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Your data matches 52 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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Mp00166: Signed permutations —even cycle type⟶ 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 = 0 + 1
[1,2] => [1,1]
=> 2 = 1 + 1
[1,-2] => [1]
=> 1 = 0 + 1
[-1,2] => [1]
=> 1 = 0 + 1
[2,1] => [2]
=> 1 = 0 + 1
[-2,-1] => [2]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> 3 = 2 + 1
[1,2,-3] => [1,1]
=> 2 = 1 + 1
[1,-2,3] => [1,1]
=> 2 = 1 + 1
[1,-2,-3] => [1]
=> 1 = 0 + 1
[-1,2,3] => [1,1]
=> 2 = 1 + 1
[-1,2,-3] => [1]
=> 1 = 0 + 1
[-1,-2,3] => [1]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> 2 = 1 + 1
[1,3,-2] => [1]
=> 1 = 0 + 1
[1,-3,2] => [1]
=> 1 = 0 + 1
[1,-3,-2] => [2,1]
=> 2 = 1 + 1
[-1,3,2] => [2]
=> 1 = 0 + 1
[-1,-3,-2] => [2]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> 2 = 1 + 1
[2,1,-3] => [2]
=> 1 = 0 + 1
[2,-1,3] => [1]
=> 1 = 0 + 1
[-2,1,3] => [1]
=> 1 = 0 + 1
[-2,-1,3] => [2,1]
=> 2 = 1 + 1
[-2,-1,-3] => [2]
=> 1 = 0 + 1
[2,3,1] => [3]
=> 1 = 0 + 1
[2,-3,-1] => [3]
=> 1 = 0 + 1
[-2,3,-1] => [3]
=> 1 = 0 + 1
[-2,-3,1] => [3]
=> 1 = 0 + 1
[3,1,2] => [3]
=> 1 = 0 + 1
[3,-1,-2] => [3]
=> 1 = 0 + 1
[-3,1,-2] => [3]
=> 1 = 0 + 1
[-3,-1,2] => [3]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> 2 = 1 + 1
[3,2,-1] => [1]
=> 1 = 0 + 1
[3,-2,1] => [2]
=> 1 = 0 + 1
[-3,2,1] => [1]
=> 1 = 0 + 1
[-3,2,-1] => [2,1]
=> 2 = 1 + 1
[-3,-2,-1] => [2]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> 4 = 3 + 1
[1,2,3,-4] => [1,1,1]
=> 3 = 2 + 1
[1,2,-3,4] => [1,1,1]
=> 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> 2 = 1 + 1
[1,-2,3,4] => [1,1,1]
=> 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> 2 = 1 + 1
[1,-2,-3,4] => [1,1]
=> 2 = 1 + 1
[1,-2,-3,-4] => [1]
=> 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
=> 3 = 2 + 1
[-1,2,3,-4] => [1,1]
=> 2 = 1 + 1
[-1,2,-3,4] => [1,1]
=> 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000157
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(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 0
[1,2] => [1,1]
=> [[1],[2]]
=> 1
[1,-2] => [1]
=> [[1]]
=> 0
[-1,2] => [1]
=> [[1]]
=> 0
[2,1] => [2]
=> [[1,2]]
=> 0
[-2,-1] => [2]
=> [[1,2]]
=> 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,2,-3] => [1,1]
=> [[1],[2]]
=> 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> 1
[1,-2,-3] => [1]
=> [[1]]
=> 0
[-1,2,3] => [1,1]
=> [[1],[2]]
=> 1
[-1,2,-3] => [1]
=> [[1]]
=> 0
[-1,-2,3] => [1]
=> [[1]]
=> 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> 1
[1,3,-2] => [1]
=> [[1]]
=> 0
[1,-3,2] => [1]
=> [[1]]
=> 0
[1,-3,-2] => [2,1]
=> [[1,2],[3]]
=> 1
[-1,3,2] => [2]
=> [[1,2]]
=> 0
[-1,-3,-2] => [2]
=> [[1,2]]
=> 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> 1
[2,1,-3] => [2]
=> [[1,2]]
=> 0
[2,-1,3] => [1]
=> [[1]]
=> 0
[-2,1,3] => [1]
=> [[1]]
=> 0
[-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> 0
[2,3,1] => [3]
=> [[1,2,3]]
=> 0
[2,-3,-1] => [3]
