Your data matches 9 different statistics following compositions of up to 3 maps.
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Matching statistic: St000377
Mapping
St000377: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => 0
[2]
 => 0
[1,1]
 => 1
[3]
 => 1
[2,1]
 => 0
[1,1,1]
 => 2
[4]
 => 2
[3,1]
 => 0
[2,2]
 => 1
[2,1,1]
 => 2
[1,1,1,1]
 => 3
[5]
 => 3
[4,1]
 => 2
[3,2]
 => 0
[3,1,1]
 => 1
[2,2,1]
 => 2
[2,1,1,1]
 => 3
[1,1,1,1,1]
 => 4
[6]
 => 4
[5,1]
 => 3
[4,2]
 => 1
[4,1,1]
 => 2
[3,3]
 => 2
[3,2,1]
 => 0
[3,1,1,1]
 => 3
[2,2,2]
 => 3
[2,2,1,1]
 => 4
[2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => 5
[7]
 => 5
[6,1]
 => 4
[5,2]
 => 3
[5,1,1]
 => 4
[4,3]
 => 2
[4,2,1]
 => 0
[4,1,1,1]
 => 3
[3,3,1]
 => 1
[3,2,2]
 => 2
[3,2,1,1]
 => 3
[3,1,1,1,1]
 => 4
[2,2,2,1]
 => 4
[2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => 6
[8]
 => 6
[7,1]
 => 5
[6,2]
 => 4
[6,1,1]
 => 5
[5,3]
 => 3
[5,2,1]
 => 3
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: St001176
Mapping
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%
Values
[1]
 => [1]
 => [1]
 => 0
[2]
 => [1,1]
 => [2]
 => 0
[1,1]
 => [2]
 => [1,1]
 => 1
[3]
 => [2,1]
 => [2,1]
 => 1
[2,1]
 => [1,1,1]
 => [3]
 => 0
[1,1,1]
 => [3]
 => [1,1,1]
 => 2
[4]
 => [2,2]
 => [2,2]
 => 2
[3,1]
 => [1,1,1,1]
 => [4]
 => 0
[2,2]
 => [2,1,1]
 => [3,1]
 => 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => 2
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => 3
[5]
 => [3,2]
 => [2,2,1]
 => 3
[4,1]
 => [3,1,1]
 => [3,1,1]
 => 2
[3,2]
 => [1,1,1,1,1]
 => [5]
 => 0
[3,1,1]
 => [2,1,1,1]
 => [4,1]
 => 1
[2,2,1]
 => [2,2,1]
 => [3,2]
 => 2
[2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1]
 => 4
[6]
 => [3,3]
 => [2,2,2]
 => 4
[5,1]
 => [3,2,1]
 => [3,2,1]
 => 3
[4,2]
 => [2,1,1,1,1]
 => [5,1]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [4,2]
 => 2
[3,3]
 => [3,1,1,1]
 => [4,1,1]
 => 2
[3,2,1]
 => [1,1,1,1,1,1]
 => [6]
 => 0
[3,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => 3
[2,2,2]
 => [2,2,2]
 => [3,3]
 => 3
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => 4
[2,1,1,1,1]
 => [5,1]
 => [2,1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1]
 => 5
[7]
 => [4,3]
 => [2,2,2,1]
 => 5
[6,1]
 => [3,3,1]
 => [3,2,2]
 => 4
[5,2]
 => [3,2,1,1]
 => [4,2,1]
 => 3
[5,1,1]
 => [4,2,1]
 => [3,2,1,1]
 => 4
[4,3]
 => [2,2,1,1,1]
 => [5,2]
 => 2
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => 0
[4,1,1,1]
 => [2,2,2,1]
 => [4,3]
 => 3
[3,3,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => 2
[3,2,1,1]
 => [4,1,1,1]
 => [4,1,1,1]
 => 3
[3,1,1,1,1]
 => [5,1,1]
 => [3,1,1,1,1]
 => 4
[2,2,2,1]
 => [3,2,2]
 => [3,3,1]
 => 4
[2,2,1,1,1]
 => [5,2]
 => [2,2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => [6,1]
 => [2,1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => [7]
 => [1,1,1,1,1,1,1]
 => 6
[8]
 => [4,4]
 => [2,2,2,2]
 => 6
[7,1]
 => [4,3,1]
 => [3,2,2,1]
 => 5
[6,2]
 => [3,3,1,1]
 => [4,2,2]
 => 4
[6,1,1]
 => [4,2,2]
 => [3,3,1,1]
 => 5
[5,3]
 => [2,2,2,1,1]
 => [5,3]
 => 3
[5,2,1]
 => [4,1,1,1,1]
 => [5,1,1,1]
 => 3
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: St000157
(load all 2 compositions to match this statistic)
Mapping
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [[1]]
 => [[1]]
 => 0
[2]
 => [1,1]
 => [[1],[2]]
 => [[1,2]]
