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Your data matches 43 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
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Mp00276: Graphs —to edge-partition of biconnected components⟶ 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
([(0,1)],2)
=> [1]
=> 1 = 2 - 1
([(1,2)],3)
=> [1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> 2 = 3 - 1
([(4,5)],6)
=> [1]
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 3 = 4 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 3 = 4 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 4 = 5 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 4 = 5 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 5 = 6 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 4 = 5 - 1
Description
The length of the partition.
Matching statistic: St000147
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(load all 3 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ 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
([(0,1)],2)
=> [1]
=> [1]
=> 1 = 2 - 1
([(1,2)],3)
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [2,2,2]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 3 - 1
([(4,5)],6)
=> [1]
=> [1]
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [3]
=> 3 = 4 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [4]
=> 4 = 5 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [5]
=> 5 = 6 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [3,1,1]
=> 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> 4 = 5 - 1
Description
The largest part of an integer partition.
Matching statistic: St000288
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(load all 3 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ 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
([(0,1)],2)
=> [1]
=> 10 => 1 = 2 - 1
([(1,2)],3)
=> [1]
=> 10 => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1000 => 1 = 2 - 1
([(2,3)],4)
=> [1]
=> 10 => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> 110 => 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1000 => 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 10000 => 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 100000 => 1 = 2 - 1
([(3,4)],5)
=> [1]
=> 10 => 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> 110 => 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> 110 => 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1000 => 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 10010 => 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 10000 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 100000 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> 1000010 => 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 10010 => 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 11000 => 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 100000 => 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> 1000010 => 2 = 3 - 1
([(4,5)],6)
=> [1]
=> 10 => 1 = 2 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> 110 => 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 111110 => 5 = 6 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> 110 => 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 1110 => 3 = 4 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 1000 => 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 11110 => 4 = 5 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 10010 => 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 111110 => 5 = 6 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 100110 => 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => 4 = 5 - 1
Description
The number of ones in a binary word.
This is also known as the Hamming weight of the word.
Matching statistic: St000378
Mp00276: Graphs —to edge-partition of biconnected components⟶ 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
([(0,1)],2)
=> [1]
=> [1]
=> 1 = 2 - 1
([(1,2)],3)
=> [1]
=> [1]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> [1]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [4,1]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [6]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [2,1,1,1,1]
=> 2 = 3 - 1
([(4,5)],6)
=> [1]
=> [1]
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [3,2]
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [2]
=> 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [2,1]
=> 3 = 4 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [3,1]
=> 4 = 5 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [2,1,1]
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [3,2]
=> 5 = 6 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [4,1]
=> 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [3,3]
=> 4 = 5 - 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
Mp00276: Graphs —to edge-partition of biconnected components⟶ 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
([(0,1)],2)
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(1,2)],3)
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [[1,2,3]]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [[1,2,3]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [[1,2,3],[4]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [[1,2,3,4]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [[1,2,3],[4]]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 2 = 3 - 1
([(4,5)],6)
=> [1]
=> [[1]]
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [[1],[2]]
=> 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> 3 = 4 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [[1,2,3]]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4 = 5 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [[1,2,3],[4]]
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5 = 6 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 4 = 5 - 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001227
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
St001227: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> 2 = 3 - 1
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> 3 = 4 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> 4 = 5 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 5 = 6 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> 4 = 5 - 1
Description
The vector space dimension of the first extension group between the socle of the regular module and the Jacobson radical of the corresponding Nakayama algebra.
