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Your data matches 116 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [[2,2],[1]]
=> [1]
=> []
=> 0
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> []
=> 0
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> []
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> []
=> 0
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> []
=> 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> []
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> []
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> []
=> 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> []
=> 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> []
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [2]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> []
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> []
=> 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,1]
=> 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,1]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,1]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> []
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [2]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [2]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> []
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [3]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> []
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> []
=> 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> []
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> []
=> 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1]
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> []
=> 0
Description
The length of the partition.
Matching statistic: St000288
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(load all 3 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[2,1] => [[2,2],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 110 => 2 = 1 + 1
[2,2] => [[3,2],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[3,1] => [[3,3],[2]]
=> [2]
=> 100 => 1 = 0 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2 = 1 + 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> 100 => 1 = 0 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 3 = 2 + 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 110 => 2 = 1 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 1010 => 2 = 1 + 1
[2,3] => [[4,2],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 1100 => 2 = 1 + 1
[3,2] => [[4,3],[2]]
=> [2]
=> 100 => 1 = 0 + 1
[4,1] => [[4,4],[3]]
=> [3]
=> 1000 => 1 = 0 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 2 = 1 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 100 => 1 = 0 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 3 = 2 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 110 => 2 = 1 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 1010 => 2 = 1 + 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 1100 => 2 = 1 + 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> 100 => 1 = 0 + 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> 1000 => 1 = 0 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 11110 => 4 = 3 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1110 => 3 = 2 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 10110 => 3 = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 110 => 2 = 1 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 11010 => 3 = 2 + 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 1010 => 2 = 1 + 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 10010 => 2 = 1 + 1
[2,4] => [[5,2],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 11100 => 3 = 2 + 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 1100 => 2 = 1 + 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 10100 => 2 = 1 + 1
[3,3] => [[5,3],[2]]
=> [2]
=> 100 => 1 = 0 + 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 11000 => 2 = 1 + 1
[4,2] => [[5,4],[3]]
=> [3]
=> 1000 => 1 = 0 + 1
[5,1] => [[5,5],[4]]
=> [4]
=> 10000 => 1 = 0 + 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> 110 => 2 = 1 + 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> 100 => 1 = 0 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> 1110 => 3 = 2 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> 110 => 2 = 1 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> 1010 => 2 = 1 + 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 10 => 1 = 0 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000378
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 98% ●values known / values provided: 98%●distinct values known / distinct values provided: 100%
Values
[2,1] => [[2,2],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[2,2] => [[3,2],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[3,1] => [[3,3],[2]]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [3]
=> 2 = 1 + 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [4]
=> 2 = 1 + 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[4,1] => [[4,4],[3]]
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [3]
=> 2 = 1 + 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [4]
=> 2 = 1 + 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [3,1]
=> 4 = 3 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [2,2]
=> 3 = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [2,2,1]
=> 3 = 2 + 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [3]
=> 2 = 1 + 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [2,1,1]
=> 2 = 1 + 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [2,2,2]
=> 3 = 2 + 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [4]
=> 2 = 1 + 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [5]
=> 2 = 1 + 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [6]
=> 2 = 1 + 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,1,1]
=> 1 = 0 + 1
[5,1] => [[5,5],[4]]
=> [4]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,1]
=> 1 = 0 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [2]
=> 2 = 1 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [3]
=> 2 = 1 + 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1]
=> 1 = 0 + 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [9,1,1,1,1,1,1,1]
=> ? = 3 + 1
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [2,2,2,2,2,2,2,1]
=> ? = 2 + 1
[4,1,1,1,1,1,1] => [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [3,3,3,3,3,2,1]
=> ? = 5 + 1
[5,1,1,1,1,1] => [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> ?
=> ? = 4 + 1
[6,1,1,1,1] => [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [11,1,1,1,1,1,1,1,1,1]
=> ? = 3 + 1
Description
The diagonal inversion number of an integer partition.
The dinv of a partition is the number of cells $c$ in the diagram of an integer partition $\lambda$ for which $\operatorname{arm}(c)-\operatorname{leg}(c) \in \{0,1\}$.
See also exercise 3.19 of [2].
This statistic is equidistributed with the length of the partition, see [3].
