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Your data matches 36 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
St000010: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> 1
[2]
=> [1,1]
=> 2
[1,1]
=> [2]
=> 1
[3]
=> [2,1]
=> 2
[2,1]
=> [1,1,1]
=> 3
[1,1,1]
=> [3]
=> 1
[4]
=> [2,2]
=> 2
[3,1]
=> [1,1,1,1]
=> 4
[2,2]
=> [2,1,1]
=> 3
[2,1,1]
=> [3,1]
=> 2
[1,1,1,1]
=> [4]
=> 1
[5]
=> [3,2]
=> 2
[4,1]
=> [3,1,1]
=> 3
[3,2]
=> [1,1,1,1,1]
=> 5
[3,1,1]
=> [2,1,1,1]
=> 4
[2,2,1]
=> [2,2,1]
=> 3
[2,1,1,1]
=> [4,1]
=> 2
[1,1,1,1,1]
=> [5]
=> 1
[6]
=> [3,3]
=> 2
[5,1]
=> [3,2,1]
=> 3
[4,2]
=> [2,1,1,1,1]
=> 5
[4,1,1]
=> [2,2,1,1]
=> 4
[3,3]
=> [3,1,1,1]
=> 4
[3,2,1]
=> [1,1,1,1,1,1]
=> 6
[3,1,1,1]
=> [4,1,1]
=> 3
[2,2,2]
=> [2,2,2]
=> 3
[2,2,1,1]
=> [4,2]
=> 2
[2,1,1,1,1]
=> [5,1]
=> 2
[1,1,1,1,1,1]
=> [6]
=> 1
[7]
=> [4,3]
=> 2
[6,1]
=> [3,3,1]
=> 3
[5,2]
=> [3,2,1,1]
=> 4
[5,1,1]
=> [4,2,1]
=> 3
[4,3]
=> [2,2,1,1,1]
=> 5
[4,2,1]
=> [1,1,1,1,1,1,1]
=> 7
[4,1,1,1]
=> [2,2,2,1]
=> 4
[3,3,1]
=> [2,1,1,1,1,1]
=> 6
[3,2,2]
=> [3,1,1,1,1]
=> 5
[3,2,1,1]
=> [4,1,1,1]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> 3
[2,2,2,1]
=> [3,2,2]
=> 3
[2,2,1,1,1]
=> [5,2]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> 1
[8]
=> [4,4]
=> 2
[7,1]
=> [4,3,1]
=> 3
[6,2]
=> [3,3,1,1]
=> 4
[6,1,1]
=> [4,2,2]
=> 3
[5,3]
=> [2,2,2,1,1]
=> 5
[5,2,1]
=> [4,1,1,1,1]
=> 5
Description
The length of the partition.
Matching statistic: St000288
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(load all 3 compositions to match this statistic)
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ 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%
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
[1]
=> [1]
=> 10 => 1
[2]
=> [1,1]
=> 110 => 2
[1,1]
=> [2]
=> 100 => 1
[3]
=> [2,1]
=> 1010 => 2
[2,1]
=> [1,1,1]
=> 1110 => 3
[1,1,1]
=> [3]
=> 1000 => 1
[4]
=> [2,2]
=> 1100 => 2
[3,1]
=> [1,1,1,1]
=> 11110 => 4
[2,2]
=> [2,1,1]
=> 10110 => 3
[2,1,1]
=> [3,1]
=> 10010 => 2
[1,1,1,1]
=> [4]
=> 10000 => 1
[5]
=> [3,2]
=> 10100 => 2
[4,1]
=> [3,1,1]
=> 100110 => 3
[3,2]
=> [1,1,1,1,1]
=> 111110 => 5
[3,1,1]
=> [2,1,1,1]
=> 101110 => 4
[2,2,1]
=> [2,2,1]
=> 11010 => 3
[2,1,1,1]
=> [4,1]
=> 100010 => 2
[1,1,1,1,1]
=> [5]
=> 100000 => 1
[6]
=> [3,3]
=> 11000 => 2
[5,1]
=> [3,2,1]
=> 101010 => 3
[4,2]
=> [2,1,1,1,1]
=> 1011110 => 5
[4,1,1]
=> [2,2,1,1]
=> 110110 => 4
[3,3]
=> [3,1,1,1]
=> 1001110 => 4
[3,2,1]
=> [1,1,1,1,1,1]
=> 1111110 => 6
[3,1,1,1]
=> [4,1,1]
=> 1000110 => 3
[2,2,2]
=> [2,2,2]
=> 11100 => 3
[2,2,1,1]
=> [4,2]
=> 100100 => 2
[2,1,1,1,1]
=> [5,1]
=> 1000010 => 2
[1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
[7]
=> [4,3]
=> 101000 => 2
[6,1]
=> [3,3,1]
=> 110010 => 3
[5,2]
=> [3,2,1,1]
=> 1010110 => 4
[5,1,1]
=> [4,2,1]
=> 1001010 => 3
[4,3]
=> [2,2,1,1,1]
=> 1101110 => 5
[4,2,1]
=> [1,1,1,1,1,1,1]
=> 11111110 => 7
[4,1,1,1]
=> [2,2,2,1]
=> 111010 => 4
[3,3,1]
=> [2,1,1,1,1,1]
=> 10111110 => 6
[3,2,2]
=> [3,1,1,1,1]
=> 10011110 => 5
[3,2,1,1]
=> [4,1,1,1]
=> 10001110 => 4
[3,1,1,1,1]
=> [5,1,1]
=> 10000110 => 3
[2,2,2,1]
=> [3,2,2]
=> 101100 => 3
[2,2,1,1,1]
=> [5,2]
=> 1000100 => 2
[2,1,1,1,1,1]
=> [6,1]
=> 10000010 => 2
[1,1,1,1,1,1,1]
=> [7]
=> 10000000 => 1
[8]
=> [4,4]
=> 110000 => 2
[7,1]
=> [4,3,1]
=> 1010010 => 3
[6,2]
=> [3,3,1,1]
=> 1100110 => 4
[6,1,1]
=> [4,2,2]
=> 1001100 => 3
[5,3]
=> [2,2,2,1,1]
=> 1110110 => 5
[5,2,1]
=> [4,1,1,1,1]
=> 100011110 => 5
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000147
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(load all 2 compositions to match this statistic)
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000147: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> []
=> []
=> 0 = 1 - 1
[2]
=> [1,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1]
=> [2]
=> []
=> []
=> 0 = 1 - 1
[3]
=> [2,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[2,1]
=> [1,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[1,1,1]
=> [3]
=> []
=> []
=> 0 = 1 - 1
[4]
=> [2,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[3,1]
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[2,2]
=> [2,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> []
=> []
=> 0 = 1 - 1
[5]
=> [3,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[4,1]
=> [3,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[3,2]
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
[3,1,1]
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[2,2,1]
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> []
=> []
=> 0 = 1 - 1
[6]
=> [3,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[5,1]
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[4,2]
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
[4,1,1]
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[3,3]
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[3,2,1]
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
[3,1,1,1]
=> [4,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,2,2]
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
[2,2,1,1]
=> [4,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [5,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> []
=> []
=> 0 = 1 - 1
[7]
=> [4,3]
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
[6,1]
=> [3,3,1]
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[5,2]
=> [3,2,1,1]
=> [2,1,1]
=> [3,1]
=> 3 = 4 - 1
[5,1,1]
=> [4,2,1]
=> [2,1]
=> [2,1]
=> 2 = 3 - 1
[4,3]
=> [2,2,1,1,1]
=> [2,1,1,1]
=> [4,1]
=> 4 = 5 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [1,1,1,1,1,1]
=> [6]
=> 6 = 7 - 1
[4,1,1,1]
=> [2,2,2,1]
=> [2,2,1]
=> [3,2]
=> 3 = 4 - 1
[3,3,1]
=> [2,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
[3,2,2]
=> [3,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
[3,2,1,1]
=> [4,1,1,1]
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [1,1]
=> [2]
=> 2 = 3 - 1
[2,2,2,1]
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
[2,2,1,1,1]
=> [5,2]
=> [2]
=> [1,1]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [1]
=> [1]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> []
=> []
=> 0 = 1 - 1
[8]
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
[7,1]
=> [4,3,1]
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
[6,2]
=> [3,3,1,1]
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
[6,1,1]
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2 = 3 - 1
[5,3]
=> [2,2,2,1,1]
=> [2,2,1,1]
=> [4,2]
=> 4 = 5 - 1
[5,2,1]
=> [4,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
Description
The largest part of an integer partition.
