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Your data matches 22 different statistics following compositions of up to 3 maps.
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Matching statistic: St000993
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(load all 3 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: 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
St000993: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [2,2]
=> [2]
=> [1,1]
=> 2
([],3)
=> [2,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
([(0,1),(0,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 2
([(0,2),(1,2)],3)
=> [3,2]
=> [2]
=> [1,1]
=> 2
([(2,3)],4)
=> [6,6]
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> [4]
=> [1,1,1,1]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2]
=> [1,1]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3]
=> [2,2,2]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [3]
=> [1,1,1]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3]
=> [2,2,2]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4]
=> [3,3,3,3]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,2,2]
=> [3,3,1,1]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [2,2,2,2,2,2]
=> [6,6]
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [2,2,2]
=> [3,3]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,2]
=> [2,2,1]
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [3,2,2]
=> [3,3,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2]
=> [1,1]
=> 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(1,4),(2,3)],5)
=> [6,6,6]
=> [6,6]
=> [2,2,2,2,2,2]
=> 6
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [6]
=> [1,1,1,1,1,1]
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [3]
=> [1,1,1]
=> 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [4]
=> [1,1,1,1]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [2,2,2,2]
=> [4,4]
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2]
=> [1,1]
=> 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [2,2]
=> [2,2]
=> 2
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [3,3,3]
=> [3,3,3]
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [5,3]
=> [2,2,2,1,1]
=> 3
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000642
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(load all 2 compositions to match this statistic)
Values
([],2)
=> 2
([],3)
=> 2
([(0,1),(0,2)],3)
=> 2
([(0,2),(1,2)],3)
=> 2
([(2,3)],4)
=> 6
([(1,2),(1,3)],4)
=> 2
([(0,1),(0,2),(0,3)],4)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(2,3)],4)
=> 4
([(0,3),(3,1),(3,2)],4)
=> 2
([(1,3),(2,3)],4)
=> 2
([(0,3),(1,3),(3,2)],4)
=> 2
([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2)],4)
=> 3
([(0,3),(1,2),(1,3)],4)
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(1,2),(1,3),(1,4)],5)
=> 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> 6
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(2,3),(3,4)],5)
=> 4
([(1,4),(4,2),(4,3)],5)
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> 2
([(1,4),(2,4),(4,3)],5)
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> 2
([(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> 2
([(0,4),(1,4),(2,3)],5)
=> 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> 6
([(1,4),(2,3)],5)
=> 6
([(1,4),(2,3),(2,4)],5)
=> 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> 2
([(0,4),(1,2),(1,3)],5)
=> 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> 3
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,5),(6,1)],7)
=> ? = 2
([(0,2),(0,3),(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,1)],7)
=> ? = 3
([(0,2),(0,3),(0,4),(1,5),(2,5),(2,6),(3,5),(3,6),(4,1),(4,6)],7)
=> ? = 6
([(0,2),(0,3),(0,4),(1,5),(2,6),(3,6),(4,1),(4,6),(6,5)],7)
=> ? = 5
([(0,2),(0,3),(0,4),(1,6),(2,5),(3,5),(4,1),(5,6)],7)
=> ? = 3
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 2
([(0,2),(0,3),(0,4),(3,6),(4,6),(5,1),(6,5)],7)
=> ? = 2
([(0,3),(1,2),(1,5),(1,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 2
([(0,3),(1,4),(1,5),(1,6),(3,4),(3,5),(3,6),(6,2)],7)
=> ? = 6
([(0,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(5,3),(6,3)],7)
=> ? = 2
([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,3),(1,2),(1,4),(1,6),(2,5),(3,4),(3,6),(6,5)],7)
=> ? = 2
([(0,3),(1,2),(1,4),(1,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ? = 2
([(0,3),(0,4),(3,6),(4,5),(4,6),(5,1),(5,2)],7)
=> ? = 3
([(0,2),(0,4),(2,5),(3,1),(3,6),(4,3),(4,5),(5,6)],7)
=> ? = 3
([(0,2),(0,3),(1,5),(1,6),(2,4),(3,1),(3,4),(4,5),(4,6)],7)
=> ? = 2
([(1,3),(1,4),(2,6),(3,5),(4,2),(4,5),(5,6)],7)
=> ? = 10
([(0,3),(0,4),(2,6),(3,5),(4,2),(4,5),(5,6),(6,1)],7)
=> ? = 5
([(0,3),(0,5),(3,6),(4,1),(5,4),(5,6),(6,2)],7)
=> ? = 5
([(0,3),(0,4),(2,5),(3,6),(4,2),(4,6),(6,1),(6,5)],7)
=> ? = 5
([(0,3),(0,5),(3,6),(4,1),(4,6),(5,4),(6,2)],7)
=> ? = 6
([(0,3),(0,4),(3,6),(4,5),(5,1),(5,2),(5,6)],7)
=> ? = 6
([(0,4),(0,5),(3,2),(3,6),(4,3),(5,1),(5,6)],7)
=> ? = 2
([(0,2),(0,4),(2,5),(2,6),(3,1),(3,6),(4,3),(4,5)],7)
=> ? = 2
([(0,2),(0,3),(1,4),(1,6),(2,4),(2,5),(3,1),(3,5),(5,6)],7)
=> ? = 5
([(0,3),(0,4),(2,5),(2,6),(3,2),(4,1),(4,5),(4,6)],7)
=> ? = 2
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(2,6),(3,1),(3,6)],7)
=> ? = 2
([(0,3),(0,4),(2,6),(3,5),(3,6),(4,2),(4,5),(6,1)],7)
=> ? = 6
([(0,3),(0,4),(2,5),(3,5),(3,6),(4,2),(4,6),(6,1)],7)