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[2,1] => [[2,2],[1]]
=> [1]
=> [[1]]
=> 0
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [[1]]
=> 0
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> [[1]]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [[1,2]]
=> 0
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [[1]]
=> 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [[1]]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [[1,2]]
=> 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> [[1]]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> [[1,2]]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [[1,2,3]]
=> 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [[1]]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [[1,2]]
=> 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [[1]]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [[1,2]]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [[1,2,3]]
=> 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [[1]]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [[1,2]]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> [[1,2,3]]
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> [[1,2,3,4]]
=> 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [[1]]
=> 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [[1,2]]
=> 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [[1]]
=> 0
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 3
[5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 2
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 4
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 3
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 2
[3,1,1,1,1,1,1,1] => [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 6
[4,1,1,1,1,1,1] => [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 5
[5,1,1,1,1,1] => [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> ? = 4
[6,1,1,1,1] => [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20]]
=> ? = 3
[7,1,1,1] => [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 2
[8,1,1] => [[8,8,8],[7,7]]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 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: St000733
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
[2,1] => [[2,2],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,2] => [[3,2],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,1] => [[3,3],[2]]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,3] => [[4,2],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[3,2] => [[4,3],[2]]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[4,1] => [[4,4],[3]]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3 = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3 = 2 + 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [[1,2,3],[4]]
=> 2 = 1 + 1
[2,4] => [[5,2],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 3 = 2 + 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [[1,2],[3,4]]
=> 2 = 1 + 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2 = 1 + 1
[3,3] => [[5,3],[2]]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 1 + 1
[4,2] => [[5,4],[3]]
=> [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[5,1] => [[5,5],[4]]
=> [4]
=> [[1,2,3,4]]
=> 1 = 0 + 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [[1]]
=> 1 = 0 + 1
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 3 + 1
[5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 2 + 1
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15]]
=> ? = 4 + 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 3 + 1
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 2 + 1
[3,1,1,1,1,1,1,1] => [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 6 + 1
[4,1,1,1,1,1,1] => [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 5 + 1
[5,1,1,1,1,1] => [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16],[17,18,19,20]]
=> ? = 4 + 1
[6,1,1,1,1] => [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15],[16,17,18,19,20]]
=> ? = 3 + 1
[7,1,1,1] => [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ? = 2 + 1
[8,1,1] => [[8,8,8],[7,7]]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 1 + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001280
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00182: Skew partitions —outer shape⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St001280: Integer partitions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
[2,1] => [[2,2],[1]]
=> [2,2]
=> [2]
=> 1 = 0 + 1
[1,2,1] => [[2,2,1],[1]]
=> [2,2,1]
=> [2,1]
=> 1 = 0 + 1
[2,1,1] => [[2,2,2],[1,1]]
=> [2,2,2]
=> [2,2]
=> 2 = 1 + 1
[2,2] => [[3,2],[1]]
=> [3,2]
=> [2]
=> 1 = 0 + 1
[3,1] => [[3,3],[2]]
=> [3,3]
=> [3]
=> 1 = 0 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [2,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [2,2,2,1]
=> [2,2,1]
=> 2 = 1 + 1
[1,2,2] => [[3,2,1],[1]]
=> [3,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,3,1] => [[3,3,1],[2]]
=> [3,3,1]
=> [3,1]
=> 1 = 0 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [2,2,2,2]
=> [2,2,2]
=> 3 = 2 + 1
[2,1,2] => [[3,2,2],[1,1]]
=> [3,2,2]
=> [2,2]
=> 2 = 1 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> [3,3,2]
=> [3,2]
=> 2 = 1 + 1
[2,3] => [[4,2],[1]]
=> [4,2]
=> [2]
=> 1 = 0 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> [3,3,3]
=> [3,3]
=> 2 = 1 + 1
[3,2] => [[4,3],[2]]
=> [4,3]
=> [3]
=> 1 = 0 + 1
[4,1] => [[4,4],[3]]
=> [4,4]
=> [4]
=> 1 = 0 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [3,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [3,3,1,1]
=> [3,1,1]
=> 1 = 0 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [2,2,2,2,1]
=> [2,2,2,1]