=> [[1,2,3]]
=> 0
[-2,3,-1] => [3]
=> [[1,2,3]]
=> 0
[-2,-3,1] => [3]
=> [[1,2,3]]
=> 0
[3,1,2] => [3]
=> [[1,2,3]]
=> 0
[3,-1,-2] => [3]
=> [[1,2,3]]
=> 0
[-3,1,-2] => [3]
=> [[1,2,3]]
=> 0
[-3,-1,2] => [3]
=> [[1,2,3]]
=> 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> 1
[3,2,-1] => [1]
=> [[1]]
=> 0
[3,-2,1] => [2]
=> [[1,2]]
=> 0
[-3,2,1] => [1]
=> [[1]]
=> 0
[-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> 0
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> 2
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> 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: St000147
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(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ 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%
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]
=> [1]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2] => [1]
=> [1]
=> 1 = 0 + 1
[-1,2] => [1]
=> [1]
=> 1 = 0 + 1
[2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[-2,-1] => [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [3]
=> 3 = 2 + 1
[1,2,-3] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,3] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,-3] => [1]
=> [1]
=> 1 = 0 + 1
[-1,2,3] => [1,1]
=> [2]
=> 2 = 1 + 1
[-1,2,-3] => [1]
=> [1]
=> 1 = 0 + 1
[-1,-2,3] => [1]
=> [1]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[1,3,-2] => [1]
=> [1]
=> 1 = 0 + 1
[1,-3,2] => [1]
=> [1]
=> 1 = 0 + 1
[1,-3,-2] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[-1,3,2] => [2]
=> [1,1]
=> 1 = 0 + 1
[-1,-3,-2] => [2]
=> [1,1]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[2,1,-3] => [2]
=> [1,1]
=> 1 = 0 + 1
[2,-1,3] => [1]
=> [1]
=> 1 = 0 + 1
[-2,1,3] => [1]
=> [1]
=> 1 = 0 + 1
[-2,-1,3] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[-2,-1,-3] => [2]
=> [1,1]
=> 1 = 0 + 1
[2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[2,-3,-1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[-2,3,-1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[-2,-3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,-1,-2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[-3,1,-2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[-3,-1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[3,2,-1] => [1]
=> [1]
=> 1 = 0 + 1
[3,-2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[-3,2,1] => [1]
=> [1]
=> 1 = 0 + 1
[-3,2,-1] => [2,1]
=> [2,1]
=> 2 = 1 + 1
[-3,-2,-1] => [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [4]
=> 4 = 3 + 1
[1,2,3,-4] => [1,1,1]
=> [3]
=> 3 = 2 + 1
[1,2,-3,4] => [1,1,1]
=> [3]
=> 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,3,4] => [1,1,1]
=> [3]
=> 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,-3,4] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,-3,-4] => [1]
=> [1]
=> 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
=> [3]
=> 3 = 2 + 1
[-1,2,3,-4] => [1,1]
=> [2]
=> 2 = 1 + 1
[-1,2,-3,4] => [1,1]
=> [2]
=> 2 = 1 + 1
Description
The largest part of an integer partition.
