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Your data matches 216 different statistics following compositions of up to 3 maps.
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Matching statistic: St000010
(load all 14 compositions to match this statistic)
(load all 14 compositions to match this statistic)
Mp00037: Graphs —to partition of connected 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
([],2)
=> [1,1]
=> 2 = 1 + 1
([],3)
=> [1,1,1]
=> 3 = 2 + 1
([(1,2)],3)
=> [2,1]
=> 2 = 1 + 1
([],4)
=> [1,1,1,1]
=> 4 = 3 + 1
([(2,3)],4)
=> [2,1,1]
=> 3 = 2 + 1
([(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> 2 = 1 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 2 = 1 + 1
([],5)
=> [1,1,1,1,1]
=> 5 = 4 + 1
([(3,4)],5)
=> [2,1,1,1]
=> 4 = 3 + 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 2 = 1 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 2 = 1 + 1
([],6)
=> [1,1,1,1,1,1]
=> 6 = 5 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> 5 = 4 + 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 4 = 3 + 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 2 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> 4 = 3 + 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 3 = 2 + 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> 3 = 2 + 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 4 = 3 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> 2 = 1 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 2 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 3 = 2 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> 2 = 1 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> 2 = 1 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 3 = 2 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> 2 = 1 + 1
Description
The length of the partition.
Matching statistic: St000147
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ 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
([],2)
=> [1,1]
=> [1]
=> [1]
=> 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [5]
=> 5
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [4]
=> 4
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [3,1]
=> 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [3]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [2,1]
=> 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
Description
The largest part of an integer partition.
Matching statistic: St000288
(load all 15 compositions to match this statistic)
(load all 15 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ 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%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000288: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [1]
=> 10 => 1
([],3)
=> [1,1,1]
=> [1,1]
=> 110 => 2
([(1,2)],3)
=> [2,1]
=> [1]
=> 10 => 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1110 => 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 110 => 2
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 100 => 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 10 => 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1110 => 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1010 => 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 100 => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 110 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 10 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 100 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 10 => 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 111110 => 5
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 11110 => 4
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 10110 => 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1010 => 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1110 => 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1000 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 110 => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 1100 => 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1010 => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 100 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 10 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 10 => 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)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ 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%
Mp00202: Integer partitions —first row removal⟶ 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%
Values
([],2)
=> [1,1]
=> [1]
=> [1]
=> 1
([],3)
=> [1,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [1]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [2]
=> 2
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [3]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [2]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [1]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [3,2]
=> 5
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [3,1]
=> 4
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [2,2]
=> 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [2,1]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [1,1,1]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [2]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [4]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [3]
=> 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [1,1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [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 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ 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%
Mp00202: Integer partitions —first row removal⟶ 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%
Values
([],2)
=> [1,1]
=> [1]
=> [[1]]
=> 1
([],3)
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(1,2)],3)
=> [2,1]
=> [1]
=> [[1]]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [[1,2]]
=> 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [[1,2]]
=> 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [[1,2]]
=> 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 5
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 4
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [[1,2],[3],[4]]
=> 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [[1,2]]
=> 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [[1,2,3]]
=> 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [[1,2]]
=> 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [[1,2],[3]]
=> 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [[1,2]]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000876
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000876: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [1]
=> 1 => 1
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 2
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 2
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 5
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 4
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 1
Description
The number of factors in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of factors in the Catalan factorisation, that is, $\ell + m$ if the middle Dyck word is empty and $\ell + 1 + m$ otherwise.
Matching statistic: St000885
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000885: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
St000885: Binary words ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [1]
=> 1 => 1
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 2
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 3
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 2
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 4
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 2
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 5
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 4
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 2
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 1
Description
The number of critical steps in the Catalan decomposition of a binary word.
Every binary word can be written in a unique way as $(\mathcal D 0)^\ell \mathcal D (1 \mathcal D)^m$, where $\mathcal D$ is the set of Dyck words. This is the Catalan factorisation, see [1, sec.9.1.2].
This statistic records the number of critical steps $\ell + m$ in the Catalan factorisation.