 => 0
[1,1]
 => [2]
 => [[1,2]]
 => [[1],[2]]
 => 1
[3]
 => [2,1]
 => [[1,3],[2]]
 => [[1,2],[3]]
 => 1
[2,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => [[1,2,3]]
 => 0
[1,1,1]
 => [3]
 => [[1,2,3]]
 => [[1],[2],[3]]
 => 2
[4]
 => [2,2]
 => [[1,2],[3,4]]
 => [[1,3],[2,4]]
 => 2
[3,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => [[1,2,3,4]]
 => 0
[2,2]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => [[1,2,3],[4]]
 => 1
[2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => [[1,2],[3],[4]]
 => 2
[1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => [[1],[2],[3],[4]]
 => 3
[5]
 => [3,2]
 => [[1,2,5],[3,4]]
 => [[1,3],[2,4],[5]]
 => 3
[4,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => [[1,2,3],[4],[5]]
 => 2
[3,2]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => [[1,2,3,4,5]]
 => 0
[3,1,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => [[1,2,3,4],[5]]
 => 1
[2,2,1]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => [[1,2,4],[3,5]]
 => 2
[2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => [[1,2],[3],[4],[5]]
 => 3
[1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => [[1],[2],[3],[4],[5]]
 => 4
[6]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => [[1,4],[2,5],[3,6]]
 => 4
[5,1]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => [[1,2,4],[3,5],[6]]
 => 3
[4,2]
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6]]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6]]
 => 2
[3,3]
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => [[1,2,3,4],[5],[6]]
 => 2
[3,2,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6]]
 => 0
[3,1,1,1]
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => [[1,2,3],[4],[5],[6]]
 => 3
[2,2,2]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => [[1,3,5],[2,4,6]]
 => 3
[2,2,1,1]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => [[1,3],[2,4],[5],[6]]
 => 4
[2,1,1,1,1]
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => [[1,2],[3],[4],[5],[6]]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => [[1],[2],[3],[4],[5],[6]]
 => 5
[7]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => [[1,4],[2,5],[3,6],[7]]
 => 5
[6,1]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7]]
 => 4
[5,2]
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => [[1,2,3,5],[4,6],[7]]
 => 3
[5,1,1]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => [[1,2,4],[3,5],[6],[7]]
 => 4
[4,3]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => [[1,2,3,4,6],[5,7]]
 => 2
[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
[4,1,1,1]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => [[1,2,4,6],[3,5,7]]
 => 3
[3,3,1]
 => [2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => [[1,2,3,4,5,6],[7]]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => [[1,2,3,4,5],[6],[7]]
 => 2
[3,2,1,1]
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => [[1,2,3,4],[5],[6],[7]]
 => 3
[3,1,1,1,1]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => [[1,2,3],[4],[5],[6],[7]]
 => 4
[2,2,2,1]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7]]
 => 4
[2,2,1,1,1]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => [[1,3],[2,4],[5],[6],[7]]
 => 5
[2,1,1,1,1,1]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => [[1,2],[3],[4],[5],[6],[7]]
 => 5
[1,1,1,1,1,1,1]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 6