=> ? = 3
([(0,4),(0,5),(2,6),(4,2),(5,1),(5,6),(6,3)],7)
=> ? = 3
([(0,6),(1,2),(1,3),(2,6),(3,6),(6,4),(6,5)],7)
=> ? = 2
([(0,3),(1,4),(1,6),(3,6),(4,2),(4,5),(6,5)],7)
=> ? = 7
([(0,2),(1,3),(1,6),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ? = 2
([(0,4),(1,3),(1,6),(3,5),(4,5),(4,6),(6,2)],7)
=> ? = 3
([(0,3),(1,2),(1,5),(2,6),(3,5),(3,6),(5,4),(6,4)],7)
=> ? = 3
([(0,4),(1,3),(1,5),(3,6),(4,5),(4,6),(6,2)],7)
=> ? = 3
([(0,4),(1,3),(1,5),(3,6),(4,5),(5,6),(6,2)],7)
=> ? = 3
([(0,3),(1,4),(1,6),(3,6),(4,5),(6,2),(6,5)],7)
=> ? = 3
([(0,3),(0,5),(1,6),(3,6),(4,1),(5,4),(6,2)],7)
=> ? = 4
([(0,3),(0,5),(2,6),(3,6),(4,2),(5,1),(5,4)],7)
=> ? = 6
([(0,2),(0,4),(1,5),(1,6),(2,5),(2,6),(3,1),(4,3)],7)
=> ? = 2
([(0,3),(1,2),(1,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)
=> ? = 2
([(0,3),(1,2),(1,6),(2,4),(2,5),(3,4),(3,6),(6,5)],7)
=> ? = 3
([(0,3),(0,6),(1,4),(3,5),(4,2),(4,5),(4,6)],7)
=> ? = 4
([(0,2),(0,5),(2,6),(3,4),(4,1),(4,6),(5,3)],7)
=> ? = 6
([(0,4),(0,5),(2,6),(4,6),(5,1),(5,2),(6,3)],7)
=> ? = 2
([(0,2),(0,3),(1,6),(2,6),(3,4),(3,5),(6,4),(6,5)],7)
=> ? = 2
([(0,5),(1,3),(1,4),(3,6),(4,5),(5,6),(6,2)],7)
=> ? = 2
([(0,6),(1,3),(1,4),(3,6),(4,5),(6,2),(6,5)],7)
=> ? = 2
([(0,6),(1,3),(1,4),(3,5),(3,6),(4,5),(4,6),(6,2)],7)
=> ? = 2
Description
The size of the smallest orbit of antichains under Panyushev complementation.
Matching statistic: St000326
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 88%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 42% ●values known / values provided: 42%●distinct values known / distinct values provided: 88%
Values
([],2)
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
([],3)
=> [2,2,2,2]
=> 111100 => 001111 => 3 = 2 + 1
([(0,1),(0,2)],3)
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
([(0,2),(1,2)],3)
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
([(2,3)],4)
=> [6,6]
=> 11000000 => 00000011 => 7 = 6 + 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> 100001100 => 001100001 => 3 = 2 + 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> 1011100 => 0011101 => 3 = 2 + 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> 100100 => 001001 => 3 = 2 + 1
([(1,2),(2,3)],4)
=> [4,4]
=> 110000 => 000011 => 5 = 4 + 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> 100100 => 001001 => 3 = 2 + 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> 100001100 => 001100001 => 3 = 2 + 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> 100100 => 001001 => 3 = 2 + 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> 1011100 => 0011101 => 3 = 2 + 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> 111000 => 000111 => 4 = 3 + 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> 1001000 => 0001001 => 4 = 3 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> 101100 => 001101 => 3 = 2 + 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> 1000011111100 => 0011111100001 => ? = 2 + 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> 101000000 => 000000101 => 7 = 6 + 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 1000001100 => 0011000001 => ? = 2 + 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> 10011100 => 00111001 => 3 = 2 + 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> 1000001100 => 0011000001 => ? = 2 + 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> 11001100 => 00110011 => 3 = 2 + 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> 1000100 => 0010001 => 3 = 2 + 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> 1011000 => 0001101 => 4 = 3 + 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> 1010000 => 0000101 => 5 = 4 + 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> 1001100 => 0011001 => 3 = 2 + 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> 11110000 => 00001111 => 5 = 4 + 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> 11001100 => 00110011 => 3 = 2 + 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> 10011100 => 00111001 => 3 = 2 + 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> 11001100 => 00110011 => 3 = 2 + 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> 1001100 => 0011001 => 3 = 2 + 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> 1000011111100 => 0011111100001 => ? = 2 + 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> 10011100 => 00111001 => 3 = 2 + 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> 1000111000 => 0001110001 => 4 = 3 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> 10000010100 => 00101000001 => ? = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> 100101100 => 001101001 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> 10111100 => 00111101 => 3 = 2 + 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> 1000100 => 0010001 => 3 = 2 + 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> 1000001100 => 0011000001 => ? = 2 + 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> 101001000 => 000100101 => 4 = 3 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> 101000000 => 000000101 => 7 = 6 + 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> 111000000 => 000000111 => 7 = 6 + 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> 100001000000 => 000000100001 => ? = 6 + 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> 1000001000 => 0001000001 => 4 = 3 + 