=> 3 = 2 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [3,2,2,1]
=> [2,2,1]
=> 2 = 1 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [3,3,2,1]
=> [3,2,1]
=> 2 = 1 + 1
[1,2,3] => [[4,2,1],[1]]
=> [4,2,1]
=> [2,1]
=> 1 = 0 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [3,3,3,1]
=> [3,3,1]
=> 2 = 1 + 1
[1,3,2] => [[4,3,1],[2]]
=> [4,3,1]
=> [3,1]
=> 1 = 0 + 1
[1,4,1] => [[4,4,1],[3]]
=> [4,4,1]
=> [4,1]
=> 1 = 0 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [2,2,2,2,2]
=> [2,2,2,2]
=> 4 = 3 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [3,2,2,2]
=> [2,2,2]
=> 3 = 2 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [3,3,2,2]
=> [3,2,2]
=> 3 = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
=> [4,2,2]
=> [2,2]
=> 2 = 1 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [3,3,3,2]
=> [3,3,2]
=> 3 = 2 + 1
[2,2,2] => [[4,3,2],[2,1]]
=> [4,3,2]
=> [3,2]
=> 2 = 1 + 1
[2,3,1] => [[4,4,2],[3,1]]
=> [4,4,2]
=> [4,2]
=> 2 = 1 + 1
[2,4] => [[5,2],[1]]
=> [5,2]
=> [2]
=> 1 = 0 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [3,3,3,3]
=> [3,3,3]
=> 3 = 2 + 1
[3,1,2] => [[4,3,3],[2,2]]
=> [4,3,3]
=> [3,3]
=> 2 = 1 + 1
[3,2,1] => [[4,4,3],[3,2]]
=> [4,4,3]
=> [4,3]
=> 2 = 1 + 1
[3,3] => [[5,3],[2]]
=> [5,3]
=> [3]
=> 1 = 0 + 1
[4,1,1] => [[4,4,4],[3,3]]
=> [4,4,4]
=> [4,4]
=> 2 = 1 + 1
[4,2] => [[5,4],[3]]
=> [5,4]
=> [4]
=> 1 = 0 + 1
[5,1] => [[5,5],[4]]
=> [5,5]
=> [5]
=> 1 = 0 + 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [2,2,1,1,1,1]
=> [2,1,1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [2,2,2,1,1,1]
=> [2,2,1,1,1]
=> 2 = 1 + 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [3,2,1,1,1]
=> [2,1,1,1]
=> 1 = 0 + 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [3,3,1,1,1]
=> [3,1,1,1]
=> 1 = 0 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [2,2,2,2,1,1]
=> [2,2,2,1,1]
=> 3 = 2 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [3,2,2,1,1]
=> [2,2,1,1]
=> 2 = 1 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [3,3,2,1,1]
=> [3,2,1,1]
=> 2 = 1 + 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [4,2,1,1]
=> [2,1,1]
=> 1 = 0 + 1
[3,1,2,1,1] => [[4,4,4,3,3],[3,3,2,2]]
=> [4,4,4,3,3]
=> ?
=> ? = 3 + 1
[3,2,1,1,1] => [[4,4,4,4,3],[3,3,3,2]]
=> [4,4,4,4,3]
=> ?
=> ? = 3 + 1
[3,3,1,1] => [[5,5,5,3],[4,4,2]]
=> [5,5,5,3]
=> ?
=> ? = 2 + 1
[4,1,2,1] => [[5,5,4,4],[4,3,3]]
=> [5,5,4,4]
=> ?
=> ? = 2 + 1
[4,2,1,1] => [[5,5,5,4],[4,4,3]]
=> [5,5,5,4]
=> ?
=> ? = 2 + 1
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [3,3,3,3,3,3,3]
=> [3,3,3,3,3,3]
=> ? = 5 + 1
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [4,4,4,4,4,4]
=> [4,4,4,4,4]
=> ? = 4 + 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [5,5,5,5,5]
=> [5,5,5,5]
=> ? = 3 + 1
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [6,6,6,6]
=> [6,6,6]
=> ? = 2 + 1
[3,1,1,1,1,1,1,1] => [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [3,3,3,3,3,3,3,3]
=> [3,3,3,3,3,3,3]
=> ? = 6 + 1
[4,1,1,1,1,1,1] => [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [4,4,4,4,4,4,4]
=> [4,4,4,4,4,4]
=> ? = 5 + 1
[5,1,1,1,1,1] => [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [5,5,5,5,5,5]
=> [5,5,5,5,5]
=> ? = 4 + 1
[6,1,1,1,1] => [[6,6,6,6,6],[5,5,5,5]]
=> [6,6,6,6,6]
=> [6,6,6,6]
=> ? = 3 + 1
[7,1,1,1] => [[7,7,7,7],[6,6,6]]
=> [7,7,7,7]
=> [7,7,7]
=> ? = 2 + 1
Description
The number of parts of an integer partition that are at least two.
Matching statistic: St000329
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 93%●distinct values known / distinct values provided: 75%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000329: Dyck paths ⟶ ℤResult quality: 75% ●values known / values provided: 93%●distinct values known / distinct values provided: 75%
Values
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0]
=> 0
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1] => [[3,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1] => [[4,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 3
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0]
=> 0
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,1,0,0,0]
=> 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,0,1,0,1,0]
=> 0
[5,1] => [[5,5],[4]]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 0
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,0,1,0]
=> 0
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 2
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,1,0,0]
=> 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,1,0,0]
=> 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0]
=> 0
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 6
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 5
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> ? = 4
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 3
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [1,1,1,1,1,0,1,0,1,0,0,0,0,0]
=> ? = 2
[7,1,1] => [[7,7,7],[6,6]]
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[8,1] => [[8,8],[7]]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
[2,1,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 7
[3,1,1,1,1,1,1,1] => [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 6
[4,1,1,1,1,1,1] => [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 5
[5,1,1,1,1,1] => [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 4
[6,1,1,1,1] => [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0]
=> ? = 3
[7,1,1,1] => [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 2
[8,1,1] => [[8,8,8],[7,7]]
=> [7,7]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 1
[9,1] => [[9,9],[8]]
=> [8]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 0
Description
The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1.