Matching statistic: St000378
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
St000378: Integer partitions ⟶ ℤResult quality: 80% ●values known / values provided: 80%●distinct values known / distinct values provided: 100%
Values
[1]
=> 1
[2]
=> 2
[1,1]
=> 1
[3]
=> 2
[2,1]
=> 3
[1,1,1]
=> 1
[4]
=> 2
[3,1]
=> 4
[2,2]
=> 3
[2,1,1]
=> 2
[1,1,1,1]
=> 1
[5]
=> 2
[4,1]
=> 3
[3,2]
=> 5
[3,1,1]
=> 4
[2,2,1]
=> 3
[2,1,1,1]
=> 2
[1,1,1,1,1]
=> 1
[6]
=> 2
[5,1]
=> 3
[4,2]
=> 5
[4,1,1]
=> 4
[3,3]
=> 4
[3,2,1]
=> 6
[3,1,1,1]
=> 3
[2,2,2]
=> 3
[2,2,1,1]
=> 2
[2,1,1,1,1]
=> 2
[1,1,1,1,1,1]
=> 1
[7]
=> 2
[6,1]
=> 3
[5,2]
=> 4
[5,1,1]
=> 3
[4,3]
=> 5
[4,2,1]
=> 7
[4,1,1,1]
=> 4
[3,3,1]
=> 6
[3,2,2]
=> 5
[3,2,1,1]
=> 4
[3,1,1,1,1]
=> 3
[2,2,2,1]
=> 3
[2,2,1,1,1]
=> 2
[2,1,1,1,1,1]
=> 2
[1,1,1,1,1,1,1]
=> 1
[8]
=> 2
[7,1]
=> 3
[6,2]
=> 4
[6,1,1]
=> 3
[5,3]
=> 5
[5,2,1]
=> 5
[11,1,1]
=> ? = 3
[10,1,1,1]
=> ? = 3
[9,4]
=> ? = 4
[9,2,1,1]
=> ? = 4
[8,2,2,1]
=> ? = 5
[7,2,2,2]
=> ? = 5
[7,2,1,1,1,1]
=> ? = 5
[7,1,1,1,1,1,1]
=> ? = 4
[6,2,2,2,1]
=> ? = 7
[4,2,2,2,2,1]
=> ? = 5
[3,2,2,2,1,1,1,1]
=> ? = 3
[11,1,1,1]
=> ? = 3
[10,2,1,1]
=> ? = 4
[9,4,1]
=> ? = 5
[9,2,2,1]
=> ? = 5
[8,1,1,1,1,1,1]
=> ? = 4
[6,2,2,1,1,1,1]
=> ? = 4
[3,3,2,2,1,1,1,1]
=> ? = 3
[12,1,1,1]
=> ? = 3
[11,1,1,1,1]
=> ? = 3
[10,5]
=> ? = 4
[10,4,1]
=> ? = 5
[9,2,1,1,1,1]
=> ? = 4
[9,1,1,1,1,1,1]
=> ? = 3
[8,6,1]
=> ? = 6
[8,1,1,1,1,1,1,1]
=> ? = 4
[5,2,2,2,2,1,1]
=> ? = 4
[3,3,2,2,2,1,1,1]
=> ? = 3
[2,2,2,2,2,2,2,1]
=> ? = 3
[12,2,1,1]
=> ? = 4
[10,5,1]
=> ? = 5
[9,5,1,1]
=> ? = 5
[9,2,2,1,1,1]
=> ? = 4
[9,1,1,1,1,1,1,1]
=> ? = 4
[8,5,1,1,1]
=> ? = 6
[7,2,2,2,1,1,1]
=> ? = 5
[5,2,2,2,2,2,1]
=> ? = 5
[4,3,3,2,2,2]
=> ? = 5
[4,3,3,2,2,1,1]
=> ? = 4
[3,3,3,2,2,1,1,1]
=> ? = 3
[3,3,2,2,2,2,1,1]
=> ? = 3
[14,1,1,1]
=> ? = 3
[13,2,1,1]
=> ? = 4
[11,5,1]
=> ? = 5
[11,2,1,1,1,1]
=> ? = 4
[10,6,1]
=> ? = 5
[9,2,2,2,2]
=> ? = 5
[9,2,2,2,1,1]
=> ? = 4
[8,2,2,2,1,1,1]
=> ? = 5
[8,2,2,1,1,1,1,1]
=> ? = 5
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: St000734
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000734: Standard tableaux ⟶ ℤResult quality: 73% ●values known / values provided: 73%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> [1]
=> [[1]]
=> 1
[2]
=> [1,1]
=> [2]
=> [[1,2]]
=> 2
[1,1]
=> [2]
=> [1,1]
=> [[1],[2]]
=> 1
[3]
=> [2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1]
=> [1,1,1]
=> [3]
=> [[1,2,3]]
=> 3
[1,1,1]
=> [3]
=> [1,1,1]
=> [[1],[2],[3]]
=> 1
[4]
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,1]
=> [1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 4
[2,2]
=> [2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 3
[2,1,1]
=> [3,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 2
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 1
[5]
=> [3,2]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 2
[4,1]
=> [3,1,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[3,2]
=> [1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 5
[3,1,1]
=> [2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 4
[2,2,1]
=> [2,2,1]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 3
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 2
[1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 1
[6]
=> [3,3]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2
[5,1]
=> [3,2,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[4,2]
=> [2,1,1,1,1]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 5
[4,1,1]
=> [2,2,1,1]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 4
[3,3]
=> [3,1,1,1]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 4
[3,2,1]
=> [1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 6
[3,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3
[2,2,2]
=> [2,2,2]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 3
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 2
[1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 1
[7]
=> [4,3]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 2
[6,1]
=> [3,3,1]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 3
[5,2]
=> [3,2,1,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 4
[5,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 3
[4,3]
=> [2,2,1,1,1]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 5
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 7
[4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 4
[3,3,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> [[1,2,3,4,5,6],[7]]
=> 6
[3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 5
[3,2,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> 3
[2,2,2,1]
=> [3,2,2]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 1
[8]
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 2
[7,1]
=> [4,3,1]
=> [3,2,2,1]
=> [[1,2,3],[4,5],[6,7],[8]]
=> 3
[6,2]
=> [3,3,1,1]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 4
[6,1,1]
=> [4,2,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 3
[5,3]
=> [2,2,2,1,1]
=> [5,3]
=> [[1,2,3,4,5],[6,7,8]]
=> 5