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> 1010000 => 0000101 => 5 = 4 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> 10000111100 => 00111100001 => ? = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> 100000100 => 001000001 => 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> 1001100 => 0011001 => 3 = 2 + 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> 1000111000 => 0001110001 => 4 = 3 + 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> 101001000 => 000100101 => 4 = 3 + 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> 1010000 => 0000101 => 5 = 4 + 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> 100000000100 => 001000000001 => ? = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> 1000001100 => 0011000001 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> 10000010100 => 00101000001 => ? = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> 100101100 => 001101001 => 3 = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> 10111100 => 00111101 => 3 = 2 + 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> 1000001000 => 0001000001 => 4 = 3 + 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> 100000100 => 001000001 => 3 = 2 + 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> 1000100 => 0010001 => 3 = 2 + 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> 1011000 => 0001101 => 4 = 3 + 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> 1010000 => 0000101 => 5 = 4 + 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> 10000001100 => 00110000001 => ? = 2 + 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> 100010000000 => 000000010001 => ? = 7 + 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> 100000111100 => 001111000001 => ? = 2 + 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> 1000000001100 => ? => ? = 2 + 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> 1000000110000 => 0000110000001 => ? = 4 + 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> 1100111100 => 0011110011 => ? = 2 + 1
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> 110000000000 => 000000000011 => ? = 10 + 1
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> 1000000001100 => ? => ? = 2 + 1
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> 1100111100 => 0011110011 => ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> [8,3,2,2]
=> 100000101100 => 001101000001 => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [7,2,2,2]
=> 10000011100 => 00111000001 => ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [10,2,2]
=> 1000000001100 => ? => ? = 2 + 1
([(1,5),(2,5),(3,4),(5,3)],6)
=> [10,2,2]
=> 1000000001100 => ? => ? = 2 + 1
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> 10000001100 => 00110000001 => ? = 2 + 1
([(0,5),(1,5),(2,4),(5,3),(5,4)],6)
=> [10,2,2]
=> 1000000001100 => ? => ? = 2 + 1
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> [10,6]
=> 100001000000 => 000000100001 => ? = 6 + 1
([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2]
=> 10000011100 => 00111000001 => ? = 2 + 1
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> [11,5]
=> 1000000100000 => ? => ? = 5 + 1
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> 10000001100 => 00110000001 => ? = 2 + 1
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> 10000001100 => 00110000001 => ? = 2 + 1
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> [10,4,4]
=> 1000000110000 => 0000110000001 => ? = 4 + 1
([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [10,4,4]
=> 1000000110000 => 0000110000001 => ? = 4 + 1
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> [10,4]
=> 100000010000 => 000010000001 => ? = 4 + 1
([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6)
=> [7,2,2,2]
=> 10000011100 => 00111000001 => ? = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [4,4,2,2,2,2]
=> 1100111100 => 0011110011 => ? = 2 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(4,5)],6)
=> [10,2,2,2]
=> 10000000011100 => ? => ? = 2 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [7,2,2,2,2]
=> 100000111100 => 001111000001 => ? = 2 + 1
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,5),(3,4)],6)
=> [10,2,2]
=> 1000000001100 => ? => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [6,5,3,2,2]
=> 10100101100 => ? => ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(3,5)],6)
=> [10,6]
=> 100001000000 => 000000100001 => ? = 6 + 1
([(1,5),(2,3),(2,5),(5,4)],6)
=> [10,10]
=> 110000000000 => 000000000011 => ? = 10 + 1
([(0,5),(1,2),(1,5),(5,3),(5,4)],6)
=> [10,2,2]
=> 1000000001100 => ? => ? = 2 + 1
([(0,4),(1,2),(1,3),(1,4),(3,5),(4,5)],6)
=> [11,5]
=> 1000000100000 => ? => ? = 5 + 1
([(0,5),(1,2),(1,3),(1,5),(5,4)],6)
=> [10,4,4]
=> 1000000110000 => 0000110000001 => ? = 4 + 1
([(0,3),(1,2),(1,4),(1,5),(3,4),(3,5)],6)
=> [8,3,2,2]
=> 100000101100 => 001101000001 => ? = 2 + 1
([(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [7,2,2,2]
=> 10000011100 => 00111000001 => ? = 2 + 1
([(1,3),(1,5),(4,2),(5,4)],6)
=> [10,4,4]
=> 1000000110000 => 0000110000001 => ? = 4 + 1
([(0,2),(0,4),(2,5),(3,1),(4,3),(4,5)],6)
=> [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
([(0,3),(0,4),(2,5),(3,2),(4,1),(4,5)],6)
=> [9,3]
=> 10000001000 => 00010000001 => ? = 3 + 1
Description
The position of the first one in a binary word after appending a 1 at the end.