[8]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => [[1,5],[2,6],[3,7],[4,8]]
 => 6
[7,1]
 => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => [[1,2,5],[3,6],[4,7],[8]]
 => 5
[6,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => [[1,2,3,6],[4,7],[5,8]]
 => 4
[6,1,1]
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => [[1,3,5],[2,4,6],[7],[8]]
 => 5
[5,3]
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => [[1,2,3,5,7],[4,6,8]]
 => 3
[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
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: St000228
Mapping
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1]
 => []
 => 0
[2]
 => [1,1]
 => [2]
 => []
 => 0
[1,1]
 => [2]
 => [1,1]
 => [1]
 => 1
[3]
 => [2,1]
 => [2,1]
 => [1]
 => 1
[2,1]
 => [1,1,1]
 => [3]
 => []
 => 0
[1,1,1]
 => [3]
 => [1,1,1]
 => [1,1]
 => 2
[4]
 => [2,2]
 => [2,2]
 => [2]
 => 2
[3,1]
 => [1,1,1,1]
 => [4]
 => []
 => 0
[2,2]
 => [2,1,1]
 => [3,1]
 => [1]
 => 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => [1,1]
 => 2
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => [1,1,1]
 => 3
[5]
 => [3,2]
 => [2,2,1]
 => [2,1]
 => 3
[4,1]
 => [3,1,1]
 => [3,1,1]
 => [1,1]
 => 2
[3,2]
 => [1,1,1,1,1]
 => [5]
 => []
 => 0
[3,1,1]
 => [2,1,1,1]
 => [4,1]
 => [1]
 => 1
[2,2,1]
 => [2,2,1]
 => [3,2]
 => [2]
 => 2
[2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => [1,1,1]
 => 3
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1]
 => [1,1,1,1]
 => 4
[6]
 => [3,3]
 => [2,2,2]
 => [2,2]
 => 4
[5,1]
 => [3,2,1]
 => [3,2,1]
 => [2,1]
 => 3
[4,2]
 => [2,1,1,1,1]
 => [5,1]
 => [1]
 => 1
[4,1,1]
 => [2,2,1,1]
 => [4,2]
 => [2]
 => 2
[3,3]
 => [3,1,1,1]
 => [4,1,1]
 => [1,1]
 => 2
[3,2,1]
 => [1,1,1,1,1,1]
 => [6]
 => []
 => 0
[3,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => [1,1,1]
 => 3
[2,2,2]
 => [2,2,2]
 => [3,3]
 => [3]
 => 3
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => [2,1,1]
 => 4
[2,1,1,1,1]
 => [5,1]
 => [2,1,1,1,1]
 => [1,1,1,1]
 => 4
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5
[7]
 => [4,3]
 => [2,2,2,1]
 => [2,2,1]
 => 5
[6,1]
 => [3,3,1]
 => [3,2,2]
 => [2,2]
 => 4
[5,2]
 => [3,2,1,1]
 => [4,2,1]
 => [2,1]
 => 3
[5,1,1]
 => [4,2,1]
 => [3,2,1,1]
 => [2,1,1]
 => 4
[4,3]
 => [2,2,1,1,1]
 => [5,2]
 => [2]
 => 2
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => []
 => 0
[4,1,1,1]
 => [2,2,2,1]
 => [4,3]
 => [3]
 => 3
[3,3,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => [1]
 => 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => [1,1]
 => 2
[3,2,1,1]
 => [4,1,1,1]
 => [4,1,1,1]
 => [1,1,1]
 => 3
[3,1,1,1,1]
 => [5,1,1]
 => [3,1,1,1,1]
 => [1,1,1,1]
 => 4
[2,2,2,1]
 => [3,2,2]
 => [3,3,1]
 => [3,1]
 => 4
[2,2,1,1,1]
 => [5,2]
 => [2,2,1,1,1]
 => [2,1,1,1]
 => 5
[2,1,1,1,1,1]
 => [6,1]
 => [2,1,1,1,1,1]
 => [1,1,1,1,1]
 => 5
[1,1,1,1,1,1,1]
 => [7]
 => [1,1,1,1,1,1,1]
 => [1,1,1,1,1,1]
 => 6
[8]
 => [4,4]
 => [2,2,2,2]
 => [2,2,2]
 => 6
[7,1]
 => [4,3,1]
 => [3,2,2,1]
 => [2,2,1]
 => 5
[6,2]
 => [3,3,1,1]
 => [4,2,2]
 => [2,2]
 => 4
[6,1,1]
 => [4,2,2]
 => [3,3,1,1]
 => [3,1,1]
 => 5
[5,3]
 => [2,2,2,1,1]
 => [5,3]
 => [3]
 => 3
[5,2,1]
 => [4,1,1,1,1]
 => [5,1,1,1]
 => [1,1,1]
 => 3
Description
The size of a partition. This statistic is the constant statistic of the level sets.