[5,2,1]
=> [4,1,1,1,1]
=> [5,1,1,1]
=> [[1,2,3,4,5],[6],[7],[8]]
=> 5
[7,1,1,1,1]
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9],[10],[11]]
=> ? = 3
[6,5]
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 5
[6,3,2]
=> [4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? = 7
[6,3,1,1]
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? = 6
[5,4,2]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 9
[5,4,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [[1,2,3,4,5,6,7,8],[9,10,11]]
=> ? = 8
[5,3,3]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 8
[5,2,2,2]
=> [2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 7
[4,2,2,1,1,1]
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? = 5
[4,1,1,1,1,1,1,1]
=> [8,2,1]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 3
[3,3,3,2]
=> [2,2,2,2,2,1]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 6
[3,3,1,1,1,1,1]
=> [7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 4
[3,2,2,2,1,1]
=> [6,4,1]
=> [3,2,2,2,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11]]
=> ? = 3
[2,2,2,2,1,1,1]
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 2
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 2
[7,5]
=> [3,3,2,2,2]
=> [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? = 5
[4,3,2,2,1]
=> [3,3,3,1,1,1]
=> [6,3,3]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12]]
=> ? = 6
[2,2,2,2,2,1,1]
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11],[12]]
=> ? = 2
[11,1,1]
=> [6,5,2]
=> [3,3,2,2,2,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13]]
=> ? = 3
[8,2,2,1]
=> [5,2,2,2,2]
=> [5,5,1,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11],[12],[13]]
=> ? = 5
[7,6]
=> [3,3,3,2,2]
=> [5,5,3]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13]]
=> ? = 5
[7,2,1,1,1,1]
=> [3,3,3,3,1]
=> [5,4,4]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12,13]]
=> ? = 5
[7,1,1,1,1,1,1]
=> [4,3,3,3]
=> [4,4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13]]
=> ? = 4
[6,3,1,1,1,1]
=> [3,3,3,2,1,1]
=> [6,4,3]
=> [[1,2,3,4,5,6],[7,8,9,10],[11,12,13]]
=> ? = 6
[6,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [7,6]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13]]
=> ? = 7
[4,3,3,3]
=> [3,3,3,1,1,1,1]
=> [7,3,3]
=> [[1,2,3,4,5,6,7],[8,9,10],[11,12,13]]
=> ? = 7
[3,2,2,2,1,1,1,1]
=> [8,4,1]
=> [3,2,2,2,1,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10],[11],[12],[13]]
=> ? = 3
[2,2,2,2,1,1,1,1,1]
=> [9,4]
=> [2,2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11],[12],[13]]
=> ? = 2
[11,1,1,1]
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13],[14]]
=> ? = 3
[10,2,1,1]
=> [5,5,3,1]
=> [4,3,3,2,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13,14]]
=> ? = 4
[8,6]
=> [3,3,3,3,2]
=> [5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? = 5
[8,1,1,1,1,1,1]
=> [4,4,3,3]
=> [4,4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14]]
=> ? = 4
[6,2,2,1,1,1,1]
=> [7,4,2,1]
=> [4,3,2,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12],[13],[14]]
=> ? = 4
[4,3,2,2,2,1]
=> [4,4,3,1,1,1]
=> [6,3,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? = 6
[3,3,2,2,1,1,1,1]
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11],[12],[13],[14]]
=> ? = 3
[3,2,2,2,2,1,1,1]
=> [8,5,1]
=> [3,2,2,2,2,1,1,1]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11],[12],[13],[14]]
=> ? = 3
[11,1,1,1,1]
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14],[15]]
=> ? = 3
[10,5]
=> [5,5,3,2]
=> [4,4,3,2,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13],[14,15]]
=> ? = 4
[10,4,1]
=> [5,5,3,1,1]
=> [5,3,3,2,2]
=> [[1,2,3,4,5],[6,7,8],[9,10,11],[12,13],[14,15]]
=> ? = 5
[9,2,1,1,1,1]
=> [7,4,3,1]
=> [4,3,3,2,1,1,1]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13],[14],[15]]
=> ? = 4
[9,1,1,1,1,1,1]
=> [8,4,3]
=> [3,3,3,2,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12],[13],[14],[15]]
=> ? = 3
[8,7]
=> [3,3,3,3,3]
=> [5,5,5]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14,15]]
=> ? = 5
[8,6,1]
=> [3,3,3,3,2,1]
=> [6,5,4]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14,15]]
=> ? = 6
[8,1,1,1,1,1,1,1]
=> [4,4,4,3]
=> [4,4,4,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15]]
=> ? = 4
[5,2,2,2,2,1,1]
=> [7,5,2,1]
=> [4,3,2,2,2,1,1]
=> [[1,2,3,4],[5,6,7],[8,9],[10,11],[12,13],[14],[15]]
=> ? = 4
[3,3,3,3,3]
=> [5,3,3,2,2]
=> [5,5,3,1,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13],[14],[15]]
=> ? = 5
[3,3,2,2,2,1,1,1]
=> [8,5,2]
=> [3,3,2,2,2,1,1,1]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12],[13],[14],[15]]
=> ? = 3
[12,2,1,1]
=> [6,6,3,1]
=> [4,3,3,2,2,2]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12],[13,14],[15,16]]
=> ? = 4
[10,5,1]
=> [5,5,3,2,1]
=> [5,4,3,2,2]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15,16]]
=> ? = 5
[9,5,1,1]
=> [6,4,3,2,1]
=> [5,4,3,2,1,1]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12],[13,14],[15],[16]]
=> ? = 5
Description
The last entry in the first row of a standard tableau.