Matching statistic: St000147
Mp00041: Integer compositions —conjugate⟶ Integer compositions
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 90% ●values known / values provided: 90%●distinct values known / distinct values provided: 100%
Values
[2,1] => [2,1] => [[2,2],[1]]
=> [1]
=> 1 = 0 + 1
[1,2,1] => [2,2] => [[3,2],[1]]
=> [1]
=> 1 = 0 + 1
[2,1,1] => [3,1] => [[3,3],[2]]
=> [2]
=> 2 = 1 + 1
[2,2] => [1,2,1] => [[2,2,1],[1]]
=> [1]
=> 1 = 0 + 1
[3,1] => [2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[1,1,2,1] => [2,3] => [[4,2],[1]]
=> [1]
=> 1 = 0 + 1
[1,2,1,1] => [3,2] => [[4,3],[2]]
=> [2]
=> 2 = 1 + 1
[1,2,2] => [1,2,2] => [[3,2,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,3,1] => [2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[2,1,1,1] => [4,1] => [[4,4],[3]]
=> [3]
=> 3 = 2 + 1
[2,1,2] => [1,3,1] => [[3,3,1],[2]]
=> [2]
=> 2 = 1 + 1
[2,2,1] => [2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[2,3] => [1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> 1 = 0 + 1
[3,1,1] => [3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> 2 = 1 + 1
[3,2] => [1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[4,1] => [2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
[1,1,1,2,1] => [2,4] => [[5,2],[1]]
=> [1]
=> 1 = 0 + 1
[1,1,2,1,1] => [3,3] => [[5,3],[2]]
=> [2]
=> 2 = 1 + 1
[1,1,2,2] => [1,2,3] => [[4,2,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,1,3,1] => [2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[1,2,1,1,1] => [4,2] => [[5,4],[3]]
=> [3]
=> 3 = 2 + 1
[1,2,1,2] => [1,3,2] => [[4,3,1],[2]]
=> [2]
=> 2 = 1 + 1
[1,2,2,1] => [2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[1,2,3] => [1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,3,1,1] => [3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> 2 = 1 + 1
[1,3,2] => [1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[1,4,1] => [2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
[2,1,1,1,1] => [5,1] => [[5,5],[4]]
=> [4]
=> 4 = 3 + 1
[2,1,1,2] => [1,4,1] => [[4,4,1],[3]]
=> [3]
=> 3 = 2 + 1
[2,1,2,1] => [2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> 3 = 2 + 1
[2,1,3] => [1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> 2 = 1 + 1
[2,2,1,1] => [3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> 3 = 2 + 1
[2,2,2] => [1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[2,3,1] => [2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> 2 = 1 + 1
[2,4] => [1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> 1 = 0 + 1
[3,1,1,1] => [4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> 3 = 2 + 1
[3,1,2] => [1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> 2 = 1 + 1
[3,2,1] => [2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> 2 = 1 + 1
[3,3] => [1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[4,1,1] => [3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> 2 = 1 + 1
[4,2] => [1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> 1 = 0 + 1
[5,1] => [2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> 1 = 0 + 1
[1,1,1,1,2,1] => [2,5] => [[6,2],[1]]
=> [1]
=> 1 = 0 + 1
[1,1,1,2,1,1] => [3,4] => [[6,3],[2]]
=> [2]
=> 2 = 1 + 1
[1,1,1,2,2] => [1,2,4] => [[5,2,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,1,1,3,1] => [2,1,4] => [[5,2,2],[1,1]]
=> [1,1]
=> 1 = 0 + 1
[1,1,2,1,1,1] => [4,3] => [[6,4],[3]]
=> [3]
=> 3 = 2 + 1
[1,1,2,1,2] => [1,3,3] => [[5,3,1],[2]]
=> [2]
=> 2 = 1 + 1
[1,1,2,2,1] => [2,2,3] => [[5,3,2],[2,1]]
=> [2,1]
=> 2 = 1 + 1
[1,1,2,3] => [1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> 1 = 0 + 1
[1,1,1,1,1,2,1] => [2,6] => [[7,2],[1]]
=> ?
=> ? = 0 + 1
[1,1,1,1,2,1,1] => [3,5] => [[7,3],[2]]
=> ?
=> ? = 1 + 1
[1,1,1,1,2,2] => [1,2,5] => [[6,2,1],[1]]
=> ?
=> ? = 0 + 1
[1,1,1,2,1,2] => [1,3,4] => [[6,3,1],[2]]
=> ?
=> ? = 1 + 1
[1,1,1,2,2,1] => [2,2,4] => [[6,3,2],[2,1]]
=> ?
=> ? = 1 + 1
[1,1,1,3,2] => [1,2,1,4] => [[5,2,2,1],[1,1]]
=> ?
=> ? = 0 + 1
[1,1,2,1,1,1,1] => [5,3] => [[7,5],[4]]
=> ?
=> ? = 3 + 1
[1,1,3,3] => [1,1,2,1,3] => [[4,2,2,1,1],[1,1]]
=> ?
=> ? = 0 + 1
[1,2,1,1,1,1,1] => [6,2] => [[7,6],[5]]
=> ?