Regarding the binary word as a subset of $\{1,\dots,n,n+1\}$ that contains $n+1$, this is the minimal element of the set.
Matching statistic: St000297
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 62%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 30% ●values known / values provided: 30%●distinct values known / distinct values provided: 62%
Values
([],2)
=> [2,2]
=> [2,2]
=> 1100 => 2
([],3)
=> [2,2,2,2]
=> [4,4]
=> 110000 => 2
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 2
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> 11010 => 2
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> 11111100 => 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> 111100 => 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> 110011110 => 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> 1100010 => 2
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> 111000 => 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> 1110110 => 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> 110010 => 2
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,1,1,1,1]
=> 1100000011110 => ? = 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 111111010 => 6
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 1110010 => 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> 11110000 => 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 2
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> 11001100 => 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,1,1,1,1]
=> 1100000011110 => ? = 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> 11000110 => 2
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> 1110001110 => ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> 11010111110 => ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 11000010 => 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> 111011010 => 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> 111111010 => 6
([(1,4),(2,3)],5)
=> [6,6,6]
=> [3,3,3,3,3,3]
=> 111111000 => 6
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> 111111011110 => ? = 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [5,5,1,1,1,1]
=> 11000011110 => ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 110111110 => 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> 1100110 => 2
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> 1110001110 => ? = 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> 111011010 => 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 4
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> 110111111110 => ? = 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> 1100111110 => ? = 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> 11010111110 => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> 110010110 => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> 11000010 => 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> 1110111110 => 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> 110111110 => 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> 1110010 => 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> 1111010 => 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> 1101110 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> 110001110 => 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1110010110 => ? = 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 2
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [3,3,3,3,3,3,1]
=> 1111110010 => ? = 6
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [2,2,2,2,2,2,2,1,1,1]
=> 111111101110 => ? = 7
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [5,5,1,1,1,1,1]
=> 110000111110 => ? = 2
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> 1100111111110 => ? = 2
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> 1111001111110 => ? = 4
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [6,6,2,2]
=> 1100001100 => ? = 2
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1110010110 => ? = 3
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 2
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> 111111111100 => ? = 10
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> 1100111111110 => ? = 2
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [6,6,2,2]
=> 1100001100 => ? = 2
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [3,3,2,2,2,2,1,1]
=> 11011110110 => ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> [8,3,2,2]
=> [4,4,2,1,1,1,1,1]
=> 110010111110 => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [7,2,2,2]
=> [4,4,1,1,1,1,1]
=> 11000111110 => ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> 1100111111110 => ? = 2
([(1,5),(2,5),(3,4),(5,3)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> 1100111111110 => ? = 2
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 2
([(0,5),(1,5),(2,4),(5,3),(5,4)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> 1100111111110 => ? = 2
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> [10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> 111111011110 => ? = 6
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 3
([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2]
=> [4,4,1,1,1,1,1]
=> 11000111110 => ? = 2
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> [11,5]
=> [2,2,2,2,2,1,1,1,1,1,1]
=> 1111101111110 => ? = 5
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 2
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [2,2,2,2,2,2,1,1]
=> 1111110110 => ? = 6
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> 1110010110 => ? = 3
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> 11011011110 => ? = 2
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> 11001111110 => ? = 2
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> 1111001111110 => ? = 4
([(0,5),(1,4),(1,5),(4,2),(4,3)],6)
=> [8,6,3]
=> [3,3,3,2,2,2,1,1]
=> 11101110110 => ? = 3
([(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> 1111001111110 => ? = 4
([(0,5),(1,4),(1,5),(4,2),(5,3)],6)
=> [10,4]
=> [2,2,2,2,1,1,1,1,1,1]
=> 111101111110 => ? = 4
([(0,4),(1,2),(1,4),(2,5),(4,3),(4,5)],6)
=> [8,4]
=> [2,2,2,2,1,1,1,1]
=> 1111011110 => ? = 4
([(0,4),(0,5),(1,4),(1,5),(5,2),(5,3)],6)
=> [7,2,2,2]
=> [4,4,1,1,1,1,1]
=> 11000111110 => ? = 2
Description
The number of leading ones in a binary word.