Matching statistic: St000012
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00030: Dyck paths —zeta map⟶ Dyck paths
St000012: Dyck paths ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1,0,1,0]
=> [1,1,0,0]
=> 1
[2]
=> [1,1,0,0,1,0]
=> [1,1,0,1,0,0]
=> 2
[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]
=> 2
[2,1]
=> [1,0,1,0,1,0]
=> [1,1,1,0,0,0]
=> 3
[1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,0,0]
=> 1
[4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0]
=> 2
[3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,1,1,0,0,1,0,0]
=> 4
[2,2]
=> [1,1,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,0]
=> 3
[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]
=> 1
[5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0]
=> 3
[3,2]
=> [1,1,0,0,1,0,1,0]
=> [1,1,1,0,1,0,0,0]
=> 5
[3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,1,0,1,1,0,0,0]
=> 4
[2,2,1]
=> [1,0,1,0,1,1,0,0]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,1,1]
=> [1,0,1,1,1,0,1,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0]
=> 2
[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]
=> 1
[6]
=> [1,1,1,1,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[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]
=> 5
[4,1,1]
=> [1,1,0,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4
[3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 4
[3,2,1]
=> [1,0,1,0,1,0,1,0]
=> [1,1,1,1,0,0,0,0]
=> 6
[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]
=> 2
[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]
=> 2
[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]
=> 1
[7]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 2
[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]
=> 3
[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]
=> 4
[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]
=> 3
[4,3]
=> [1,1,1,0,0,0,1,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 5
[4,2,1]
=> [1,1,0,1,0,1,0,0,1,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 7
[4,1,1,1]
=> [1,0,1,1,1,0,0,0,1,0]
=> [1,0,1,1,0,1,1,0,0,0]
=> 4
[3,3,1]
=> [1,1,0,1,0,0,1,1,0,0]
=> [1,1,1,0,1,0,0,1,0,0]
=> 6
[3,2,2]
=> [1,1,0,0,1,1,0,1,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 5
[3,2,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> [1,1,1,0,0,0,1,1,0,0]
=> 4
[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]
=> 3
[2,2,2,1]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,0,1,0,1,1,1,0,0,0]
=> 3
[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]
=> 2
[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]
=> 2
[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]
=> 1
[8]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[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]
=> 3
[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]
=> 3
[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]
=> 5
[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]
=> 5
[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]
=> ? = 1
[9]
=> [1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[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]
=> ? = 3
[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]
=> ? = 2
[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]
=> ? = 1
[10]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[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]
=> ? = 3
[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]
=> ? = 4
[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]
=> ? = 3
[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]
=> ? = 4
[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]
=> ? = 3
[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]
=> ? = 2
[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]
=> ? = 2
[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]
=> ? = 1
[11]
=> [1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[10,1]
=> [1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[9,2]
=> [1,1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4
[9,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[8,3]
=> [1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4
[8,2,1]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[8,1,1,1]
=> [1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[7,2,2]
=> [1,1,1,1,1,0,0,1,1,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 5
[4,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 3
[3,3,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,1,1,0,0,0,0,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 4
[2,2,1,1,1,1,1,1,1]
=> [1,0,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[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]
=> ? = 2
[10,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[9,3]
=> [1,1,1,1,1,1,1,1,0,0,0,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,0,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4
[9,2,1]
=> [1,1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,1,0,0,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[9,1,1,1]
=> [1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 3
[8,4]
=> [1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 4
[8,2,2]
=> [1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,1,0]
=> [1,1,0,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[8,2,1,1]
=> [1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4
[7,3,1,1]
=> [1,1,1,1,0,1,1,0,0,1,0,0,0,0,1,0]
=> [1,1,0,1,1,0,0,0,1,0,1,1,0,1,0,0]
=> ? = 6
[13]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[11,1,1]
=> [1,1,1,1,1,1,1,1,1,0,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 3
[10,1,1,1]
=> [1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 3
[9,4]
=> [1,1,1,1,1,1,1,1,0,0,0,0,1,0,0,0,0,0,1,0]
=> [1,0,1,0,1,1,0,1,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4
[9,2,2]
=> [1,1,1,1,1,1,1,0,0,1,1,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 5
[9,2,1,1]
=> [1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 4
[8,5]
=> [1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0]
=> [1,0,1,0,1,0,1,1,0,1,0,0,1,1,0,1,0,0]
=> ? = 4
[8,4,1]
=> [1,1,1,1,1,1,0,1,0,0,0,1,0,0,0,0,1,0]
=> [1,1,0,0,1,1,0,1,0,0,1,0,1,1,0,1,0,0]
=> ? = 5
[8,2,2,1]
=> [1,1,1,1,1,0,1,0,1,1,0,0,0,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 5
[3,2,2,2,1,1,1,1]
=> [1,0,1,1,1,1,0,1,1,1,0,1,0,0,0,0,0,0]
=> [1,1,0,0,1,0,1,1,0,0,1,0,1,0,1,1,0,0]
=> ? = 3
[2,2,2,2,1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,1,1,1,1,0,0,0,0,0,0,0,0]
=> [1,0,1,0,1,0,1,1,0,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 2
[14]
=> [1,1,1,1,1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,1,0]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 2
[11,1,1,1]
=> [1,1,1,1,1,1,1,1,0,1,1,1,0,0,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 3
[10,2,1,1]
=> [1,1,1,1,1,1,1,0,1,1,0,1,0,0,0,0,0,0,0,0,1,0]
=> ?
=> ? = 4
Description
The area of a Dyck path.
This is the number of complete squares in the integer lattice which are below the path and above the x-axis. The 'half-squares' directly above the axis do not contribute to this statistic.
1. Dyck paths are bijection with '''area sequences''' $(a_1,\ldots,a_n)$ such that $a_1 = 0, a_{k+1} \leq a_k + 1$.
2. The generating function $\mathbf{D}_n(q) = \sum_{D \in \mathfrak{D}_n} q^{\operatorname{area}(D)}$ satisfy the recurrence $$\mathbf{D}_{n+1}(q) = \sum q^k \mathbf{D}_k(q) \mathbf{D}_{n-k}(q).$$
3. The area is equidistributed with [[St000005]] and [[St000006]]. Pairs of these statistics play an important role in the theory of $q,t$-Catalan numbers.