=> ? = 4 + 1
[1,2,1,1,1,2] => [1,5,2] => [[6,5,1],[4]]
=> ?
=> ? = 3 + 1
[2,1,1,1,1,2] => [1,6,1] => [[6,6,1],[5]]
=> ?
=> ? = 4 + 1
[2,1,1,1,2,1] => [2,5,1] => [[6,6,2],[5,1]]
=> ?
=> ? = 4 + 1
[2,1,1,2,2] => [1,2,4,1] => [[5,5,2,1],[4,1]]
=> ?
=> ? = 3 + 1
[2,1,1,3,1] => [2,1,4,1] => [[5,5,2,2],[4,1,1]]
=> ?
=> ? = 3 + 1
[2,2,1,1,1,1] => [5,2,1] => [[6,6,5],[5,4]]
=> ?
=> ? = 4 + 1
[3,1,1,3] => [1,1,4,1,1] => [[4,4,4,1,1],[3,3]]
=> ?
=> ? = 2 + 1
[4,1,3] => [1,1,3,1,1,1] => [[3,3,3,3,1,1],[2,2,2]]
=> ?
=> ? = 1 + 1
[4,1,1,1,1,1,1] => [7,1,1,1] => [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> ? = 5 + 1
[5,1,1,1,1,1] => [6,1,1,1,1] => [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> ? = 4 + 1
[6,1,1,1,1] => [5,1,1,1,1,1] => [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> ? = 3 + 1
[7,1,1,1] => [4,1,1,1,1,1,1] => [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> ? = 2 + 1
Description
The largest part of an integer partition.
Matching statistic: St001227
Mp00180: Integer compositions —to ribbon⟶ Skew partitions
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 90%●distinct values known / distinct values provided: 62%
Mp00183: Skew partitions —inner shape⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 62% ●values known / values provided: 90%●distinct values known / distinct values provided: 62%
Values
[2,1] => [[2,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,1] => [[2,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1,1] => [[2,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2] => [[3,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[3,1] => [[3,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,2,1] => [[2,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,2,1,1] => [[2,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,2] => [[3,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,3,1] => [[3,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[2,1,1,1] => [[2,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,2] => [[3,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [[3,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3] => [[4,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[3,1,1] => [[3,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2] => [[4,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[4,1] => [[4,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [[2,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,2,1,1] => [[2,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,2] => [[3,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,3,1] => [[3,3,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,2,1,1,1] => [[2,2,2,2,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,2] => [[3,2,2,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [[3,3,2,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,2,3] => [[4,2,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,3,1,1] => [[3,3,3,1],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[1,3,2] => [[4,3,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,4,1] => [[4,4,1],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[2,1,1,1,1] => [[2,2,2,2,2],[1,1,1,1]]
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 3 + 1
[2,1,1,2] => [[3,2,2,2],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,2,1] => [[3,3,2,2],[2,1,1]]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 3 = 2 + 1
[2,1,3] => [[4,2,2],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [[3,3,3,2],[2,2,1]]
=> [2,2,1]
=> [1,0,1,0,1,1,0,0]
=> 3 = 2 + 1
[2,2,2] => [[4,3,2],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[2,3,1] => [[4,4,2],[3,1]]
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 1 + 1
[2,4] => [[5,2],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[3,1,1,1] => [[3,3,3,3],[2,2,2]]
=> [2,2,2]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[3,1,2] => [[4,3,3],[2,2]]
=> [2,2]
=> [1,1,0,0,1,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [[4,4,3],[3,2]]
=> [3,2]
=> [1,1,0,0,1,0,1,0]
=> 2 = 1 + 1
[3,3] => [[5,3],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[4,1,1] => [[4,4,4],[3,3]]
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 1 + 1
[4,2] => [[5,4],[3]]
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 0 + 1
[5,1] => [[5,5],[4]]
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 0 + 1
[1,1,1,1,2,1] => [[2,2,1,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,1,2,1,1] => [[2,2,2,1,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,1,2,2] => [[3,2,1,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[1,1,1,3,1] => [[3,3,1,1,1],[2]]
=> [2]
=> [1,1,0,0,1,0]
=> 1 = 0 + 1
[1,1,2,1,1,1] => [[2,2,2,2,1,1],[1,1,1]]
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1,2] => [[3,2,2,1,1],[1,1]]
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 1 + 1
[1,1,2,2,1] => [[3,3,2,1,1],[2,1]]
=> [2,1]
=> [1,0,1,0,1,0]
=> 2 = 1 + 1
[1,1,2,3] => [[4,2,1,1],[1]]
=> [1]
=> [1,0,1,0]
=> 1 = 0 + 1
[2,1,1,1,1,1,1] => [[2,2,2,2,2,2,2],[1,1,1,1,1,1]]
=> [1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5 + 1
[3,1,1,1,1,1] => [[3,3,3,3,3,3],[2,2,2,2,2]]
=> [2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[4,1,1,1,1] => [[4,4,4,4,4],[3,3,3,3]]
=> [3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[5,1,1,1] => [[5,5,5,5],[4,4,4]]
=> [4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[6,1,1] => [[6,6,6],[5,5]]
=> [5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[7,1] => [[7,7],[6]]