Matching statistic: St000738
Mapping
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: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[2]
 => [1,1]
 => [2]
 => [[1,2]]
 => 1 = 0 + 1
[1,1]
 => [2]
 => [1,1]
 => [[1],[2]]
 => 2 = 1 + 1
[3]
 => [2,1]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[2,1]
 => [1,1,1]
 => [3]
 => [[1,2,3]]
 => 1 = 0 + 1
[1,1,1]
 => [3]
 => [1,1,1]
 => [[1],[2],[3]]
 => 3 = 2 + 1
[4]
 => [2,2]
 => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[3,1]
 => [1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 1 = 0 + 1
[2,2]
 => [2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 2 = 1 + 1
[2,1,1]
 => [3,1]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 3 = 2 + 1
[1,1,1,1]
 => [4]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 4 = 3 + 1
[5]
 => [3,2]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 4 = 3 + 1
[4,1]
 => [3,1,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[3,2]
 => [1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 1 = 0 + 1
[3,1,1]
 => [2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 2 = 1 + 1
[2,2,1]
 => [2,2,1]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 3 = 2 + 1
[2,1,1,1]
 => [4,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [5]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 5 = 4 + 1
[6]
 => [3,3]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 5 = 4 + 1
[5,1]
 => [3,2,1]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 4 = 3 + 1
[4,2]
 => [2,1,1,1,1]
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => 2 = 1 + 1
[4,1,1]
 => [2,2,1,1]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 3 = 2 + 1
[3,3]
 => [3,1,1,1]
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 3 = 2 + 1
[3,2,1]
 => [1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => 1 = 0 + 1
[3,1,1,1]
 => [4,1,1]
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 4 = 3 + 1
[2,2,2]
 => [2,2,2]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 4 = 3 + 1
[2,2,1,1]
 => [4,2]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 5 = 4 + 1
[2,1,1,1,1]
 => [5,1]
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 5 = 4 + 1
[1,1,1,1,1,1]
 => [6]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
[7]
 => [4,3]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 6 = 5 + 1
[6,1]
 => [3,3,1]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 5 = 4 + 1
[5,2]
 => [3,2,1,1]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 4 = 3 + 1
[5,1,1]
 => [4,2,1]
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 5 = 4 + 1
[4,3]
 => [2,2,1,1,1]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 1 = 0 + 1
[4,1,1,1]
 => [2,2,2,1]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 4 = 3 + 1
[3,3,1]
 => [2,1,1,1,1,1]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => 2 = 1 + 1
[3,2,2]
 => [3,1,1,1,1]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => 3 = 2 + 1
[3,2,1,1]
 => [4,1,1,1]
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [5,1,1]
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => 5 = 4 + 1
[2,2,2,1]
 => [3,2,2]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 5 = 4 + 1
[2,2,1,1,1]
 => [5,2]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 6 = 5 + 1
[2,1,1,1,1,1]
 => [6,1]
 => [2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 6 = 5 + 1
[1,1,1,1,1,1,1]
 => [7]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 7 = 6 + 1
[8]
 => [4,4]
 => [2,2,2,2]
 => [[1,2],[3,4],[5,6],[7,8]]
 => 7 = 6 + 1
[7,1]
 => [4,3,1]
 => [3,2,2,1]
 => [[1,3,8],[2,5],[4,7],[6]]
 => 6 = 5 + 1
[6,2]
 => [3,3,1,1]
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 5 = 4 + 1
[6,1,1]
 => [4,2,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 6 = 5 + 1
[5,3]
 => [2,2,2,1,1]
 => [5,3]
 => [[1,2,3,7,8],[4,5,6]]
 => 4 = 3 + 1
[5,2,1]
 => [4,1,1,1,1]
 => [5,1,1,1]
 => [[1,5,6,7,8],[2],[3],[4]]
 => 4 = 3 + 1
Description
The first entry in the last row of a standard tableau. For the last entry in the first row, see [[St000734]].