Matching statistic: St000733
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> [[1]]
=> 1
[2]
=> [1,1]
=> [[1],[2]]
=> 2
[1,1]
=> [2]
=> [[1,2]]
=> 1
[3]
=> [2,1]
=> [[1,2],[3]]
=> 2
[2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
[1,1,1]
=> [3]
=> [[1,2,3]]
=> 1
[4]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
[3,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
[2,2]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
[2,1,1]
=> [3,1]
=> [[1,2,3],[4]]
=> 2
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 1
[5]
=> [3,2]
=> [[1,2,3],[4,5]]
=> 2
[4,1]
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3
[3,2]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
[3,1,1]
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> 4
[2,2,1]
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> 3
[2,1,1,1]
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 1
[6]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 2
[5,1]
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> 3
[4,2]
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> 5
[4,1,1]
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> 4
[3,3]
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 4
[3,2,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 6
[3,1,1,1]
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> 3
[2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 3
[2,2,1,1]
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2
[1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 1
[7]
=> [4,3]
=> [[1,2,3,4],[5,6,7]]
=> 2
[6,1]
=> [3,3,1]
=> [[1,2,3],[4,5,6],[7]]
=> 3
[5,2]
=> [3,2,1,1]
=> [[1,2,3],[4,5],[6],[7]]
=> 4
[5,1,1]
=> [4,2,1]
=> [[1,2,3,4],[5,6],[7]]
=> 3
[4,3]
=> [2,2,1,1,1]
=> [[1,2],[3,4],[5],[6],[7]]
=> 5
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 7
[4,1,1,1]
=> [2,2,2,1]
=> [[1,2],[3,4],[5,6],[7]]
=> 4
[3,3,1]
=> [2,1,1,1,1,1]
=> [[1,2],[3],[4],[5],[6],[7]]
=> 6
[3,2,2]
=> [3,1,1,1,1]
=> [[1,2,3],[4],[5],[6],[7]]
=> 5
[3,2,1,1]
=> [4,1,1,1]
=> [[1,2,3,4],[5],[6],[7]]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [[1,2,3,4,5],[6],[7]]
=> 3
[2,2,2,1]
=> [3,2,2]
=> [[1,2,3],[4,5],[6,7]]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [[1,2,3,4,5],[6,7]]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [[1,2,3,4,5,6],[7]]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 1
[8]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2
[7,1]
=> [4,3,1]
=> [[1,2,3,4],[5,6,7],[8]]
=> 3
[6,2]
=> [3,3,1,1]
=> [[1,2,3],[4,5,6],[7],[8]]
=> 4
[6,1,1]
=> [4,2,2]
=> [[1,2,3,4],[5,6],[7,8]]
=> 3
[5,3]
=> [2,2,2,1,1]
=> [[1,2],[3,4],[5,6],[7],[8]]
=> 5
[5,2,1]
=> [4,1,1,1,1]
=> [[1,2,3,4],[5],[6],[7],[8]]
=> 5
[11]
=> [6,5]
=> [[1,2,3,4,5,6],[7,8,9,10,11]]
=> ? = 2
[10,1]
=> [5,5,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11]]
=> ? = 3
[7,1,1,1,1]
=> [6,3,2]
=> [[1,2,3,4,5,6],[7,8,9],[10,11]]
=> ? = 3
[6,5]
=> [3,2,2,2,2]
=> [[1,2,3],[4,5],[6,7],[8,9],[10,11]]
=> ? = 5
[6,3,2]
=> [4,2,1,1,1,1,1]
=> [[1,2,3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 7
[6,3,1,1]
=> [5,2,1,1,1,1]
=> [[1,2,3,4,5],[6,7],[8],[9],[10],[11]]
=> ? = 6
[5,4,2]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,2],[3,4],[5],[6],[7],[8],[9],[10],[11]]
=> ? = 9
[5,4,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,2],[3,4],[5,6],[7],[8],[9],[10],[11]]
=> ? = 8
[5,3,3]
=> [3,2,1,1,1,1,1,1]
=> [[1,2,3],[4,5],[6],[7],[8],[9],[10],[11]]
=> ? = 8
[5,2,2,2]
=> [2,2,2,2,1,1,1]
=> [[1,2],[3,4],[5,6],[7,8],[9],[10],[11]]
=> ? = 7
[4,2,2,1,1,1]
=> [6,2,1,1,1]
=> [[1,2,3,4,5,6],[7,8],[9],[10],[11]]
=> ? = 5
[4,1,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,2,3,4,5,6,7,8],[9,10],[11]]
=> ? = 3
[3,3,3,2]
=> [2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11]]
=> ? = 6
[3,3,1,1,1,1,1]
=> [7,2,1,1]
=> [[1,2,3,4,5,6,7],[8,9],[10],[11]]
=> ? = 4
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,2,3,4,5,6],[7,8,9,10],[11]]
=> ? = 3
[2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,5,6,7],[8,9,10,11]]
=> ? = 2
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,3,4,5,6,7,8,9],[10,11]]
=> ? = 2
[10,1,1]
=> [5,5,2]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12]]
=> ? = 3
[7,5]
=> [3,3,2,2,2]
=> [[1,2,3],[4,5,6],[7,8],[9,10],[11,12]]
=> ? = 5
[7,1,1,1,1,1]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4
[4,3,2,2,1]
=> [3,3,3,1,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11],[12]]
=> ? = 6
[2,2,2,2,2,2]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3
[2,2,2,2,2,1,1]
=> [7,5]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12]]
=> ? = 2
[13]
=> [7,6]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13]]
=> ? = 2
[11,1,1]
=> [6,5,2]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13]]
=> ? = 3
[10,1,1,1]
=> [5,5,3]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13]]
=> ? = 3
[8,2,2,1]
=> [5,2,2,2,2]
=> [[1,2,3,4,5],[6,7],[8,9],[10,11],[12,13]]
=> ? = 5
[7,6]
=> [3,3,3,2,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12,13]]
=> ? = 5
[7,2,1,1,1,1]
=> [3,3,3,3,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13]]
=> ? = 5
[7,1,1,1,1,1,1]
=> [4,3,3,3]
=> [[1,2,3,4],[5,6,7],[8,9,10],[11,12,13]]
=> ? = 4
[6,3,1,1,1,1]
=> [3,3,3,2,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11],[12],[13]]
=> ? = 6
[6,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13]]
=> ? = 7
[4,3,3,3]
=> [3,3,3,1,1,1,1]
=> [[1,2,3],[4,5,6],[7,8,9],[10],[11],[12],[13]]
=> ? = 7
[3,2,2,2,2,2]
=> [4,4,4,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13]]
=> ? = 4
[3,2,2,2,1,1,1,1]
=> [8,4,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12],[13]]
=> ? = 3
[2,2,2,2,2,2,1]
=> [5,4,4]
=> [[1,2,3,4,5],[6,7,8,9],[10,11,12,13]]
=> ? = 3
[2,2,2,2,1,1,1,1,1]
=> [9,4]
=> [[1,2,3,4,5,6,7,8,9],[10,11,12,13]]
=> ? = 2
[14]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 2
[11,1,1,1]
=> [6,5,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11],[12,13,14]]
=> ? = 3
[10,2,1,1]
=> [5,5,3,1]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13],[14]]
=> ? = 4
[8,6]
=> [3,3,3,3,2]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14]]
=> ? = 5
[8,1,1,1,1,1,1]
=> [4,4,3,3]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12,13,14]]
=> ? = 4
[6,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 7
[6,2,2,1,1,1,1]
=> [7,4,2,1]
=> [[1,2,3,4,5,6,7],[8,9,10,11],[12,13],[14]]
=> ? = 4
[4,3,2,2,2,1]
=> [4,4,3,1,1,1]
=> [[1,2,3,4],[5,6,7,8],[9,10,11],[12],[13],[14]]
=> ? = 6
[3,3,2,2,2,2]
=> [4,4,4,2]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14]]
=> ? = 4
[3,3,2,2,1,1,1,1]
=> [8,4,2]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12],[13,14]]
=> ? = 3
[3,2,2,2,2,1,1,1]
=> [8,5,1]
=> [[1,2,3,4,5,6,7,8],[9,10,11,12,13],[14]]
=> ? = 3
[2,2,2,2,2,2,2]
=> [5,5,4]
=> [[1,2,3,4,5],[6,7,8,9,10],[11,12,13,14]]
=> ? = 3
[12,1,1,1]
=> [6,6,3]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15]]
=> ? = 3
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 67% ●values known / values provided: 67%●distinct values known / distinct values provided: 100%
Values
[1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
[2]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
[1,1]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
[3]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
[2,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
[1,1,1]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
[4]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
[3,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