Matching statistic: St001038
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 62%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
St001038: Dyck paths ⟶ ℤResult quality: 25% ●values known / values provided: 25%●distinct values known / distinct values provided: 62%
Values
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> 2
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> 2
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> 6
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> 2
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> ? = 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> ? = 6
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 7
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ? = 2
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 10
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> [8,3,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [7,2,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(1,5),(2,5),(3,4),(5,3)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,5),(1,4),(2,4),(4,5),(5,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> ? = 2
([(0,5),(1,5),(2,4),(5,3),(5,4)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,5),(1,5),(2,3),(2,5),(5,4)],6)
=> [10,6]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
([(0,4),(1,4),(2,3),(2,5),(4,5)],6)
=> [8,6,3]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,1,1,0,0,0,0,0]
=> ? = 3
([(0,3),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> ? = 2
([(0,5),(1,2),(1,4),(3,5),(4,3)],6)
=> [11,5]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,0,0]
=> ? = 5
([(0,3),(0,4),(1,5),(2,5),(3,5),(4,1),(4,2)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,5),(1,5),(2,3),(3,5),(5,4)],6)
=> [8,6]
=> [1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> ? = 6
([(0,4),(1,4),(2,3),(3,5),(4,5)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ? = 3
([(0,5),(1,4),(3,5),(4,2),(4,3)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> ? = 2
([(0,4),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ? = 2
([(0,5),(1,5),(2,3),(3,4),(3,5)],6)
=> [10,4,4]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> ? = 4
Description
The minimal height of a column in the parallelogram polyomino associated with the Dyck path.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 62%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 16% ●values known / values provided: 16%●distinct values known / distinct values provided: 62%
Values
([],2)
=> [2,2]
=> [2,2]
=> [[1,2],[3,4]]
=> 2
([],3)
=> [2,2,2,2]
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> 2
([(0,1),(0,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
([(0,2),(1,2)],3)
=> [3,2]
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> 2
([(2,3)],4)
=> [6,6]
=> [2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12]]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [3,3,1,1,1,1]
=> [[1,6,7],[2,9,10],[3],[4],[5],[8]]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [4,4,1]
=> [[1,3,4,5],[2,7,8,9],[6]]
=> 2
([(0,3),(1,2)],4)
=> [3,3,3]
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [2,2,2,1,1]
=> [[1,4],[2,6],[3,8],[5],[7]]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [3,3,1]
=> [[1,3,4],[2,6,7],[5]]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2,13,14,15,16,17,18],[3],[4],[5],[12]]
=> ? = 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 6
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? = 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? = 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> ? = 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [7,7,1,1,1,1]
=> [[1,6,7,8,9,10,11],[2,13,14,15,16,17,18],[3],[4],[5],[12]]
=> ? = 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [4,4,1,1]
=> [[1,4,5,6],[2,8,9,10],[3],[7]]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3,13,14,15],[4],[8],[12]]
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> [[1,7,10],[2,9,13],[3,12],[4],[5],[6],[8],[11]]
=> ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? = 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? = 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [2,2,2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8,11],[10,13],[12]]
=> ? = 6
([(1,4),(2,3)],5)
=> [6,6,6]
=> [3,3,3,3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9],[10,11,12],[13,14,15],[16,17,18]]
=> ? = 6
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [2,2,2,2,2,2,1,1,1,1]
=> [[1,6],[2,8],[3,10],[4,12],[5,14],[7,16],[9],[11],[13],[15]]
=> ? = 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [5,5,1,1,1,1]
=> [[1,6,7,8,9],[2,11,12,13,14],[3],[4],[5],[10]]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [3,3,1,1]
=> [[1,4,5],[2,7,8],[3],[6]]
=> 2
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [4,4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3,13,14,15],[4],[8],[12]]
=> ? = 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [3,3,3,2,2,1]
=> [[1,3,8],[2,5,11],[4,7,14],[6,10],[9,13],[12]]
=> ? = 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> 4
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [2,2,1,1,1,1,1,1,1,1]
=> [[1,10],[2,12],[3],[4],[5],[6],[7],[8],[9],[11]]
=> ? = 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [3,3,1,1,1,1,1]
=> [[1,7,8],[2,10,11],[3],[4],[5],[6],[9]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [3,3,2,1,1,1,1,1]
=> [[1,7,10],[2,9,13],[3,12],[4],[5],[6],[8],[11]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10],[5],[9]]
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [5,5,1]
=> [[1,3,4,5,6],[2,8,9,10,11],[7]]
=> ? = 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [2,2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3,11],[4],[5],[6],[8],[10]]
=> ? = 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [2,2,1,1,1,1,1]
=> [[1,7],[2,9],[3],[4],[5],[6],[8]]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8]]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [2,2,2,2,1]
=> [[1,3],[2,5],[4,7],[6,9],[8]]