Matching statistic: St000507
(load all 2 compositions to match this statistic)
Mapping
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [[1]]
 => 1 = 0 + 1
[2]
 => [1,1]
 => [[1],[2]]
 => 1 = 0 + 1
[1,1]
 => [2]
 => [[1,2]]
 => 2 = 1 + 1
[3]
 => [2,1]
 => [[1,3],[2]]
 => 2 = 1 + 1
[2,1]
 => [1,1,1]
 => [[1],[2],[3]]
 => 1 = 0 + 1
[1,1,1]
 => [3]
 => [[1,2,3]]
 => 3 = 2 + 1
[4]
 => [2,2]
 => [[1,2],[3,4]]
 => 3 = 2 + 1
[3,1]
 => [1,1,1,1]
 => [[1],[2],[3],[4]]
 => 1 = 0 + 1
[2,2]
 => [2,1,1]
 => [[1,4],[2],[3]]
 => 2 = 1 + 1
[2,1,1]
 => [3,1]
 => [[1,3,4],[2]]
 => 3 = 2 + 1
[1,1,1,1]
 => [4]
 => [[1,2,3,4]]
 => 4 = 3 + 1
[5]
 => [3,2]
 => [[1,2,5],[3,4]]
 => 4 = 3 + 1
[4,1]
 => [3,1,1]
 => [[1,4,5],[2],[3]]
 => 3 = 2 + 1
[3,2]
 => [1,1,1,1,1]
 => [[1],[2],[3],[4],[5]]
 => 1 = 0 + 1
[3,1,1]
 => [2,1,1,1]
 => [[1,5],[2],[3],[4]]
 => 2 = 1 + 1
[2,2,1]
 => [2,2,1]
 => [[1,3],[2,5],[4]]
 => 3 = 2 + 1
[2,1,1,1]
 => [4,1]
 => [[1,3,4,5],[2]]
 => 4 = 3 + 1
[1,1,1,1,1]
 => [5]
 => [[1,2,3,4,5]]
 => 5 = 4 + 1
[6]
 => [3,3]
 => [[1,2,3],[4,5,6]]
 => 5 = 4 + 1
[5,1]
 => [3,2,1]
 => [[1,3,6],[2,5],[4]]
 => 4 = 3 + 1
[4,2]
 => [2,1,1,1,1]
 => [[1,6],[2],[3],[4],[5]]
 => 2 = 1 + 1
[4,1,1]
 => [2,2,1,1]
 => [[1,4],[2,6],[3],[5]]
 => 3 = 2 + 1
[3,3]
 => [3,1,1,1]
 => [[1,5,6],[2],[3],[4]]
 => 3 = 2 + 1
[3,2,1]
 => [1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6]]
 => 1 = 0 + 1
[3,1,1,1]
 => [4,1,1]
 => [[1,4,5,6],[2],[3]]
 => 4 = 3 + 1
[2,2,2]
 => [2,2,2]
 => [[1,2],[3,4],[5,6]]
 => 4 = 3 + 1
[2,2,1,1]
 => [4,2]
 => [[1,2,5,6],[3,4]]
 => 5 = 4 + 1
[2,1,1,1,1]
 => [5,1]
 => [[1,3,4,5,6],[2]]
 => 5 = 4 + 1
[1,1,1,1,1,1]
 => [6]
 => [[1,2,3,4,5,6]]
 => 6 = 5 + 1
[7]
 => [4,3]
 => [[1,2,3,7],[4,5,6]]
 => 6 = 5 + 1
[6,1]
 => [3,3,1]
 => [[1,3,4],[2,6,7],[5]]
 => 5 = 4 + 1
[5,2]
 => [3,2,1,1]
 => [[1,4,7],[2,6],[3],[5]]
 => 4 = 3 + 1
[5,1,1]
 => [4,2,1]
 => [[1,3,6,7],[2,5],[4]]
 => 5 = 4 + 1
[4,3]
 => [2,2,1,1,1]
 => [[1,5],[2,7],[3],[4],[6]]
 => 3 = 2 + 1
[4,2,1]
 => [1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7]]
 => 1 = 0 + 1
[4,1,1,1]
 => [2,2,2,1]
 => [[1,3],[2,5],[4,7],[6]]
 => 4 = 3 + 1
[3,3,1]
 => [2,1,1,1,1,1]
 => [[1,7],[2],[3],[4],[5],[6]]
 => 2 = 1 + 1
[3,2,2]
 => [3,1,1,1,1]
 => [[1,6,7],[2],[3],[4],[5]]
 => 3 = 2 + 1
[3,2,1,1]
 => [4,1,1,1]
 => [[1,5,6,7],[2],[3],[4]]
 => 4 = 3 + 1
[3,1,1,1,1]
 => [5,1,1]
 => [[1,4,5,6,7],[2],[3]]
 => 5 = 4 + 1
[2,2,2,1]
 => [3,2,2]
 => [[1,2,7],[3,4],[5,6]]
 => 5 = 4 + 1
[2,2,1,1,1]
 => [5,2]
 => [[1,2,5,6,7],[3,4]]
 => 6 = 5 + 1
[2,1,1,1,1,1]
 => [6,1]
 => [[1,3,4,5,6,7],[2]]
 => 6 = 5 + 1
[1,1,1,1,1,1,1]
 => [7]
 => [[1,2,3,4,5,6,7]]
 => 7 = 6 + 1
[8]
 => [4,4]
 => [[1,2,3,4],[5,6,7,8]]
 => 7 = 6 + 1
[7,1]
 => [4,3,1]
 => [[1,3,4,8],[2,6,7],[5]]
 => 6 = 5 + 1
[6,2]
 => [3,3,1,1]
 => [[1,4,5],[2,7,8],[3],[6]]
 => 5 = 4 + 1
[6,1,1]
 => [4,2,2]
 => [[1,2,7,8],[3,4],[5,6]]
 => 6 = 5 + 1
[5,3]
 => [2,2,2,1,1]
 => [[1,4],[2,6],[3,8],[5],[7]]
 => 4 = 3 + 1
[5,2,1]
 => [4,1,1,1,1]
 => [[1,6,7,8],[2],[3],[4],[5]]
 => 4 = 3 + 1
[5,4,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [[1],[2],[3],[4],[5],[6],[7],[8],[9],[10],[11],[12]]
 => ? = 0 + 1
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: St000394
Mapping