[2,2]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
[2,1,1]
=> [3,1]
=> [[1,3,4],[2]]
=> 1 = 2 - 1
[1,1,1,1]
=> [4]
=> [[1,2,3,4]]
=> 0 = 1 - 1
[5]
=> [3,2]
=> [[1,2,5],[3,4]]
=> 1 = 2 - 1
[4,1]
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> 2 = 3 - 1
[3,2]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
[3,1,1]
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> 3 = 4 - 1
[2,2,1]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2 = 3 - 1
[2,1,1,1]
=> [4,1]
=> [[1,3,4,5],[2]]
=> 1 = 2 - 1
[1,1,1,1,1]
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 1 - 1
[6]
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 2 - 1
[5,1]
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> 2 = 3 - 1
[4,2]
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> 4 = 5 - 1
[4,1,1]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 3 = 4 - 1
[3,3]
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> 3 = 4 - 1
[3,2,1]
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> 5 = 6 - 1
[3,1,1,1]
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> 2 = 3 - 1
[2,2,2]
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> 2 = 3 - 1
[2,2,1,1]
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> 1 = 2 - 1
[2,1,1,1,1]
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> 1 = 2 - 1
[1,1,1,1,1,1]
=> [6]
=> [[1,2,3,4,5,6]]
=> 0 = 1 - 1
[7]
=> [4,3]
=> [[1,2,3,7],[4,5,6]]
=> 1 = 2 - 1
[6,1]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2 = 3 - 1
[5,2]
=> [3,2,1,1]
=> [[1,4,7],[2,6],[3],[5]]
=> 3 = 4 - 1
[5,1,1]
=> [4,2,1]
=> [[1,3,6,7],[2,5],[4]]
=> 2 = 3 - 1
[4,3]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 4 = 5 - 1
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6],[7]]
=> 6 = 7 - 1
[4,1,1,1]
=> [2,2,2,1]
=> [[1,3],[2,5],[4,7],[6]]
=> 3 = 4 - 1
[3,3,1]
=> [2,1,1,1,1,1]
=> [[1,7],[2],[3],[4],[5],[6]]
=> 5 = 6 - 1
[3,2,2]
=> [3,1,1,1,1]
=> [[1,6,7],[2],[3],[4],[5]]
=> 4 = 5 - 1
[3,2,1,1]
=> [4,1,1,1]
=> [[1,5,6,7],[2],[3],[4]]
=> 3 = 4 - 1
[3,1,1,1,1]
=> [5,1,1]
=> [[1,4,5,6,7],[2],[3]]
=> 2 = 3 - 1
[2,2,2,1]
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> 2 = 3 - 1
[2,2,1,1,1]
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> 1 = 2 - 1
[2,1,1,1,1,1]
=> [6,1]
=> [[1,3,4,5,6,7],[2]]
=> 1 = 2 - 1
[1,1,1,1,1,1,1]
=> [7]
=> [[1,2,3,4,5,6,7]]
=> 0 = 1 - 1
[8]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 1 = 2 - 1
[7,1]
=> [4,3,1]
=> [[1,3,4,8],[2,6,7],[5]]
=> 2 = 3 - 1
[6,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 3 = 4 - 1
[6,1,1]
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> 2 = 3 - 1
[5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> 4 = 5 - 1
[5,2,1]
=> [4,1,1,1,1]
=> [[1,6,7,8],[2],[3],[4],[5]]
=> 4 = 5 - 1
[11]
=> [6,5]
=> [[1,2,3,4,5,11],[6,7,8,9,10]]
=> ? = 2 - 1
[10,1]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 3 - 1
[7,1,1,1,1]
=> [6,3,2]
=> [[1,2,5,9,10,11],[3,4,8],[6,7]]
=> ? = 3 - 1
[6,5]
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> ? = 5 - 1
[6,3,2]
=> [4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9],[3],[4],[5],[6],[8]]
=> ? = 7 - 1
[6,3,1,1]
=> [5,2,1,1,1,1]
=> [[1,6,9,10,11],[2,8],[3],[4],[5],[7]]
=> ? = 6 - 1
[5,4,2]
=> [2,2,1,1,1,1,1,1,1]
=> [[1,9],[2,11],[3],[4],[5],[6],[7],[8],[10]]
=> ? = 9 - 1
[5,4,1,1]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 8 - 1
[5,3,3]
=> [3,2,1,1,1,1,1,1]
=> [[1,8,11],[2,10],[3],[4],[5],[6],[7],[9]]
=> ? = 8 - 1
[5,2,2,2]
=> [2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6],[8],[10]]
=> ? = 7 - 1
[4,2,2,1,1,1]
=> [6,2,1,1,1]
=> [[1,5,8,9,10,11],[2,7],[3],[4],[6]]
=> ? = 5 - 1
[4,1,1,1,1,1,1,1]
=> [8,2,1]
=> [[1,3,6,7,8,9,10,11],[2,5],[4]]
=> ? = 3 - 1
[3,3,3,2]
=> [2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10]]
=> ? = 6 - 1
[3,3,1,1,1,1,1]
=> [7,2,1,1]
=> [[1,4,7,8,9,10,11],[2,6],[3],[5]]
=> ? = 4 - 1
[3,2,2,2,1,1]
=> [6,4,1]
=> [[1,3,4,5,10,11],[2,7,8,9],[6]]
=> ? = 3 - 1
[2,2,2,2,1,1,1]
=> [7,4]
=> [[1,2,3,4,9,10,11],[5,6,7,8]]
=> ? = 2 - 1
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [[1,2,5,6,7,8,9,10,11],[3,4]]
=> ? = 2 - 1
[10,1,1]
=> [5,5,2]
=> [[1,2,5,6,7],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
[7,5]
=> [3,3,2,2,2]
=> [[1,2,9],[3,4,12],[5,6],[7,8],[10,11]]
=> ? = 5 - 1
[7,1,1,1,1,1]
=> [3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4 - 1
[4,3,2,2,1]
=> [3,3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3,11,12],[4],[7],[10]]
=> ? = 6 - 1
[2,2,2,2,2,2]
=> [4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12]]
=> ? = 3 - 1
[2,2,2,2,2,1,1]
=> [7,5]
=> [[1,2,3,4,5,11,12],[6,7,8,9,10]]
=> ? = 2 - 1
[13]
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> ? = 2 - 1
[11,1,1]
=> [6,5,2]
=> [[1,2,5,6,7,13],[3,4,10,11,12],[8,9]]
=> ? = 3 - 1
[10,1,1,1]
=> [5,5,3]
=> [[1,2,3,7,8],[4,5,6,12,13],[9,10,11]]
=> ? = 3 - 1
[8,2,2,1]
=> [5,2,2,2,2]
=> [[1,2,11,12,13],[3,4],[5,6],[7,8],[9,10]]
=> ? = 5 - 1
[7,6]
=> [3,3,3,2,2]
=> [[1,2,7],[3,4,10],[5,6,13],[8,9],[11,12]]
=> ? = 5 - 1
[7,2,1,1,1,1]
=> [3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11]]
=> ? = 5 - 1
[7,1,1,1,1,1,1]
=> [4,3,3,3]
=> [[1,2,3,13],[4,5,6],[7,8,9],[10,11,12]]
=> ? = 4 - 1
[6,3,1,1,1,1]
=> [3,3,3,2,1,1]
=> [[1,4,7],[2,6,10],[3,9,13],[5,12],[8],[11]]
=> ? = 6 - 1
[6,2,2,2,1]
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 7 - 1
[4,3,3,3]
=> [3,3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3,12,13],[4],[5],[8],[11]]
=> ? = 7 - 1
[3,2,2,2,2,2]
=> [4,4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6,11,12,13],[10]]
=> ? = 4 - 1
[3,2,2,2,1,1,1,1]
=> [8,4,1]
=> [[1,3,4,5,10,11,12,13],[2,7,8,9],[6]]
=> ? = 3 - 1
[2,2,2,2,2,2,1]
=> [5,4,4]
=> [[1,2,3,4,13],[5,6,7,8],[9,10,11,12]]
=> ? = 3 - 1
[2,2,2,2,1,1,1,1,1]
=> [9,4]
=> [[1,2,3,4,9,10,11,12,13],[5,6,7,8]]
=> ? = 2 - 1
[14]
=> [7,7]
=> [[1,2,3,4,5,6,7],[8,9,10,11,12,13,14]]
=> ? = 2 - 1
[11,1,1,1]
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> ? = 3 - 1
[10,2,1,1]
=> [5,5,3,1]
=> [[1,3,4,8,9],[2,6,7,13,14],[5,11,12],[10]]
=> ? = 4 - 1
[8,6]
=> [3,3,3,3,2]
=> [[1,2,5],[3,4,8],[6,7,11],[9,10,14],[12,13]]
=> ? = 5 - 1
[8,1,1,1,1,1,1]
=> [4,4,3,3]
=> [[1,2,3,10],[4,5,6,14],[7,8,9],[11,12,13]]
=> ? = 4 - 1
[6,2,2,2,2]
=> [2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> ? = 7 - 1
[6,2,2,1,1,1,1]
=> [7,4,2,1]
=> [[1,3,6,7,12,13,14],[2,5,10,11],[4,9],[8]]
=> ? = 4 - 1
[4,3,2,2,2,1]
=> [4,4,3,1,1,1]
=> [[1,5,6,10],[2,8,9,14],[3,12,13],[4],[7],[11]]
=> ? = 6 - 1
[3,3,2,2,2,2]
=> [4,4,4,2]
=> [[1,2,5,6],[3,4,9,10],[7,8,13,14],[11,12]]
=> ? = 4 - 1
[3,3,2,2,1,1,1,1]
=> [8,4,2]
=> [[1,2,5,6,11,12,13,14],[3,4,9,10],[7,8]]
=> ? = 3 - 1
[3,2,2,2,2,1,1,1]
=> [8,5,1]
=> [[1,3,4,5,6,12,13,14],[2,8,9,10,11],[7]]
=> ? = 3 - 1
[2,2,2,2,2,2,2]
=> [5,5,4]
=> [[1,2,3,4,9],[5,6,7,8,14],[10,11,12,13]]
=> ? = 3 - 1
[12,1,1,1]
=> [6,6,3]
=> [[1,2,3,7,8,9],[4,5,6,13,14,15],[10,11,12]]
=> ? = 3 - 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: St000676
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St000676: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1]
=> [1]
=> [1,0]
=> 1
[2]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[1,1]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[3]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,1]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[4]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,2]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[5]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1]
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,2]