=> [6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6 + 1
[3,1,1,1,1,1,1] => [[3,3,3,3,3,3,3],[2,2,2,2,2,2]]
=> [2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5 + 1
[4,1,1,1,1,1] => [[4,4,4,4,4,4],[3,3,3,3,3]]
=> [3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[5,1,1,1,1] => [[5,5,5,5,5],[4,4,4,4]]
=> [4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[6,1,1,1] => [[6,6,6,6],[5,5,5]]
=> [5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[7,1,1] => [[7,7,7],[6,6]]
=> [6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[8,1] => [[8,8],[7]]
=> [7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
[2,1,1,1,1,1,1,1,1] => [[2,2,2,2,2,2,2,2,2],[1,1,1,1,1,1,1,1]]
=> [1,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0]
=> ? = 7 + 1
[3,1,1,1,1,1,1,1] => [[3,3,3,3,3,3,3,3],[2,2,2,2,2,2,2]]
=> [2,2,2,2,2,2,2]
=> [1,1,0,0,1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> ? = 6 + 1
[4,1,1,1,1,1,1] => [[4,4,4,4,4,4,4],[3,3,3,3,3,3]]
=> [3,3,3,3,3,3]
=> [1,1,1,0,0,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 5 + 1
[5,1,1,1,1,1] => [[5,5,5,5,5,5],[4,4,4,4,4]]
=> [4,4,4,4,4]
=> [1,1,1,1,0,0,0,0,1,1,1,1,1,0,0,0,0,0]
=> ? = 4 + 1
[6,1,1,1,1] => [[6,6,6,6,6],[5,5,5,5]]
=> [5,5,5,5]
=> [1,1,1,1,1,0,0,0,0,0,1,1,1,1,0,0,0,0]
=> ? = 3 + 1
[7,1,1,1] => [[7,7,7,7],[6,6,6]]
=> [6,6,6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,1,1,0,0,0]
=> ? = 2 + 1
[8,1,1] => [[8,8,8],[7,7]]
=> [7,7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,1,0,0]
=> ? = 1 + 1
[9,1] => [[9,9],[8]]
=> [8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> ? = 0 + 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St001033
Mp00231: Integer compositions —bounce path⟶ Dyck paths
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Mp00229: Dyck paths —Delest-Viennot⟶ Dyck paths
Mp00143: Dyck paths —inverse promotion⟶ Dyck paths
St001033: Dyck paths ⟶ ℤResult quality: 74% ●values known / values provided: 74%●distinct values known / distinct values provided: 100%
Values
[2,1] => [1,1,0,0,1,0]
=> [1,0,1,1,0,0]
=> [1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,1] => [1,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1] => [1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[2,2] => [1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[3,1] => [1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,2,1] => [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,1,1] => [1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,2,2] => [1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,3,1] => [1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1,1] => [1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,1,2] => [1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,2,1] => [1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,3] => [1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,1] => [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[3,2] => [1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[4,1] => [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,2,1] => [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,2,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,2] => [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,3,1] => [1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,2,1,1,1] => [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,2,1,2] => [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,2,2,1] => [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,2,3] => [1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[1,3,1,1] => [1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,0,1,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,3,2] => [1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,4,1] => [1,0,1,1,1,1,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[2,1,1,1,1] => [1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> 4 = 3 + 1
[2,1,1,2] => [1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,0,0]
=> 3 = 2 + 1
[2,1,2,1] => [1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,0]
=> 3 = 2 + 1
[2,1,3] => [1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 2 = 1 + 1
[2,2,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,0]
=> 3 = 2 + 1
[2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,0,0]
=> 2 = 1 + 1
[2,3,1] => [1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,0]
=> 2 = 1 + 1
[2,4] => [1,1,0,0,1,1,1,1,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 0 + 1
[3,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[3,1,2] => [1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[3,2,1] => [1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[3,3] => [1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[4,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[4,2] => [1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[5,1] => [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,1,2,1] => [1,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,1,2,1,1] => [1,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> 2 = 1 + 1
[1,1,1,2,2] => [1,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> 1 = 0 + 1
[1,1,1,3,1] => [1,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,1,0,0,0]
=> 1 = 0 + 1
[1,1,2,1,1,1] => [1,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,0,1,0,1,1,1,0,1,0,1,0,0,0]
=> 3 = 2 + 1
[1,1,2,1,2] => [1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,0]
=> 2 = 1 + 1