Matching statistic: St000288
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ 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 = 0 + 1
[1,2] => [1,1]
=> 110 => 2 = 1 + 1
[1,-2] => [1]
=> 10 => 1 = 0 + 1
[-1,2] => [1]
=> 10 => 1 = 0 + 1
[2,1] => [2]
=> 100 => 1 = 0 + 1
[-2,-1] => [2]
=> 100 => 1 = 0 + 1
[1,2,3] => [1,1,1]
=> 1110 => 3 = 2 + 1
[1,2,-3] => [1,1]
=> 110 => 2 = 1 + 1
[1,-2,3] => [1,1]
=> 110 => 2 = 1 + 1
[1,-2,-3] => [1]
=> 10 => 1 = 0 + 1
[-1,2,3] => [1,1]
=> 110 => 2 = 1 + 1
[-1,2,-3] => [1]
=> 10 => 1 = 0 + 1
[-1,-2,3] => [1]
=> 10 => 1 = 0 + 1
[1,3,2] => [2,1]
=> 1010 => 2 = 1 + 1
[1,3,-2] => [1]
=> 10 => 1 = 0 + 1
[1,-3,2] => [1]
=> 10 => 1 = 0 + 1
[1,-3,-2] => [2,1]
=> 1010 => 2 = 1 + 1
[-1,3,2] => [2]
=> 100 => 1 = 0 + 1
[-1,-3,-2] => [2]
=> 100 => 1 = 0 + 1
[2,1,3] => [2,1]
=> 1010 => 2 = 1 + 1
[2,1,-3] => [2]
=> 100 => 1 = 0 + 1
[2,-1,3] => [1]
=> 10 => 1 = 0 + 1
[-2,1,3] => [1]
=> 10 => 1 = 0 + 1
[-2,-1,3] => [2,1]
=> 1010 => 2 = 1 + 1
[-2,-1,-3] => [2]
=> 100 => 1 = 0 + 1
[2,3,1] => [3]
=> 1000 => 1 = 0 + 1
[2,-3,-1] => [3]
=> 1000 => 1 = 0 + 1
[-2,3,-1] => [3]
=> 1000 => 1 = 0 + 1
[-2,-3,1] => [3]
=> 1000 => 1 = 0 + 1
[3,1,2] => [3]
=> 1000 => 1 = 0 + 1
[3,-1,-2] => [3]
=> 1000 => 1 = 0 + 1
[-3,1,-2] => [3]
=> 1000 => 1 = 0 + 1
[-3,-1,2] => [3]
=> 1000 => 1 = 0 + 1
[3,2,1] => [2,1]
=> 1010 => 2 = 1 + 1
[3,2,-1] => [1]
=> 10 => 1 = 0 + 1
[3,-2,1] => [2]
=> 100 => 1 = 0 + 1
[-3,2,1] => [1]
=> 10 => 1 = 0 + 1
[-3,2,-1] => [2,1]
=> 1010 => 2 = 1 + 1
[-3,-2,-1] => [2]
=> 100 => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> 11110 => 4 = 3 + 1
[1,2,3,-4] => [1,1,1]
=> 1110 => 3 = 2 + 1
[1,2,-3,4] => [1,1,1]
=> 1110 => 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> 110 => 2 = 1 + 1
[1,-2,3,4] => [1,1,1]
=> 1110 => 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> 110 => 2 = 1 + 1
[1,-2,-3,4] => [1,1]
=> 110 => 2 = 1 + 1
[1,-2,-3,-4] => [1]
=> 10 => 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
=> 1110 => 3 = 2 + 1
[-1,2,3,-4] => [1,1]
=> 110 => 2 = 1 + 1
[-1,2,-3,4] => [1,1]
=> 110 => 2 = 1 + 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000378
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00322: Integer partitions —Loehr-Warrington⟶ Integer partitions
St000378: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [1]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2] => [1]
=> [1]
=> 1 = 0 + 1
[-1,2] => [1]
=> [1]
=> 1 = 0 + 1
[2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[-2,-1] => [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[1,2,-3] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,3] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,-3] => [1]
=> [1]
=> 1 = 0 + 1
[-1,2,3] => [1,1]
=> [2]
=> 2 = 1 + 1
[-1,2,-3] => [1]
=> [1]
=> 1 = 0 + 1
[-1,-2,3] => [1]
=> [1]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [3]
=> 2 = 1 + 1
[1,3,-2] => [1]
=> [1]
=> 1 = 0 + 1
[1,-3,2] => [1]
=> [1]
=> 1 = 0 + 1
[1,-3,-2] => [2,1]
=> [3]
=> 2 = 1 + 1
[-1,3,2] => [2]
=> [1,1]
=> 1 = 0 + 1
[-1,-3,-2] => [2]
=> [1,1]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [3]
=> 2 = 1 + 1
[2,1,-3] => [2]
=> [1,1]
=> 1 = 0 + 1
[2,-1,3] => [1]
=> [1]
=> 1 = 0 + 1
[-2,1,3] => [1]
=> [1]
=> 1 = 0 + 1
[-2,-1,3] => [2,1]
=> [3]
=> 2 = 1 + 1
[-2,-1,-3] => [2]
=> [1,1]
=> 1 = 0 + 1
[2,3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[2,-3,-1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[-2,3,-1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[-2,-3,1] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,-1,-2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[-3,1,-2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[-3,-1,2] => [3]
=> [1,1,1]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [3]
=> 2 = 1 + 1
[3,2,-1] => [1]
=> [1]
=> 1 = 0 + 1
[3,-2,1] => [2]
=> [1,1]
=> 1 = 0 + 1
[-3,2,1] => [1]
=> [1]
=> 1 = 0 + 1
[-3,2,-1] => [2,1]
=> [3]
=> 2 = 1 + 1