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [2,2,1,1,1]
=> [[1,5],[2,7],[3],[4],[6]]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? = 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? = 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3],[4],[5],[6],[7],[10]]
=> ? = 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> 2
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [3,3,3,3,3,3,1]
=> [[1,3,4],[2,6,7],[5,9,10],[8,12,13],[11,15,16],[14,18,19],[17]]
=> ? = 6
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [2,2,2,2,2,2,2,1,1,1]
=> [[1,5],[2,7],[3,9],[4,11],[6,13],[8,15],[10,17],[12],[14],[16]]
=> ? = 7
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [5,5,1,1,1,1,1]
=> [[1,7,8,9,10],[2,12,13,14,15],[3],[4],[5],[6],[11]]
=> ? = 2
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2,13,14],[3],[4],[5],[6],[7],[8],[9],[12]]
=> ? = 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> 2
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> [3,3,3,3,1,1,1,1,1,1]
=> [[1,8,9],[2,11,12],[3,14,15],[4,17,18],[5],[6],[7],[10],[13],[16]]
=> ? = 4
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [6,6,2,2]
=> [[1,2,7,8,9,10],[3,4,13,14,15,16],[5,6],[11,12]]
=> ? = 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3],[4],[5],[6],[7],[9]]
=> 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [3,3,1,1,1]
=> [[1,5,6],[2,8,9],[3],[4],[7]]
=> 2
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [4,4,4,2,1,1]
=> [[1,4,7,8],[2,6,11,12],[3,10,15,16],[5,14],[9],[13]]
=> ? = 3
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [3,3,3,3,2,1]
=> [[1,3,6],[2,5,9],[4,8,12],[7,11,15],[10,14],[13]]
=> ? = 4
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 2
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> [2,2,2,2,2,2,2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14],[15,16],[17,18],[19,20]]
=> ? = 10
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2,13,14],[3],[4],[5],[6],[7],[8],[9],[12]]
=> ? = 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> 2
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [6,6,2,2]
=> [[1,2,7,8,9,10],[3,4,13,14,15,16],[5,6],[11,12]]
=> ? = 2
([(0,5),(1,5),(5,2),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
([(0,5),(1,5),(2,5),(5,3),(5,4)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> [5,2,2,2]
=> [4,4,1,1,1]
=> [[1,5,6,7],[2,9,10,11],[3],[4],[8]]
=> 2
([(0,5),(1,4),(2,4),(2,5),(5,3)],6)
=> [8,6,2]
=> [3,3,2,2,2,2,1,1]
=> [[1,4,13],[2,6,16],[3,8],[5,10],[7,12],[9,15],[11],[14]]
=> ? = 2
([(0,4),(1,3),(2,3),(2,4),(3,5),(4,5)],6)
=> [8,4,2]
=> [3,3,2,2,1,1,1,1]
=> [[1,6,11],[2,8,14],[3,10],[4,13],[5],[7],[9],[12]]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(4,3)],6)
=> [8,3,2,2]
=> [4,4,2,1,1,1,1,1]
=> [[1,7,10,11],[2,9,14,15],[3,13],[4],[5],[6],[8],[12]]
=> ? = 2
([(0,5),(1,3),(1,5),(2,3),(2,5),(3,4),(5,4)],6)
=> [5,4,2,2]
=> [4,4,2,2,1]
=> [[1,3,8,9],[2,5,12,13],[4,7],[6,11],[10]]
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [7,2,2,2]
=> [4,4,1,1,1,1,1]
=> [[1,7,8,9],[2,11,12,13],[3],[4],[5],[6],[10]]
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [4,2,2,2,2]
=> [5,5,1,1]
=> [[1,4,5,6,7],[2,9,10,11,12],[3],[8]]
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(5,3)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2,13,14],[3],[4],[5],[6],[7],[8],[9],[12]]
=> ? = 2
([(1,5),(2,5),(3,4),(5,3)],6)
=> [10,2,2]
=> [3,3,1,1,1,1,1,1,1,1]
=> [[1,10,11],[2,13,14],[3],[4],[5],[6],[7],[8],[9],[12]]
=> ? = 2
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [2,2,1,1,1,1,1,1]
=> [[1,8],[2,10],[3],[4],[5],[6],[7],[9]]
=> 2
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St001107
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Mp00230: Integer partitions —parallelogram polyomino⟶ Dyck paths
Mp00227: Dyck paths —Delest-Viennot-inverse⟶ Dyck paths
St001107: Dyck paths ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Values
([],2)
=> [2,2]
=> [1,1,1,0,0,0]
=> [1,1,0,1,0,0]
=> 1 = 2 - 1
([],3)
=> [2,2,2,2]
=> [1,1,1,1,0,1,0,0,0,0]
=> [1,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,1),(0,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(1,2)],3)
=> [3,2]
=> [1,0,1,1,1,0,0,0]
=> [1,1,1,0,0,1,0,0]
=> 1 = 2 - 1
([(2,3)],4)
=> [6,6]
=> [1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,0,1,0,0,0,0,0,0]
=> 5 = 6 - 1
([(1,2),(1,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4,4]
=> [1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,0,1,0,0,0,0]
=> 3 = 4 - 1
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(1,3),(2,3)],4)
=> [6,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2)],4)
=> [3,3,3]
=> [1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 4 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,0]
=> [1,1,1,1,0,1,0,1,0,1,0,0,0,0]
=> 3 = 4 - 1
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
([(1,4),(2,3)],5)
=> [6,6,6]
=> [1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [1,0,1,0,1,0,1,1,1,1,1,1,0,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0,0]
=> ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [1,0,1,1,1,0,1,0,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,0,1,0,0,0]
=> ? = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 4 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,1,0,0]
=> ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [1,0,1,0,1,1,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,0,0,1,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0,0]
=> ? = 3 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,0,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [1,0,1,1,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,0]
=> 3 = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 3 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 1
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [1,0,1,1,1,1,1,0,1,0,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,0,0,1,0,1,0,0,0,0,0,0]
=> ? = 6 - 1
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [1,0,1,0,1,0,1,1,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> ?