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[3,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[2,2,1]
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[6]
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[5,1]
=> [3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[4,2]
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[4,1,1]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[3,3]
=> [3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[3,2,1]
=> [1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[3,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[2,2,2]
=> [2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[7]
=> [4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[6,1]
=> [3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[5,2]
=> [3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 4
[5,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 3
[4,3]
=> [2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[3,3,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5
[3,2,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 3
[2,2,2,1]
=> [3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[8]
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[7,1]
=> [4,3,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 3
[6,2]
=> [3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[6,1,1]
=> [4,2,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
[5,3]
=> [2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5
[5,2,1]
=> [4,1,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 5
[1,1,1,1,1,1,1,1]
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[5,3,1]
=> [3,1,1,1,1,1,1]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[4,4,1]
=> [4,1,1,1,1,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[4,3,2]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,3,1,1]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[4,2,1,1,1]
=> [5,1,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[3,2,1,1,1,1]
=> [6,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[3,1,1,1,1,1,1]
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[2,2,1,1,1,1,1]
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[6,3,1]
=> [4,2,1,1,1,1]
=> [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 6
[5,3,2]
=> [2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[5,3,1,1]
=> [2,2,1,1,1,1,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
[5,1,1,1,1,1]
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[4,4,2]
=> [3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[4,4,1,1]
=> [3,2,1,1,1,1,1]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[4,3,3]
=> [4,1,1,1,1,1,1]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[4,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[4,3,1,1,1]
=> [5,1,1,1,1,1]
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[4,2,2,1,1]
=> [5,2,1,1,1]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 5
[4,2,1,1,1,1]
=> [6,1,1,1,1]
=> [5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,1,1,1,1,1,1]
=> [7,2,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[3,3,1,1,1,1]
=> [6,2,1,1]
=> [4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[3,2,1,1,1,1,1]
=> [7,1,1,1]
=> [4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[2,2,2,1,1,1,1]
=> [7,3]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,2,1,1,1,1,1,1]
=> [8,2]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[7,4]
=> [5,2,2,2]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[7,1,1,1,1]
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
[6,3,2]
=> [4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 7
[6,3,1,1]
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6
[6,2,2,1]
=> [5,2,2,1,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5
[5,5,1]
=> [5,3,1,1,1]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 5
[5,4,2]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9
[5,4,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8
[5,3,3]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 8
[4,2,2,1,1,1]
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,1,1,1,1,1,1,1]
=> [8,2,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[3,3,1,1,1,1,1]
=> [7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[2,2,2,2,1,1,1]
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,2,1]
=> [5,4,1,1,1]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 5
[7,4,1]
=> [5,3,2,1,1]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5
[7,2,2,1]
=> [5,2,2,2,1]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5
[6,6]
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4
[3,3,2,2,1,1]
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
[2,2,2,2,2,1,1]
=> [7,5]
=> [2,2,2,2,2,1,1]
=> [1,1,1,1,0,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 2
Description
The number of odd rises of a Dyck path.
This is the number of ones at an odd position, with the initial position equal to 1.
The number of Dyck paths of semilength $n$ with $k$ up steps in odd positions and $k$ returns to the main diagonal are counted by the binomial coefficient $\binom{n-1}{k-1}$ [3,4].
Matching statistic: St001039
Mp00323: Integer partitions —Loehr-Warrington inverse⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001039: Dyck paths ⟶ ℤResult quality: 60% ●values known / values provided: 60%●distinct values known / distinct values provided: 80%
Values
[1]
=> [1]
=> [1]
=> [1,0]
=> ? = 1
[2]
=> [1,1]
=> [2]
=> [1,0,1,0]
=> 2
[1,1]
=> [2]
=> [1,1]
=> [1,1,0,0]
=> 1
[3]
=> [2,1]
=> [2,1]
=> [1,0,1,1,0,0]
=> 2
[2,1]
=> [1,1,1]
=> [3]
=> [1,0,1,0,1,0]
=> 3
[1,1,1]
=> [3]
=> [1,1,1]
=> [1,1,0,1,0,0]
=> 1
[4]
=> [2,2]
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
[3,1]
=> [1,1,1,1]
=> [4]
=> [1,0,1,0,1,0,1,0]
=> 4
[2,2]
=> [2,1,1]
=> [3,1]
=> [1,0,1,0,1,1,0,0]
=> 3
[2,1,1]
=> [3,1]
=> [2,1,1]
=> [1,0,1,1,0,1,0,0]
=> 2
[1,1,1,1]
=> [4]
=> [1,1,1,1]
=> [1,1,0,1,0,1,0,0]
=> 1
[5]
=> [3,2]
=> [2,2,1]
=> [1,1,1,0,0,1,0,0]
=> 2
[4,1]
=> [3,1,1]
=> [3,1,1]
=> [1,0,1,0,1,1,0,1,0,0]
=> 3
[3,2]
=> [1,1,1,1,1]
=> [5]
=> [1,0,1,0,1,0,1,0,1,0]
=> 5
[3,1,1]
=> [2,1,1,1]
=> [4,1]
=> [1,0,1,0,1,0,1,1,0,0]
=> 4
[2,2,1]
=> [2,2,1]
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 3
[2,1,1,1]
=> [4,1]
=> [2,1,1,1]
=> [1,0,1,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1]
=> [5]
=> [1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1
[6]
=> [3,3]
=> [2,2,2]
=> [1,1,1,1,0,0,0,0]
=> 2
[5,1]
=> [3,2,1]
=> [3,2,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 3
[4,2]
=> [2,1,1,1,1]
=> [5,1]
=> [1,0,1,0,1,0,1,0,1,1,0,0]
=> 5
[4,1,1]
=> [2,2,1,1]
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 4
[3,3]
=> [3,1,1,1]
=> [4,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,0]
=> 4
[3,2,1]
=> [1,1,1,1,1,1]
=> [6]
=> [1,0,1,0,1,0,1,0,1,0,1,0]
=> 6
[3,1,1,1]
=> [4,1,1]
=> [3,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,0]
=> 3
[2,2,2]
=> [2,2,2]
=> [3,3]
=> [1,1,1,0,1,0,0,0]
=> 3
[2,2,1,1]
=> [4,2]
=> [2,2,1,1]
=> [1,1,1,0,0,1,0,1,0,0]
=> 2
[2,1,1,1,1]
=> [5,1]
=> [2,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,1]
=> [6]
=> [1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[7]
=> [4,3]
=> [2,2,2,1]
=> [1,1,1,1,0,0,0,1,0,0]
=> 2
[6,1]
=> [3,3,1]
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 3
[5,2]
=> [3,2,1,1]
=> [4,2,1]
=> [1,0,1,0,1,1,1,0,0,1,0,0]
=> 4
[5,1,1]
=> [4,2,1]
=> [3,2,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,0]
=> 3
[4,3]
=> [2,2,1,1,1]
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 5
[4,2,1]
=> [1,1,1,1,1,1,1]
=> [7]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> 7