[-3,-2,-1] => [2]
=> [1,1]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [3,1]
=> 4 = 3 + 1
[1,2,3,-4] => [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[1,2,-3,4] => [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,3,4] => [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,-3,4] => [1,1]
=> [2]
=> 2 = 1 + 1
[1,-2,-3,-4] => [1]
=> [1]
=> 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
=> [2,1]
=> 3 = 2 + 1
[-1,2,3,-4] => [1,1]
=> [2]
=> 2 = 1 + 1
[-1,2,-3,4] => [1,1]
=> [2]
=> 2 = 1 + 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: St000733
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,2] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2] => [1]
=> [[1]]
=> 1 = 0 + 1
[-1,2] => [1]
=> [[1]]
=> 1 = 0 + 1
[2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[-2,-1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,-3] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2,-3] => [1]
=> [[1]]
=> 1 = 0 + 1
[-1,2,3] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-1,2,-3] => [1]
=> [[1]]
=> 1 = 0 + 1
[-1,-2,3] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[1,3,-2] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,-3,2] => [1]
=> [[1]]
=> 1 = 0 + 1
[1,-3,-2] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[-1,3,2] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[-1,-3,-2] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[2,1,-3] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,-1,3] => [1]
=> [[1]]
=> 1 = 0 + 1
[-2,1,3] => [1]
=> [[1]]
=> 1 = 0 + 1
[-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[2,3,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[2,-3,-1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[-2,3,-1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[-2,-3,1] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[3,1,2] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[3,-1,-2] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[-3,1,-2] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[-3,-1,2] => [3]
=> [[1,2,3]]
=> 1 = 0 + 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[3,2,-1] => [1]
=> [[1]]
=> 1 = 0 + 1
[3,-2,1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[-3,2,1] => [1]
=> [[1]]
=> 1 = 0 + 1
[-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> 2 = 1 + 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 3 + 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> 3 = 2 + 1
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> 2 = 1 + 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000546
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000546: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2] => [1]
=> [[1]]
=> [1] => 0
[-1,2] => [1]
=> [[1]]
=> [1] => 0
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[-1,2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,-2,3] => [1]
=> [[1]]
=> [1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[1,3,-2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[-1,3,2] => [2]
=> [[1,2]]
=> [1,2] => 0
[-1,-3,-2] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[2,1,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,-1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> [1,2] => 0
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[-3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[3,2,-1] => [1]
=> [[1]]
=> [1] => 0
[3,-2,1] => [2]
=> [[1,2]]
=> [1,2] => 0
[-3,2,1] => [1]
=> [[1]]
=> [1] => 0
[-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 3
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 2
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 1
Description
The number of global descents of a permutation.
The global descents are the integers in the set
$$C(\pi)=\{i\in [n-1] : \forall 1 \leq j \leq i < k \leq n :\quad \pi(j) > \pi(k)\}.$$
In particular, if $i\in C(\pi)$ then $i$ is a descent.
For the number of global ascents, see [[St000234]].