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: 99% ●values known / values provided: 99%●distinct values known / distinct values provided: 100%
Values
[1]
 => [1]
 => [1,0]
 => [1,0]
 => 0
[2]
 => [1,1]
 => [1,1,0,0]
 => [1,0,1,0]
 => 0
[1,1]
 => [2]
 => [1,0,1,0]
 => [1,1,0,0]
 => 1
[3]
 => [2,1]
 => [1,0,1,1,0,0]
 => [1,1,0,0,1,0]
 => 1
[2,1]
 => [1,1,1]
 => [1,1,0,1,0,0]
 => [1,0,1,0,1,0]
 => 0
[1,1,1]
 => [3]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 2
[4]
 => [2,2]
 => [1,1,1,0,0,0]
 => [1,1,0,1,0,0]
 => 2
[3,1]
 => [1,1,1,1]
 => [1,1,0,1,0,1,0,0]
 => [1,0,1,0,1,0,1,0]
 => 0
[2,2]
 => [2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,0,1,0]
 => 1
[2,1,1]
 => [3,1]
 => [1,0,1,0,1,1,0,0]
 => [1,1,1,0,0,0,1,0]
 => 2
[1,1,1,1]
 => [4]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 3
[5]
 => [3,2]
 => [1,0,1,1,1,0,0,0]
 => [1,1,1,0,0,1,0,0]
 => 3
[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
[3,2]
 => [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
[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
[2,2,1]
 => [2,2,1]
 => [1,1,1,0,0,1,0,0]
 => [1,1,0,1,0,0,1,0]
 => 2
[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
[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
[6]
 => [3,3]
 => [1,1,1,0,1,0,0,0]
 => [1,1,1,0,1,0,0,0]
 => 4
[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
[4,2]
 => [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
[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
[3,3]
 => [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
[3,2,1]
 => [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
[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
[2,2,2]
 => [2,2,2]
 => [1,1,1,1,0,0,0,0]
 => [1,1,0,1,0,1,0,0]
 => 3
[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
[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
[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
[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]
 => [3,3,1]
 => [1,1,1,0,1,0,0,1,0,0]
 => [1,1,1,0,1,0,0,0,1,0]
 => 4
[5,2]
 => [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
[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
[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
[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
[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
[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
[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
[3,2,1,1]
 => [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
[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
[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
[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
[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
[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
[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]
 => [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
[6,2]
 => [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
[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
[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
[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
[5,4,2,1]
 => [1,1,1,1,1,1,1,1,1,1,1,1]
 => [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
 => ?