The distribution of this statistic on words of length $n$ is
$$
(n+1)q^n+\sum_{\substack{k=0\\\text{k even}}}^{n-2} \frac{(n-1-k)^2}{1+k/2}\binom{n}{k/2}q^{n-2-k}.
$$
Matching statistic: St000157
(load all 7 compositions to match this statistic)
(load all 7 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000157: Standard tableaux ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> 4 = 5 - 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> 3 = 4 - 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> [[1,4],[2],[3]]
=> 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> [[1],[2],[3]]
=> 2 = 3 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> [[1,2,3]]
=> 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> [[1],[2]]
=> 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> [[1,3],[2]]
=> 1 = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> [[1,2]]
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> [[1]]
=> 0 = 1 - 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: St000519
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ 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%
Mp00202: Integer partitions —first row removal⟶ 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%
Values
([],2)
=> [1,1]
=> [1]
=> 1 => 0 = 1 - 1
([],3)
=> [1,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1 => 0 = 1 - 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 111 => 2 = 3 - 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0 = 1 - 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 0 => 0 = 1 - 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1 => 0 = 1 - 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 3 = 4 - 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 111 => 2 = 3 - 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0 = 1 - 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 01 => 1 = 2 - 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0 = 1 - 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0 = 1 - 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1 => 0 = 1 - 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0 = 1 - 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 0 => 0 = 1 - 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1 => 0 = 1 - 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 11111 => 4 = 5 - 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1111 => 3 = 4 - 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 2 = 3 - 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 011 => 2 = 3 - 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 1 = 2 - 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 111 => 2 = 3 - 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0 = 1 - 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 1 => 0 = 1 - 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 11 => 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 00 => 1 = 2 - 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0 = 1 - 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 01 => 1 = 2 - 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 0 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1 => 0 = 1 - 1
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: St000806
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Mp00317: Integer partitions —odd parts⟶ Binary words
Mp00178: Binary words —to composition⟶ Integer compositions
St000806: Integer compositions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([],3)
=> [1,1,1]
=> 111 => [1,1,1,1] => 5 = 2 + 3
([(1,2)],3)
=> [2,1]
=> 01 => [2,1] => 4 = 1 + 3
([],4)
=> [1,1,1,1]
=> 1111 => [1,1,1,1,1] => 6 = 3 + 3
([(2,3)],4)
=> [2,1,1]
=> 011 => [2,1,1] => 5 = 2 + 3
([(1,3),(2,3)],4)
=> [3,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(0,3),(1,2)],4)
=> [2,2]
=> 00 => [3] => 4 = 1 + 3
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([],5)
=> [1,1,1,1,1]
=> 11111 => [1,1,1,1,1,1] => 7 = 4 + 3
([(3,4)],5)
=> [2,1,1,1]
=> 0111 => [2,1,1,1] => 6 = 3 + 3
([(2,4),(3,4)],5)
=> [3,1,1]
=> 111 => [1,1,1,1] => 5 = 2 + 3
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 01 => [2,1] => 4 = 1 + 3
([(1,4),(2,3)],5)
=> [2,2,1]
=> 001 => [3,1] => 5 = 2 + 3
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 01 => [2,1] => 4 = 1 + 3
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 10 => [1,2] => 4 = 1 + 3
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 111 => [1,1,1,1] => 5 = 2 + 3
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 01 => [2,1] => 4 = 1 + 3
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 01 => [2,1] => 4 = 1 + 3
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 01 => [2,1] => 4 = 1 + 3
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10 => [1,2] => 4 = 1 + 3
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 01 => [2,1] => 4 = 1 + 3
([],6)
=> [1,1,1,1,1,1]
=> 111111 => [1,1,1,1,1,1,1] => 8 = 5 + 3
([(4,5)],6)
=> [2,1,1,1,1]
=> 01111 => [2,1,1,1,1] => 7 = 4 + 3
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1111 => [1,1,1,1,1] => 6 = 3 + 3
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 011 => [2,1,1] => 5 = 2 + 3
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> 0011 => [3,1,1] => 6 = 3 + 3
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 011 => [2,1,1] => 5 = 2 + 3
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> 101 => [1,2,1] => 5 = 2 + 3
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1111 => [1,1,1,1,1] => 6 = 3 + 3
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> 00 => [3] => 4 = 1 + 3
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 011 => [2,1,1] => 5 = 2 + 3
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 011 => [2,1,1] => 5 = 2 + 3
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> 11 => [1,1,1] => 4 = 1 + 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 011 => [2,1,1] => 5 = 2 + 3
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> 000 => [4] => 5 = 2 + 3
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> 00 => [3] => 4 = 1 + 3
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 101 => [1,2,1] => 5 = 2 + 3
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> 00 => [3] => 4 = 1 + 3
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> 11 => [1,1,1] => 4 = 1 + 3
Description
The semiperimeter of the associated bargraph.