[1,1,2,2,1] => [1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,1,0,0,1,1,0,0,0]
=> 2 = 1 + 1
[1,1,2,3] => [1,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> 1 = 0 + 1
[2,1,1,1,2,1] => [1,1,0,0,1,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 4 + 1
[2,1,1,2,1,1] => [1,1,0,0,1,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 4 + 1
[2,1,1,2,2] => [1,1,0,0,1,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 3 + 1
[2,1,1,3,1] => [1,1,0,0,1,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 3 + 1
[2,1,2,1,1,1] => [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 4 + 1
[2,1,2,1,2] => [1,1,0,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 3 + 1
[2,1,2,2,1] => [1,1,0,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
[2,1,2,3] => [1,1,0,0,1,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[2,1,3,1,1] => [1,1,0,0,1,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[2,1,3,2] => [1,1,0,0,1,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[2,2,1,1,1,1] => [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 + 1
[2,2,1,1,2] => [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 3 + 1
[2,2,1,2,1] => [1,1,0,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
[2,2,1,3] => [1,1,0,0,1,1,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[2,2,2,1,1] => [1,1,0,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[2,2,2,2] => [1,1,0,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[2,2,3,1] => [1,1,0,0,1,1,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,1,0,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[2,3,1,1,1] => [1,1,0,0,1,1,1,0,0,0,1,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[2,3,1,2] => [1,1,0,0,1,1,1,0,0,0,1,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[2,3,2,1] => [1,1,0,0,1,1,1,0,0,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[2,3,3] => [1,1,0,0,1,1,1,0,0,0,1,1,1,0,0,0]
=> [1,0,1,1,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,1,1,0,0,1,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
[2,4,1,1] => [1,1,0,0,1,1,1,1,0,0,0,0,1,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[2,4,2] => [1,1,0,0,1,1,1,1,0,0,0,0,1,1,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,1,1,0,0,1,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[3,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 + 1
[3,1,1,2,1] => [1,1,1,0,0,0,1,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,1,0,0,0]
=> ? = 3 + 1
[3,1,1,3] => [1,1,1,0,0,0,1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,1,0,1,0,0,1,0,1,0,0]
=> ? = 2 + 1
[3,1,2,1,1] => [1,1,1,0,0,0,1,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,1,0,0,0]
=> ? = 3 + 1
[3,1,2,2] => [1,1,1,0,0,0,1,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,1,0,0,1,1,0,0,1,0,0]
=> ? = 2 + 1
[3,1,3,1] => [1,1,1,0,0,0,1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,1,0,0,1,0,1,1,0,0,0]
=> ? = 2 + 1
[3,2,1,1,1] => [1,1,1,0,0,0,1,1,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[3,2,1,2] => [1,1,1,0,0,0,1,1,0,0,1,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[3,2,2,1] => [1,1,1,0,0,0,1,1,0,0,1,1,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[3,2,3] => [1,1,1,0,0,0,1,1,0,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0,1,1,0,0,1,0,1,0]
=> [1,1,0,1,1,0,0,1,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
[3,3,1,1] => [1,1,1,0,0,0,1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[3,3,2] => [1,1,1,0,0,0,1,1,1,0,0,0,1,1,0,0]
=> [1,0,1,0,1,1,0,0,1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,1,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[3,4,1] => [1,1,1,0,0,0,1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> [1,1,0,1,1,0,0,1,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
[4,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[4,1,1,2] => [1,1,1,1,0,0,0,0,1,0,1,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,0,1,0,0]
=> ? = 2 + 1
[4,1,2,1] => [1,1,1,1,0,0,0,0,1,0,1,1,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,1,0,0,0]
=> ? = 2 + 1
[4,1,3] => [1,1,1,1,0,0,0,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,0,1,0]
=> [1,1,0,1,0,1,1,0,1,0,0,1,0,1,0,0]
=> ? = 1 + 1
[4,2,1,1] => [1,1,1,1,0,0,0,0,1,1,0,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,1,0,0,0]
=> ? = 2 + 1
[4,2,2] => [1,1,1,1,0,0,0,0,1,1,0,0,1,1,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,1,0,0,1,0]
=> [1,1,0,1,0,1,1,0,0,1,1,0,0,1,0,0]
=> ? = 1 + 1
[4,3,1] => [1,1,1,1,0,0,0,0,1,1,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,1,0,0]
=> [1,1,0,1,0,1,1,0,0,1,0,1,1,0,0,0]
=> ? = 1 + 1
[3,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[4,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 + 1
[5,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 3 + 1
[6,1,1,1] => [1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0,0]
=> ? = 2 + 1
[3,1,1,1,1,1,1,1] => [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6 + 1
[4,1,1,1,1,1,1] => [1,1,1,1,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 5 + 1
[5,1,1,1,1,1] => [1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> [1,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 4 + 1
Description
The normalized area of the parallelogram polyomino associated with the Dyck path.
The area of the smallest parallelogram polyomino equals the semilength of the Dyck path. This statistic is therefore the area of the parallelogram polyomino minus the semilength of the Dyck path.
The area itself is equidistributed with [[St001034]] and with [[St000395]].
