10 - 1
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 9 - 1
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 8 - 1
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 7 - 1
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 9 - 1
[7,4]
=> [1,1,1,1,1,1,0,0,0,0,1,0,0,0,1,0]
=> ? = 8 - 1
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> ? = 7 - 1
[7,2,1,1]
=> [1,1,1,1,0,1,1,0,1,0,0,0,0,0,1,0]
=> ? = 8 - 1
[12]
=> [1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> ? = 11 - 1
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> ? = 9 - 1
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 10 - 1
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> ? = 9 - 1
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> ? = 8 - 1
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> ? = 9 - 1
[7,4,1]
=> [1,1,1,1,1,0,1,0,0,0,1,0,0,0,1,0]
=> ? = 8 - 1
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> ? = 7 - 1
[7,2,2,1]
=> [1,1,1,1,0,1,0,1,1,0,0,0,0,0,1,0]
=> ? = 8 - 1
[6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 9 - 1
[1,1,1,1,1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0]
=> ? = 12 - 1
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 10 - 1
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 9 - 1
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> ? = 10 - 1
[8,5]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> ? = 10 - 1
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> ? = 9 - 1
[7,2,2,2]
=> [1,1,1,1,0,0,1,1,1,0,0,0,0,0,1,0]
=> ? = 9 - 1
[6,6,1]
=> [1,1,1,1,1,0,1,0,0,0,0,0,1,1,0,0]
=> ? = 9 - 1
[9,5]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,1,0]
=> ? = 11 - 1
[9,4,1]
=> [1,1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,0,1,0]
=> ? = 10 - 1
Description
The dinv deficit of a Dyck path.
For a Dyck path $D$ of semilength $n$, this is defined as
$$\binom{n}{2} - \operatorname{area}(D) - \operatorname{dinv}(D).$$
In other words, this is the number of boxes in the partition traced out by $D$ for which the leg-length minus the arm-length is not in $\{0,1\}$.
See also [[St000376]] for the bounce deficit.
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