 => ? = 0
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)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000376: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 88%●distinct values known / distinct values provided: 82%
Values
[1]
 => [1,0,1,0]
 => [1,1,0,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => [1,1,0,1,0,0]
 => 0
[1,1]
 => [1,0,1,1,0,0]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => [1,0,1,1,0,1,0,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => [1,1,1,0,0,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => [1,0,1,0,1,1,0,0]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => [1,0,1,0,1,1,0,1,0,0]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => [1,1,1,0,0,1,0,0]
 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => [1,1,0,0,1,1,0,0]
 => 2
[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
[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]
 => [1,1,1,0,1,0,0,0,1,0]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => [1,1,1,0,1,0,0,0]
 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => [1,1,0,1,1,0,0,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => [1,0,1,1,1,0,0,0]
 => 2
[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
[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
[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]
 => [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
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => [1,1,0,1,1,0,0,1,0,0]
 => 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
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => [1,1,0,1,0,1,0,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => [1,1,1,1,0,0,0,0]
 => 0
[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
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[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
[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
[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
[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]
 => [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
[5,2]
 => [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
[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
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => [1,0,1,1,1,0,1,0,0,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => [1,1,1,1,0,0,0,1,0,0]
 => 0
[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
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => [1,1,1,0,1,0,0,1,0,0]
 => 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
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => [1,1,1,0,0,0,1,1,0,0]
 => 3
[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
[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
[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
[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
[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
[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]
 => ? = 6
[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
[6,2]
 => [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
[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
[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
[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
[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
[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]
 => ? = 7
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
 => ? = 7
[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]
 => ? = 6
[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]
 => ? = 7
[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]
 => ? = 8
[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]
 => ? = 8
[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]
 => ? = 7
[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]
 => ? = 6
[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]
 => ? = 7
[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]
 => ? = 5
[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]
 => ? = 6
[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]
 => ? = 7
[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]