Matching statistic: St000157
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ 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
([(0,1)],2)
=> [1]
=> [[1]]
=> 0 = 2 - 2
([(1,2)],3)
=> [1]
=> [[1]]
=> 0 = 2 - 2
([(0,2),(1,2)],3)
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> 0 = 2 - 2
([(2,3)],4)
=> [1]
=> [[1]]
=> 0 = 2 - 2
([(1,3),(2,3)],4)
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([(0,3),(1,2)],4)
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [[1,2,3]]
=> 0 = 2 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> 0 = 2 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 2 - 2
([(3,4)],5)
=> [1]
=> [[1]]
=> 0 = 2 - 2
([(2,4),(3,4)],5)
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 5 - 2
([(1,4),(2,3)],5)
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [[1,2,3]]
=> 0 = 2 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 5 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 3 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 4 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [[1,2,3,4]]
=> 0 = 2 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 2 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 4 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 1 = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 5 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> 0 = 2 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> 1 = 3 - 2
([(4,5)],6)
=> [1]
=> [[1]]
=> 0 = 2 - 2
([(3,5),(4,5)],6)
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 5 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 6 - 2
([(2,5),(3,4)],6)
=> [1,1]
=> [[1],[2]]
=> 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 4 - 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [[1,2,3]]
=> 0 = 2 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 5 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 5 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [[1,2,3],[4]]
=> 1 = 3 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 6 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> 2 = 4 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> 3 = 5 - 2
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: St000519
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000519: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00317: Integer partitions —odd parts⟶ Binary words
St000519: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> 1 => 0 = 2 - 2
([(1,2)],3)
=> [1]
=> 1 => 0 = 2 - 2
([(0,2),(1,2)],3)
=> [1,1]
=> 11 => 1 = 3 - 2
([(0,1),(0,2),(1,2)],3)
=> [3]
=> 1 => 0 = 2 - 2
([(2,3)],4)
=> [1]
=> 1 => 0 = 2 - 2
([(1,3),(2,3)],4)
=> [1,1]
=> 11 => 1 = 3 - 2
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> 111 => 2 = 4 - 2
([(0,3),(1,2)],4)
=> [1,1]
=> 11 => 1 = 3 - 2
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> 111 => 2 = 4 - 2
([(1,2),(1,3),(2,3)],4)
=> [3]
=> 1 => 0 = 2 - 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 11 => 1 = 3 - 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> 0 => 0 = 2 - 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> 1 => 0 = 2 - 2
([(3,4)],5)
=> [1]
=> 1 => 0 = 2 - 2
([(2,4),(3,4)],5)
=> [1,1]
=> 11 => 1 = 3 - 2
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> 111 => 2 = 4 - 2
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> 1111 => 3 = 5 - 2
([(1,4),(2,3)],5)
=> [1,1]
=> 11 => 1 = 3 - 2
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> 111 => 2 = 4 - 2
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> 111 => 2 = 4 - 2
([(2,3),(2,4),(3,4)],5)
=> [3]
=> 1 => 0 = 2 - 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> 1111 => 3 = 5 - 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 11 => 1 = 3 - 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 111 => 2 = 4 - 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> 0 => 0 = 2 - 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> 01 => 1 = 3 - 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> 1 => 0 = 2 - 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 111 => 2 = 4 - 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> 11 => 1 = 3 - 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> 1111 => 3 = 5 - 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> 11 => 1 = 3 - 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> 111 => 2 = 4 - 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 11 => 1 = 3 - 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> 1 => 0 = 2 - 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> 11 => 1 = 3 - 2
([(4,5)],6)
=> [1]
=> 1 => 0 = 2 - 2
([(3,5),(4,5)],6)
=> [1,1]
=> 11 => 1 = 3 - 2
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> 111 => 2 = 4 - 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 1111 => 3 = 5 - 2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> 11111 => 4 = 6 - 2
([(2,5),(3,4)],6)
=> [1,1]
=> 11 => 1 = 3 - 2
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> 111 => 2 = 4 - 2
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> 111 => 2 = 4 - 2
([(3,4),(3,5),(4,5)],6)
=> [3]
=> 1 => 0 = 2 - 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> 1111 => 3 = 5 - 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> 1111 => 3 = 5 - 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> 11 => 1 = 3 - 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> 11111 => 4 = 6 - 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> 111 => 2 = 4 - 2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1111 => 3 = 5 - 2
Description
The largest length of a factor maximising the subword complexity.
Let $p_w(n)$ be the number of distinct factors of length $n$. Then the statistic is the largest $n$ such that $p_w(n)$ is maximal:
$$
H_w = \max\{n: p_w(n)\text{ is maximal}\}
$$
A related statistic is the number of distinct factors of arbitrary length, also known as subword complexity, [[St000294]].