The following 106 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000678The number of up steps after the last double rise of a Dyck path. St000052The number of valleys of a Dyck path not on the x-axis. St000225Difference between largest and smallest parts in a partition. St000356The number of occurrences of the pattern 13-2. St000463The number of admissible inversions of a permutation. St001090The number of pop-stack-sorts needed to sort a permutation. St000451The length of the longest pattern of the form k 1 2. St000141The maximum drop size of a permutation. St000337The lec statistic, the sum of the inversion numbers of the hook factors of a permutation. St000374The number of exclusive right-to-left minima of a permutation. St000703The number of deficiencies of a permutation. St000996The number of exclusive left-to-right maxima of a permutation. St001331The size of the minimal feedback vertex set. St001336The minimal number of vertices in a graph whose complement is triangle-free. St001723The differential of a graph. St001724The 2-packing differential of a graph. St000171The degree of the graph. St000204The number of internal nodes of a binary tree. St000272The treewidth of a graph. St000362The size of a minimal vertex cover of a graph. St000454The largest eigenvalue of a graph if it is integral. St000536The pathwidth of a graph. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001120The length of a longest path in a graph. St001270The bandwidth of a graph. St001277The degeneracy of a graph. St001358The largest degree of a regular subgraph of a graph. St001644The dimension of a graph. St001962The proper pathwidth of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St000172The Grundy number of a graph. St000363The number of minimal vertex covers of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000822The Hadwiger number of the graph. St001029The size of the core of a graph. St001108The 2-dynamic chromatic number of a graph. St001110The 3-dynamic chromatic number of a graph. St001116The game chromatic number of a graph. St001302The number of minimally dominating sets of vertices of a graph. St001304The number of maximally independent sets of vertices of a graph. St001330The hat guessing number of a graph. St001368The number of vertices of maximal degree in a graph. St001458The rank of the adjacency matrix of a graph. St001459The number of zero columns in the nullspace of a graph. St001494The Alon-Tarsi number of a graph. St001580The acyclic chromatic number of a graph. St001581The achromatic number of a graph. St001670The connected partition number of a graph. St001725The harmonious chromatic number of a graph. St001883The mutual visibility number of a graph. St001963The tree-depth of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001812The biclique partition number of a graph. St001323The independence gap of a graph. St000145The Dyson rank of a partition. St001083The number of boxed occurrences of 132 in a permutation. St001642The Prague dimension of a graph. St000617The number of global maxima of a Dyck path. St000028The number of stack-sorts needed to sort a permutation. St000741The Colin de Verdière graph invariant. St000803The number of occurrences of the vincular pattern |132 in a permutation. St001683The number of distinct positions of the pattern letter 3 in occurrences of 132 in a permutation. St000371The number of mid points of decreasing subsequences of length 3 in a permutation. St000373The number of weak exceedences of a permutation that are also mid-points of a decreasing subsequence of length $3$. St001964The interval resolution global dimension of a poset. St000866The number of admissible inversions of a permutation in the sense of Shareshian-Wachs. St001687The number of distinct positions of the pattern letter 2 in occurrences of 213 in a permutation. St001411The number of patterns 321 or 3412 in a permutation. St001552The number of inversions between excedances and fixed points of a permutation. St001727The number of invisible inversions of a permutation. St000710The number of big deficiencies of a permutation. St001685The number of distinct positions of the pattern letter 1 in occurrences of 132 in a permutation. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000956The maximal displacement of a permutation. St001645The pebbling number of a connected graph. St001091The number of parts in an integer partition whose next smaller part has the same size. 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. St000223The number of nestings in the permutation. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St000299The number of nonisomorphic vertex-induced subtrees. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001292The injective dimension of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000034The maximum defect over any reduced expression for a permutation and any subexpression. St000155The number of exceedances (also excedences) of a permutation. St000358The number of occurrences of the pattern 31-2. St000711The number of big exceedences of a permutation. St000801The number of occurrences of the vincular pattern |312 in a permutation. 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. 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. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001229The vector space dimension of the first extension group between the Jacobson radical J and J^2. St001744The number of occurrences of the arrow pattern 1-2 with an arrow from 1 to 2 in a permutation. St000316The number of non-left-to-right-maxima of a permutation. St000798The makl of a permutation. St001194The injective dimension of $A/AfA$ in the corresponding Nakayama algebra $A$ when $Af$ is the minimal faithful projective-injective left $A$-module St001273The projective dimension of the first term in an injective coresolution of the regular module. St001005The number of indices for a permutation that are either left-to-right maxima or right-to-left minima but not both. St001691The number of kings in a graph. St001305The number of induced cycles on four vertices in a graph. St001324The minimal number of occurrences of the chordal-pattern in a linear ordering of the vertices of the graph. St000455The second largest eigenvalue of a graph if it is integral. St001326The minimal number of occurrences of the interval-pattern in a linear ordering of the vertices of the graph.
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