Interpret the composition as the sequence of heights of the bars of a bargraph. This statistic is the semiperimeter of the polygon determined by the axis and the bargraph. Put differently, it is the sum of the number of up steps and the number of horizontal steps when regarding the bargraph as a path with up, horizontal and down steps.
The following 206 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St000329The number of evenly positioned ascents of the Dyck path, with the initial position equal to 1. St001508The degree of the standard monomial associated to a Dyck path relative to the diagonal boundary. 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. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000546The number of global descents of a permutation. St001777The number of weak descents in an integer composition. St000007The number of saliances of the permutation. St000326The position of the first one in a binary word after appending a 1 at the end. St000382The first part of an integer composition. St000507The number of ascents of a standard tableau. St000734The last entry in the first row of a standard tableau. St000297The number of leading ones in a binary word. St000757The length of the longest weakly inreasing subsequence of parts of an integer composition. St000765The number of weak records in an integer composition. St001372The length of a longest cyclic run of ones of a binary word. St000676The number of odd rises of a Dyck path. 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. St000383The last part of an integer composition. St000439The position of the first down step of a Dyck path. St001176The size of a partition minus its first part. St000678The number of up steps after the last double rise of a Dyck path. St000925The number of topologically connected components of a set partition. St000011The number of touch points (or returns) of a Dyck path. St000053The number of valleys of the Dyck path. St001068Number of torsionless simple modules in the corresponding Nakayama algebra. St001167The number of simple modules that appear as the top of an indecomposable non-projective modules that is reflexive in the corresponding Nakayama algebra. St000809The reduced reflection length of the permutation. St000617The number of global maxima of a Dyck path. St000773The multiplicity of the largest Laplacian eigenvalue in a graph. St001733The number of weak left to right maxima of a Dyck path. St000258The burning number of a graph. St000273The domination number of a graph. St000544The cop number of a graph. St000916The packing number of a graph. St001322The size of a minimal independent dominating set in a graph. St001337The upper domination number of a graph. St001338The upper irredundance number of a graph. St001829The common independence number of a graph. St000025The number of initial rises of a Dyck path. St000026The position of the first return of a Dyck path. St000319The spin of an integer partition. St000320The dinv adjustment of an integer partition. St000090The variation of a composition. St000105The number of blocks in the set partition. St000172The Grundy number of a graph. St000722The number of different neighbourhoods in a graph. St001029The size of the core of a graph. St001135The projective dimension of the first simple module in the Nakayama algebra corresponding to the Dyck path. St001581The achromatic number of a graph. St001363The Euler characteristic of a graph according to Knill. St000363The number of minimal vertex covers of a graph. St001304The number of maximally independent sets of vertices of a graph. St001458The rank of the adjacency matrix of a graph. St000287The number of connected components of a graph. St000286The number of connected components of the complement of a graph. St001316The domatic number of a graph. St000786The maximal number of occurrences of a colour in a proper colouring of a graph. St001828The Euler characteristic of a graph. St000097The order of the largest clique of the graph. St000098The chromatic number of a graph. St001339The irredundance number of a graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000553The number of blocks of a graph. St001340The cardinality of a minimal non-edge isolating set of a graph. St000093The cardinality of a maximal independent set of vertices of a graph. St000989The number of final rises of a permutation. St001330The hat guessing number of a graph. 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. St001570The minimal number of edges to add to make a graph Hamiltonian. St000006The dinv of a Dyck path. St001373The logarithm of the number of winning configurations of the lights out game on a graph. St000155The number of exceedances (also excedences) of a permutation. St000216The absolute length 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. St001238The number of simple modules S such that the Auslander-Reiten translate of S is isomorphic to the Nakayama functor applied to the second syzygy of S. 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. St000312The number of leaves in a graph. St001250The number of parts of a partition that are not congruent 0 modulo 3. St000454The largest eigenvalue of a graph if it is integral. St001084The number of occurrences of the vincular pattern |1-23 in a permutation. St001087The number of occurrences of the vincular pattern |12-3 in a permutation. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St001645The pebbling number of a connected graph. St000838The number of terminal right-hand endpoints when the vertices are written in order. St000654The first descent of a permutation. St000482The (zero)-forcing number of a graph. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000774The maximal multiplicity of a Laplacian eigenvalue in a graph. St000776The maximal multiplicity of an eigenvalue in a graph. St000778The metric dimension of a graph. St000986The multiplicity of the eigenvalue zero of the adjacency matrix of the graph. St001323The independence gap of a graph. St001723The differential of a graph. St001724The 2-packing differential of a graph. St001798The difference of the number of edges in a graph and the number of edges in the complement of the Turán graph. St001949The rigidity index of a graph. St000081The number of edges of a graph. St000171The degree of the graph. St000271The chromatic index of a graph. St000469The distinguishing number of a graph. St000636The hull number of a graph. St000723The maximal cardinality of a set of vertices with the same neighbourhood in a graph. St000987The number of positive eigenvalues of the Laplacian matrix of the graph. St001118The acyclic chromatic index of a graph. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001286The annihilation number of a graph. St001315The dissociation number of a graph. St001345The Hamming dimension of a graph. St001366The maximal multiplicity of a degree of a vertex of a graph. St001479The number of bridges of a graph. St001654The monophonic hull number of a graph. St001655The general position number of a graph. St001656The monophonic position number of a graph. St001708The number of pairs of vertices of different degree in a graph. St001826The maximal number of leaves on a vertex of a graph. St001844The maximal degree of a generator of the invariant ring of the automorphism group of a graph. St001883The mutual visibility number of a graph. St001917The order of toric promotion on the set of labellings of a graph. St000349The number of different adjacency matrices of a graph. St000468The Hosoya index of a graph. St000718The largest Laplacian eigenvalue of a graph if it is integral. St000915The Ore degree of a graph. St001318The number of vertices of the largest induced subforest with the same number of connected components of a graph. St001321The number of vertices of the largest induced subforest of a graph. St001463The number of distinct columns in the nullspace of a graph. St001672The restrained domination number of a graph. St001674The number of vertices of the largest induced star graph in the graph. St001725The harmonious chromatic number of a graph. St001746The coalition number of a graph. St001957The number of Hasse diagrams with a given underlying undirected graph. St000021The number of descents of a permutation. St000261The edge connectivity of a graph. St000262The vertex connectivity of a graph. St000284The Plancherel distribution on integer partitions. St000310The minimal degree of a vertex of a graph. 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. St000619The number of cyclic descents of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000831The number of indices that are either descents or recoils. St000901The cube of the number of standard Young tableaux with shape given by the partition. St000993The multiplicity of the largest part of an integer partition. St001061The number of indices that are both descents and recoils of a permutation. St001128The exponens consonantiae of a partition. St001152The number of pairs with even minimum in a perfect matching. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001489The maximum of the number of descents and the number of inverse descents. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St000314The number of left-to-right-maxima of a permutation. St000325The width of the tree associated to a permutation. St000470The number of runs in a permutation. St000542The number of left-to-right-minima of a permutation. St000567The sum of the products of all pairs of parts. St000699The toughness times the least common multiple of 1,. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001201The grade of the simple module $S_0$ in the special CNakayama algebra corresponding to the Dyck path. St001202Call a CNakayama algebra (a Nakayama algebra with a cyclic quiver) with Kupisch series $L=[c_0,c_1,...,c_{n−1}]$ such that $n=c_0 < c_i$ for all $i > 0$ a special CNakayama algebra. St001390The number of bumps occurring when Schensted-inserting the letter 1 of a permutation. St001481The minimal height of a peak of a Dyck path. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001290The first natural number n such that the tensor product of n copies of D(A) is zero for the corresponding Nakayama algebra A. St001742The difference of the maximal and the minimal degree in a graph. St001060The distinguishing index of a graph. St001117The game chromatic index of a graph. St001578The minimal number of edges to add or remove to make a graph a line graph. St001827The number of two-component spanning forests of a graph. St001869The maximum cut size of a graph. St000086The number of subgraphs. St000299The number of nonisomorphic vertex-induced subtrees. St001281The normalized isoperimetric number of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001651The Frankl number of a lattice. St001812The biclique partition number of a graph. 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. St000264The girth of a graph, which is not a tree. St001960The number of descents of a permutation minus one if its first entry is not one. St001235The global dimension of the corresponding Comp-Nakayama algebra.
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