=> ? = 7 - 1
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> ?
=> ? = 2 - 1
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ?
=> ? = 2 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,1,0,0,0,0]
=> ? = 4 - 1
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [1,0,1,0,1,1,1,0,1,1,1,1,0,0,0,0,0,0]
=> ?
=> ? = 3 - 1
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [1,0,1,1,1,0,1,1,1,0,1,0,0,0,0,0]
=> [1,1,1,1,1,1,0,0,1,0,0,1,0,0,0,0]
=> ? = 4 - 1
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,1,1,1,0,0,0,0,0,1,0,0,0,1,0,0]
=> ? = 2 - 1
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [1,0,1,1,1,0,1,0,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> [1,1,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,1,1,1,1,1,0,1,0,0,0,0,0,0,0,0,0,0]
=> ? = 10 - 1
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> [1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> ?
=> ? = 2 - 1
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(1,5),(2,5),(5,3),(5,4)],6)
=> [4,4,2,2,2,2]
=> [1,1,1,0,1,0,1,1,0,1,0,1,0,1,0,0,0,0]
=> [1,1,1,1,0,1,0,0,0,1,0,1,0,1,0,1,0,0]
=> ? = 2 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(3,2),(4,1)],6)
=> [5,3,3]
=> [1,0,1,0,1,1,1,1,1,0,0,0,0,0]
=> [1,1,1,1,1,0,0,0,1,0,1,0,0,0]
=> 2 = 3 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [1,0,1,0,1,0,1,0,1,1,1,0,0,0]
=> [1,1,1,1,1,1,0,0,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,5),(1,2),(1,5),(2,3),(2,4),(5,3),(5,4)],6)
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4 = 5 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2)],6)
=> [4,3,3,2]
=> [1,0,1,1,1,1,1,0,0,1,0,0,0,0]
=> [1,1,1,1,0,0,1,0,1,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,3),(5,2),(5,3)],6)
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [4,2,2,2]
=> [1,0,1,0,1,1,1,1,0,1,0,0,0,0]
=> [1,1,1,1,0,0,0,1,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(5,2)],6)
=> [5,4,2]
=> [1,0,1,1,1,0,1,0,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,1,0,0,0,1,0,0]
=> 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [5,2,2]
=> [1,0,1,0,1,0,1,1,1,1,0,0,0,0]
=> [1,1,1,1,1,0,0,0,0,1,0,1,0,0]
=> 1 = 2 - 1
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [1,1,1,0,1,0,1,0,1,0,0,0]
=> [1,1,1,1,1,0,1,0,0,0,0,0]
=> 4 = 5 - 1
Description
The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path.
In other words, this is the lowest height of a valley of a Dyck path, or its semilength in case of the unique path without valleys.