 => ? = 8
[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]
 => ? = 8
[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]
 => ? = 9
[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]
 => ? = 10
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: St000369
(load all 3 compositions to match this statistic)
Mapping
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St000369: Dyck paths ⟶ ℤResult quality: 82% ●values known / values provided: 84%●distinct values known / distinct values provided: 82%
Values
[1]
 => [1,0,1,0]
 => 0
[2]
 => [1,1,0,0,1,0]
 => 0
[1,1]
 => [1,0,1,1,0,0]
 => 1
[3]
 => [1,1,1,0,0,0,1,0]
 => 1
[2,1]
 => [1,0,1,0,1,0]
 => 0
[1,1,1]
 => [1,0,1,1,1,0,0,0]
 => 2
[4]
 => [1,1,1,1,0,0,0,0,1,0]
 => 2
[3,1]
 => [1,1,0,1,0,0,1,0]
 => 0
[2,2]
 => [1,1,0,0,1,1,0,0]
 => 1
[2,1,1]
 => [1,0,1,1,0,1,0,0]
 => 2
[1,1,1,1]
 => [1,0,1,1,1,1,0,0,0,0]
 => 3
[5]
 => [1,1,1,1,1,0,0,0,0,0,1,0]
 => 3
[4,1]
 => [1,1,1,0,1,0,0,0,1,0]
 => 2
[3,2]
 => [1,1,0,0,1,0,1,0]
 => 0
[3,1,1]
 => [1,0,1,1,0,0,1,0]
 => 1
[2,2,1]
 => [1,0,1,0,1,1,0,0]
 => 2
[2,1,1,1]
 => [1,0,1,1,1,0,1,0,0,0]
 => 3
[1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,0,0,0,0]
 => 4
[6]
 => [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
 => 4
[5,1]
 => [1,1,1,1,0,1,0,0,0,0,1,0]
 => 3
[4,2]
 => [1,1,1,0,0,1,0,0,1,0]
 => 1
[4,1,1]
 => [1,1,0,1,1,0,0,0,1,0]
 => 2
[3,3]
 => [1,1,1,0,0,0,1,1,0,0]
 => 2
[3,2,1]
 => [1,0,1,0,1,0,1,0]
 => 0
[3,1,1,1]
 => [1,0,1,1,1,0,0,1,0,0]
 => 3
[2,2,2]
 => [1,1,0,0,1,1,1,0,0,0]
 => 3
[2,2,1,1]
 => [1,0,1,1,0,1,1,0,0,0]
 => 4
[2,1,1,1,1]
 => [1,0,1,1,1,1,0,1,0,0,0,0]
 => 4
[1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
 => 5
[7]
 => [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 5
[6,1]
 => [1,1,1,1,1,0,1,0,0,0,0,0,1,0]
 => 4
[5,2]
 => [1,1,1,1,0,0,1,0,0,0,1,0]
 => 3
[5,1,1]
 => [1,1,1,0,1,1,0,0,0,0,1,0]
 => 4
[4,3]
 => [1,1,1,0,0,0,1,0,1,0]
 => 2
[4,2,1]
 => [1,1,0,1,0,1,0,0,1,0]
 => 0
[4,1,1,1]
 => [1,0,1,1,1,0,0,0,1,0]
 => 3
[3,3,1]
 => [1,1,0,1,0,0,1,1,0,0]
 => 1
[3,2,2]
 => [1,1,0,0,1,1,0,1,0,0]
 => 2
[3,2,1,1]
 => [1,0,1,1,0,1,0,1,0,0]
 => 3
[3,1,1,1,1]
 => [1,0,1,1,1,1,0,0,1,0,0,0]
 => 4
[2,2,2,1]
 => [1,0,1,0,1,1,1,0,0,0]
 => 4
[2,2,1,1,1]
 => [1,0,1,1,1,0,1,1,0,0,0,0]
 => 5
[2,1,1,1,1,1]
 => [1,0,1,1,1,1,1,0,1,0,0,0,0,0]
 => 5
[1,1,1,1,1,1,1]
 => [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
 => 6
[8]
 => [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 6
[7,1]
 => [1,1,1,1,1,1,0,1,0,0,0,0,0,0,1,0]
 => ? = 5
[6,2]
 => [1,1,1,1,1,0,0,1,0,0,0,0,1,0]
 => 4
[6,1,1]
 => [1,1,1,1,0,1,1,0,0,0,0,0,1,0]
 => 5
[5,3]
 => [1,1,1,1,0,0,0,1,0,0,1,0]
 => 3
[5,2,1]
 => [1,1,1,0,1,0,1,0,0,0,1,0]
 => 3
[5,1,1,1]
 => [1,1,0,1,1,1,0,0,0,0,1,0]
 => 4
[4,4]
 => [1,1,1,1,0,0,0,0,1,1,0,0]
 => 4
[4,3,1]
 => [1,1,0,1,0,0,1,0,1,0]
 => 0
[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]
 => ? = 7
[9]
 => [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 7
[8,1]
 => [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,1,0]
 => ? = 6
[7,2]
 => [1,1,1,1,1,1,0,0,1,0,0,0,0,0,1,0]
 => ? = 5
[7,1,1]
 => [1,1,1,1,1,0,1,1,0,0,0,0,0,0,1,0]
 => ? = 6
[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]
 => ? = 7
[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]
 => ? = 8
[10]
 => [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
 => ? = 8
[9,1]
 => [1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
 => ? = 7
[8,2]
 => [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,1,0]
 => ? = 6
[8,1,1]
 => [1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,1,0]
 => ? = 7
[7,3]
 => [1,1,1,1,1,1,0,0,0,1,0,0,0,0,1,0]
 => ? = 6
[7,2,1]
 => [1,1,1,1,1,0,1,0,1,0,0,0,0,0,1,0]
 => ? = 5
[7,1,1,1]
 => [1,1,1,1,0,1,1,1,0,0,0,0,0,0,1,0]
 => ? = 7
[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]
 => ? = 7
[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]
 => ? = 8
[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]
 => ? = 8
[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]
 => ? = 9
[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]
 => ? = 10
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.