Matching statistic: St001291
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00043: Integer partitions —to Dyck path⟶ Dyck paths
Mp00028: Dyck paths —reverse⟶ Dyck paths
St001291: Dyck paths ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([(1,2)],3)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([(0,2),(1,2)],3)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(2,3)],4)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([(1,3),(2,3)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
([(0,3),(1,2)],4)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
([(3,4)],5)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([(2,4),(3,4)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
([(1,4),(2,3)],5)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [1,1,1,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [1,1,1,0,1,0,0,0,1,0]
=> [1,0,1,1,1,0,1,0,0,0]
=> 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [1,1,1,0,0,0,1,1,0,0]
=> [1,1,0,0,1,1,1,0,0,0]
=> 3
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [1,1,1,1,0,1,0,0,0,0,1,0]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 3
([(4,5)],6)
=> [1]
=> [1,0,1,0]
=> [1,0,1,0]
=> 2
([(3,5),(4,5)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
([(2,5),(3,4)],6)
=> [1,1]
=> [1,0,1,1,0,0]
=> [1,1,0,0,1,0]
=> 3
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,0,1,0]
=> 4
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [1,1,1,0,0,0,1,0]
=> [1,0,1,1,1,0,0,0]
=> 2
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,0,1,0]
=> 5
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [1,1,0,1,0,0,1,0]
=> [1,0,1,1,0,1,0,0]
=> 3
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,0,1,0]
=> 6
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [1,0,1,1,0,0,1,0]
=> [1,0,1,1,0,0,1,0]
=> 4
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,0,1,1,1,0,0,1,0,0]
=> [1,1,0,1,1,0,0,0,1,0]
=> 5
Description
The number of indecomposable summands of the tensor product of two copies of the dual of the Nakayama algebra associated to a Dyck path.
Let $A$ be the Nakayama algebra associated to a Dyck path as given in [[DyckPaths/NakayamaAlgebras]]. This statistics is the number of indecomposable summands of $D(A) \otimes D(A)$, where $D(A)$ is the natural dual of $A$.
Matching statistic: St000507
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00276: Graphs —to edge-partition of biconnected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000507: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([(0,1)],2)
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
([(1,2)],3)
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
([(0,1),(0,2),(1,2)],3)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
([(2,3)],4)
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
([(0,3),(1,3),(2,3)],4)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
([(0,3),(1,2)],4)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
([(0,3),(1,2),(2,3)],4)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
([(1,2),(1,3),(2,3)],4)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1 = 2 - 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1 = 2 - 1
([(3,4)],5)
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
([(2,4),(3,4)],5)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
([(1,4),(2,4),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
([(1,4),(2,3)],5)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
([(1,4),(2,3),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
([(0,1),(2,4),(3,4)],5)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
([(2,3),(2,4),(3,4)],5)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4]
=> [[1,2,3,4]]
=> [[1],[2],[3],[4]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 2 = 3 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3 = 4 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 2 = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3 = 4 - 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 2 = 3 - 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [5]
=> [[1,2,3,4,5]]
=> [[1],[2],[3],[4],[5]]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 2 = 3 - 1
([(4,5)],6)
=> [1]
=> [[1]]
=> [[1]]
=> 1 = 2 - 1
([(3,5),(4,5)],6)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5 = 6 - 1
([(2,5),(3,4)],6)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
([(1,2),(3,5),(4,5)],6)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 3 = 4 - 1
([(3,4),(3,5),(4,5)],6)
=> [3]
=> [[1,2,3]]
=> [[1],[2],[3]]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 4 = 5 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 5 = 6 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 3 = 4 - 1
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 4 = 5 - 1
Description
The number of ascents of a standard tableau.
Entry $i$ of a standard Young tableau is an '''ascent''' if $i+1$ appears to the right or above $i$ in the tableau (with respect to the English notation for tableaux).
The following 33 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000676The number of odd rises of a Dyck path. St000734The last entry in the first row of a standard tableau. St001480The number of simple summands of the module J^2/J^3. St001039The maximal height of a column in the parallelogram polyomino associated with a Dyck path. St001462The number of factors of a standard tableaux under concatenation. St001505The number of elements generated by the Dyck path as a map in the full transformation monoid. St000684The global dimension of the LNakayama algebra associated to a Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St000053The number of valleys of the Dyck path. St001247The number of parts of a partition that are not congruent 2 modulo 3. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St001918The degree of the cyclic sieving polynomial corresponding to an integer partition. St000678The number of up steps after the last double rise of a Dyck path. St000668The least common multiple of the parts of the partition. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. 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. St001530The depth of a Dyck path. St000331The number of upper interactions of a Dyck path. St001142The projective dimension of the socle of the regular module as a bimodule in the Nakayama algebra corresponding to the Dyck path. St001169Number of simple modules with projective dimension at least two in the corresponding Nakayama algebra. St001296The maximal torsionfree index of an indecomposable non-projective module in the corresponding Nakayama algebra. St001509The degree of the standard monomial associated to a Dyck path relative to the trivial lower boundary. St001622The number of join-irreducible elements of a lattice. St001613The binary logarithm of the size of the center of a lattice. St001615The number of join prime elements of a lattice. St001617The dimension of the space of valuations of a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001621The number of atoms of a lattice. St001624The breadth of a lattice. St000640The rank of the largest boolean interval in a poset.
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