Matching statistic: St001372
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00200: Binary words —twist⟶ Binary words
St001372: 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
Mp00200: Binary words —twist⟶ Binary words
St001372: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> 10 => 00 => 0
[1,2] => [1,1]
=> 110 => 010 => 1
[1,-2] => [1]
=> 10 => 00 => 0
[-1,2] => [1]
=> 10 => 00 => 0
[2,1] => [2]
=> 100 => 000 => 0
[-2,-1] => [2]
=> 100 => 000 => 0
[1,2,3] => [1,1,1]
=> 1110 => 0110 => 2
[1,2,-3] => [1,1]
=> 110 => 010 => 1
[1,-2,3] => [1,1]
=> 110 => 010 => 1
[1,-2,-3] => [1]
=> 10 => 00 => 0
[-1,2,3] => [1,1]
=> 110 => 010 => 1
[-1,2,-3] => [1]
=> 10 => 00 => 0
[-1,-2,3] => [1]
=> 10 => 00 => 0
[1,3,2] => [2,1]
=> 1010 => 0010 => 1
[1,3,-2] => [1]
=> 10 => 00 => 0
[1,-3,2] => [1]
=> 10 => 00 => 0
[1,-3,-2] => [2,1]
=> 1010 => 0010 => 1
[-1,3,2] => [2]
=> 100 => 000 => 0
[-1,-3,-2] => [2]
=> 100 => 000 => 0
[2,1,3] => [2,1]
=> 1010 => 0010 => 1
[2,1,-3] => [2]
=> 100 => 000 => 0
[2,-1,3] => [1]
=> 10 => 00 => 0
[-2,1,3] => [1]
=> 10 => 00 => 0
[-2,-1,3] => [2,1]
=> 1010 => 0010 => 1
[-2,-1,-3] => [2]
=> 100 => 000 => 0
[2,3,1] => [3]
=> 1000 => 0000 => 0
[2,-3,-1] => [3]
=> 1000 => 0000 => 0
[-2,3,-1] => [3]
=> 1000 => 0000 => 0
[-2,-3,1] => [3]
=> 1000 => 0000 => 0
[3,1,2] => [3]
=> 1000 => 0000 => 0
[3,-1,-2] => [3]
=> 1000 => 0000 => 0
[-3,1,-2] => [3]
=> 1000 => 0000 => 0
[-3,-1,2] => [3]
=> 1000 => 0000 => 0
[3,2,1] => [2,1]
=> 1010 => 0010 => 1
[3,2,-1] => [1]
=> 10 => 00 => 0
[3,-2,1] => [2]
=> 100 => 000 => 0
[-3,2,1] => [1]
=> 10 => 00 => 0
[-3,2,-1] => [2,1]
=> 1010 => 0010 => 1
[-3,-2,-1] => [2]
=> 100 => 000 => 0
[1,2,3,4] => [1,1,1,1]
=> 11110 => 01110 => 3
[1,2,3,-4] => [1,1,1]
=> 1110 => 0110 => 2
[1,2,-3,4] => [1,1,1]
=> 1110 => 0110 => 2
[1,2,-3,-4] => [1,1]
=> 110 => 010 => 1
[1,-2,3,4] => [1,1,1]
=> 1110 => 0110 => 2
[1,-2,3,-4] => [1,1]
=> 110 => 010 => 1
[1,-2,-3,4] => [1,1]
=> 110 => 010 => 1
[1,-2,-3,-4] => [1]
=> 10 => 00 => 0
[-1,2,3,4] => [1,1,1]
=> 1110 => 0110 => 2
[-1,2,3,-4] => [1,1]
=> 110 => 010 => 1
[-1,2,-3,4] => [1,1]
=> 110 => 010 => 1
Description
The length of a longest cyclic run of ones of a binary word.
Consider the binary word as a cyclic arrangement of ones and zeros. Then this statistic is the length of the longest continuous sequence of ones in this arrangement.
Matching statistic: St001777
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St001777: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 0
[1,2] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,-2] => [1]
=> [[1]]
=> [1] => 0
[-1,2] => [1]
=> [[1]]
=> [1] => 0
[2,1] => [2]
=> [[1,2]]
=> [2] => 0
[-2,-1] => [2]
=> [[1,2]]
=> [2] => 0
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 2
[1,2,-3] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,-2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[-1,2,-3] => [1]
=> [[1]]
=> [1] => 0
[-1,-2,3] => [1]
=> [[1]]
=> [1] => 0
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[1,3,-2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,2] => [1]
=> [[1]]
=> [1] => 0
[1,-3,-2] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[-1,3,2] => [2]
=> [[1,2]]
=> [2] => 0
[-1,-3,-2] => [2]
=> [[1,2]]
=> [2] => 0
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[2,1,-3] => [2]
=> [[1,2]]
=> [2] => 0
[2,-1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,1,3] => [1]
=> [[1]]
=> [1] => 0
[-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> [2] => 0
[2,3,1] => [3]
=> [[1,2,3]]
=> [3] => 0
[2,-3,-1] => [3]
=> [[1,2,3]]
=> [3] => 0
[-2,3,-1] => [3]
=> [[1,2,3]]
=> [3] => 0
[-2,-3,1] => [3]
=> [[1,2,3]]
=> [3] => 0
[3,1,2] => [3]
=> [[1,2,3]]
=> [3] => 0
[3,-1,-2] => [3]
=> [[1,2,3]]
=> [3] => 0
[-3,1,-2] => [3]
=> [[1,2,3]]
=> [3] => 0
[-3,-1,2] => [3]
=> [[1,2,3]]
=> [3] => 0
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[3,2,-1] => [1]
=> [[1]]
=> [1] => 0
[3,-2,1] => [2]
=> [[1,2]]
=> [2] => 0
[-3,2,1] => [1]
=> [[1]]
=> [1] => 0
[-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> [2] => 0
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 3
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 2
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 2
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 2
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> [1] => 0
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 2
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [1,1] => 1
Description
The number of weak descents in an integer composition.