[4,1,1,1]
=> [2,2,2,1]
=> [4,3]
=> [1,0,1,1,1,0,1,0,0,0]
=> 4
[3,3,1]
=> [2,1,1,1,1,1]
=> [6,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> 6
[3,2,2]
=> [3,1,1,1,1]
=> [5,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> 5
[3,2,1,1]
=> [4,1,1,1]
=> [4,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 4
[3,1,1,1,1]
=> [5,1,1]
=> [3,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> 3
[2,2,2,1]
=> [3,2,2]
=> [3,3,1]
=> [1,1,1,0,1,0,0,1,0,0]
=> 3
[2,2,1,1,1]
=> [5,2]
=> [2,2,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 2
[2,1,1,1,1,1]
=> [6,1]
=> [2,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> 2
[1,1,1,1,1,1,1]
=> [7]
=> [1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> 1
[8]
=> [4,4]
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
[7,1]
=> [4,3,1]
=> [3,2,2,1]
=> [1,0,1,1,1,1,0,0,0,1,0,0]
=> 3
[6,2]
=> [3,3,1,1]
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 4
[6,1,1]
=> [4,2,2]
=> [3,3,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,0]
=> 3
[5,3]
=> [2,2,2,1,1]
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 5
[5,2,1]
=> [4,1,1,1,1]
=> [5,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> 5
[5,1,1,1]
=> [2,2,2,2]
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
[1,1,1,1,1,1,1,1]
=> [8]
=> [1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[5,3,1]
=> [3,1,1,1,1,1,1]
=> [7,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 7
[4,4,1]
=> [4,1,1,1,1,1]
=> [6,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 6
[4,3,2]
=> [1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 9
[4,3,1,1]
=> [2,1,1,1,1,1,1,1]
=> [8,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 8
[4,2,1,1,1]
=> [5,1,1,1,1]
=> [5,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 5
[3,2,1,1,1,1]
=> [6,1,1,1]
=> [4,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[3,1,1,1,1,1,1]
=> [7,1,1]
=> [3,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[2,2,1,1,1,1,1]
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1]
=> [8,1]
=> [2,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1]
=> [9]
=> [1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[6,3,1]
=> [4,2,1,1,1,1]
=> [6,2,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 6
[5,3,2]
=> [2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,0]
=> ? = 9
[5,3,1,1]
=> [2,2,1,1,1,1,1,1]
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 8
[5,1,1,1,1,1]
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [1,1,1,0,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[4,4,2]
=> [3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,0]
=> ? = 8
[4,4,1,1]
=> [3,2,1,1,1,1,1]
=> [7,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 7
[4,3,3]
=> [4,1,1,1,1,1,1]
=> [7,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,0]
=> ? = 7
[4,3,2,1]
=> [1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0]
=> ? = 10
[4,3,1,1,1]
=> [5,1,1,1,1,1]
=> [6,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,0]
=> ? = 6
[4,2,2,1,1]
=> [5,2,1,1,1]
=> [5,2,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 5
[4,2,1,1,1,1]
=> [6,1,1,1,1]
=> [5,1,1,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,1,1,1,1,1,1]
=> [7,2,1]
=> [3,2,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[3,3,1,1,1,1]
=> [6,2,1,1]
=> [4,2,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[3,2,1,1,1,1,1]
=> [7,1,1,1]
=> [4,1,1,1,1,1,1]
=> [1,0,1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[3,1,1,1,1,1,1,1]
=> [8,1,1]
=> [3,1,1,1,1,1,1,1]
=> [1,0,1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[2,2,2,1,1,1,1]
=> [7,3]
=> [2,2,2,1,1,1,1]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,2,1,1,1,1,1,1]
=> [8,2]
=> [2,2,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[2,1,1,1,1,1,1,1,1]
=> [9,1]
=> [2,1,1,1,1,1,1,1,1]
=> [1,0,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[1,1,1,1,1,1,1,1,1,1]
=> [10]
=> [1,1,1,1,1,1,1,1,1,1]
=> [1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 1
[7,4]
=> [5,2,2,2]
=> [4,4,1,1,1]
=> [1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 4
[7,1,1,1,1]
=> [6,3,2]
=> [3,3,2,1,1,1]
=> [1,1,1,0,1,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 3
[6,3,2]
=> [4,2,1,1,1,1,1]
=> [7,2,1,1]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,0]
=> ? = 7
[6,3,1,1]
=> [5,2,1,1,1,1]
=> [6,2,1,1,1]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,0]
=> ? = 6
[6,2,2,1]
=> [5,2,2,1,1]
=> [5,3,1,1,1]
=> [1,0,1,0,1,1,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5
[5,5,1]
=> [5,3,1,1,1]
=> [5,2,2,1,1]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,1,0,1,0,0]
=> ? = 5
[5,4,2]
=> [2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 9
[5,4,1,1]
=> [2,2,2,1,1,1,1,1]
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 8
[5,3,3]
=> [3,2,1,1,1,1,1,1]
=> [8,2,1]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,1,0,0]
=> ? = 8
[4,2,2,1,1,1]
=> [6,2,1,1,1]
=> [5,2,1,1,1,1]
=> [1,0,1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 5
[4,1,1,1,1,1,1,1]
=> [8,2,1]
=> [3,2,1,1,1,1,1,1]
=> [1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 3
[3,3,1,1,1,1,1]
=> [7,2,1,1]
=> [4,2,1,1,1,1,1]
=> [1,0,1,0,1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 4
[2,2,2,2,1,1,1]
=> [7,4]
=> [2,2,2,2,1,1,1]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,0]
=> ? = 2
[2,2,1,1,1,1,1,1,1]
=> [9,2]
=> [2,2,1,1,1,1,1,1,1]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2
[9,2,1]
=> [5,4,1,1,1]
=> [5,2,2,2,1]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,1,0,0]
=> ? = 5
[7,4,1]
=> [5,3,2,1,1]
=> [5,3,2,1,1]
=> [1,0,1,0,1,1,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 5
[7,2,2,1]
=> [5,2,2,2,1]
=> [5,4,1,1,1]
=> [1,0,1,1,1,0,1,0,1,0,0,1,0,1,0,1,0,0]
=> ? = 5
[6,6]
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [1,1,1,0,1,0,1,1,0,0,0,1,0,1,0,0]
=> ? = 4
[3,3,2,2,1,1]
=> [6,4,2]
=> [3,3,2,2,1,1]
=> [1,1,1,0,1,1,0,1,0,0,0,1,0,1,0,0]
=> ? = 3
Description
The maximal height of a column in the parallelogram polyomino associated with a Dyck path.
The following 26 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000161The sum of the sizes of the right subtrees of a binary tree. St000507The number of ascents of a standard tableau. St001227The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra. St000041The number of nestings of a perfect matching. St001480The number of simple summands of the module J^2/J^3. St001291The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path. St000678The number of up steps after the last double rise of a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000053The number of valleys of the Dyck path. St001397Number of pairs of incomparable elements in a finite poset. St001558The number of transpositions that are smaller or equal to a permutation in Bruhat order. St000018The number of inversions of a permutation. St000246The number of non-inversions of a permutation. St000067The inversion number of the alternating sign matrix. St000076The rank of the alternating sign matrix in the alternating sign matrix poset. St000006The dinv of a Dyck path. St001295Gives the vector space dimension of the homomorphism space between J^2 and J^2. St000005The bounce statistic of a Dyck path. St000057The Shynar inversion number of a standard tableau. St001462The number of factors of a standard tableaux under concatenation. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000015The number of peaks of a Dyck path. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2.
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