A weak descent of an integer composition $\alpha=(a_1, \dots, a_n)$ is an index $1\leq i < n$ such that $a_i \geq a_{i+1}$.
Matching statistic: St000007
Mp00166: Signed permutations —even cycle type⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000007: Permutations ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
[1] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,2] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-1,2] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[2,1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[1,2,3] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[1,2,-3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,-3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-1,2,3] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[-1,2,-3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-1,-2,3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,3,2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[1,3,-2] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,-3,2] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[1,-3,-2] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[-1,3,2] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[-1,-3,-2] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[2,1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[2,1,-3] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[2,-1,3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-2,1,3] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-2,-1,3] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[-2,-1,-3] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[2,3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[2,-3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[-2,3,-1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[-2,-3,1] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[3,1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[3,-1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[-3,1,-2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[-3,-1,2] => [3]
=> [[1,2,3]]
=> [1,2,3] => 1 = 0 + 1
[3,2,1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[3,2,-1] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[3,-2,1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[-3,2,1] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-3,2,-1] => [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 2 = 1 + 1
[-3,-2,-1] => [2]
=> [[1,2]]
=> [1,2] => 1 = 0 + 1
[1,2,3,4] => [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 4 = 3 + 1
[1,2,3,-4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[1,2,-3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[1,2,-3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[1,-2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[1,-2,-3,-4] => [1]
=> [[1]]
=> [1] => 1 = 0 + 1
[-1,2,3,4] => [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 3 = 2 + 1
[-1,2,3,-4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
[-1,2,-3,4] => [1,1]
=> [[1],[2]]
=> [2,1] => 2 = 1 + 1
Description
The number of saliances of the permutation.
A saliance is a right-to-left maximum. This can be described as an occurrence of the mesh pattern $([1], {(1,1)})$, i.e., the upper right quadrant is shaded, see [1].
The following 42 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000507The number of ascents of a standard tableau. St000734The last entry in the first row of a standard tableau. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001462The number of factors of a standard tableaux under concatenation. St000806The semiperimeter of the associated bargraph. St000676The number of odd rises of a Dyck path. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St000105The number of blocks in the set partition. St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St000155The number of exceedances (also excedences) of a permutation. St000331The number of upper interactions of a Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. 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. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St000015The number of peaks of a Dyck path. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001489The maximum of the number of descents and the number of inverse descents. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000653The last descent of a permutation. 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. St000021The number of descents of a permutation. St000325The width of the tree associated to a permutation. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St000006The dinv of a Dyck path. St001152The number of pairs with even minimum in a perfect matching. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000925The number of topologically connected components of a set partition. St000809The reduced reflection length of the permutation. St000678The number of up steps after the last double rise of a Dyck path. St000216The absolute length of a permutation. St000354The number of recoils of a permutation. St000541The number of indices greater than or equal to 2 of a permutation such that all smaller indices appear to its right. St000831The number of indices that are either descents or recoils. St001061The number of indices that are both descents and recoils of a permutation. St000519The largest length of a factor maximising the subword complexity. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. St001491The number of indecomposable projective-injective modules in the algebra corresponding to a subset. St001250The number of parts of a partition that are not congruent 0 modulo 3.
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