Matching statistic: St000745
Mp00306: Posets —rowmotion cycle type⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 10% ●values known / values provided: 10%●distinct values known / distinct values provided: 62%
Values
([],2)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
([],3)
=> [2,2,2,2]
=> [[1,2],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8]]
=> 2
([(0,1),(0,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
([(0,2),(1,2)],3)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
([(2,3)],4)
=> [6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12]]
=> 6
([(1,2),(1,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> 2
([(0,1),(0,2),(0,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2
([(1,2),(2,3)],4)
=> [4,4]
=> [[1,2,3,4],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8]]
=> 4
([(0,3),(3,1),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2
([(1,3),(2,3)],4)
=> [6,2,2]
=> [[1,2,7,8,9,10],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10]]
=> 2
([(0,3),(1,3),(3,2)],4)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2
([(0,3),(1,3),(2,3)],4)
=> [3,2,2,2]
=> [[1,2,9],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9]]
=> 2
([(0,3),(1,2)],4)
=> [3,3,3]
=> [[1,2,3],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9]]
=> 3
([(0,3),(1,2),(1,3)],4)
=> [5,3]
=> [[1,2,3,7,8],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8]]
=> 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> [3,2,2]
=> [[1,2,7],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7]]
=> 2
([(1,2),(1,3),(1,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,15,16,17,18],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14],[15],[16],[17],[18]]
=> ? = 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13]]
=> ? = 6
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
=> ? = 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
=> ? = 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 2
([(2,3),(3,4)],5)
=> [4,4,4,4]
=> [[1,2,3,4],[5,6,7,8],[9,10,11,12],[13,14,15,16]]
=> [[1,5,9,13],[2,6,10,14],[3,7,11,15],[4,8,12,16]]
=> ? = 4
([(1,4),(4,2),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 2
([(1,4),(2,4),(4,3)],5)
=> [4,4,2,2]
=> [[1,2,7,8],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12]]
=> ? = 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 2
([(1,4),(2,4),(3,4)],5)
=> [6,2,2,2,2,2,2]
=> [[1,2,15,16,17,18],[3,4],[5,6],[7,8],[9,10],[11,12],[13,14]]
=> [[1,3,5,7,9,11,13],[2,4,6,8,10,12,14],[15],[16],[17],[18]]
=> ? = 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 2
([(0,4),(1,4),(2,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13],[14],[15]]
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> [8,3,2]
=> [[1,2,5,9,10,11,12,13],[3,4,8],[6,7]]
=> [[1,3,6],[2,4,7],[5,8],[9],[10],[11],[12],[13]]
=> ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
=> ? = 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13],[14]]
=> ? = 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> [7,6]
=> [[1,2,3,4,5,6,13],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13]]
=> ? = 6
([(1,4),(2,3)],5)
=> [6,6,6]
=> [[1,2,3,4,5,6],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> [[1,7,13],[2,8,14],[3,9,15],[4,10,16],[5,11,17],[6,12,18]]
=> ? = 6
([(1,4),(2,3),(2,4)],5)
=> [10,6]
=> [[1,2,3,4,5,6,13,14,15,16],[7,8,9,10,11,12]]
=> [[1,7],[2,8],[3,9],[4,10],[5,11],[6,12],[13],[14],[15],[16]]
=> ? = 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ? = 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> [6,2,2,2,2]
=> [[1,2,11,12,13,14],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11],[12],[13],[14]]
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [4,2,2]
=> [[1,2,7,8],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8]]
=> 2
([(0,4),(1,2),(1,3)],5)
=> [6,3,3,3]
=> [[1,2,3,13,14,15],[4,5,6],[7,8,9],[10,11,12]]
=> [[1,4,7,10],[2,5,8,11],[3,6,9,12],[13],[14],[15]]
=> ? = 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> [6,5,3]
=> [[1,2,3,7,8,14],[4,5,6,12,13],[9,10,11]]
=> [[1,4,9],[2,5,10],[3,6,11],[7,12],[8,13],[14]]
=> ? = 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 4
([(0,4),(1,2),(1,3),(3,4)],5)
=> [10,2]
=> [[1,2,5,6,7,8,9,10,11,12],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10],[11],[12]]
=> ? = 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [7,2,2]
=> [[1,2,7,8,9,10,11],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9],[10],[11]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> [8,3,2]
=> [[1,2,5,9,10,11,12,13],[3,4,8],[6,7]]
=> [[1,3,6],[2,4,7],[5,8],[9],[10],[11],[12],[13]]
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [5,3,2,2]
=> [[1,2,7,11,12],[3,4,10],[5,6],[8,9]]
=> [[1,3,5,8],[2,4,6,9],[7,10],[11],[12]]
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [3,2,2,2,2]
=> [[1,2,11],[3,4],[5,6],[7,8],[9,10]]
=> [[1,3,5,7,9],[2,4,6,8,10],[11]]
=> ? = 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> [8,3]
=> [[1,2,3,7,8,9,10,11],[4,5,6]]
=> [[1,4],[2,5],[3,6],[7],[8],[9],[10],[11]]
=> ? = 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [7,2]
=> [[1,2,5,6,7,8,9],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9]]
=> 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,3,3]
=> [[1,2,3,10],[4,5,6],[7,8,9]]
=> [[1,4,7],[2,5,8],[3,6,9],[10]]
=> 3
([(0,4),(1,2),(2,3),(3,4)],5)
=> [5,4]
=> [[1,2,3,4,9],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9]]
=> 4
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [5,2]
=> [[1,2,5,6,7],[3,4]]
=> [[1,3],[2,4],[5],[6],[7]]
=> 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(3,5)],6)
=> [8,4,2]
=> [[1,2,5,6,11,12,13,14],[3,4,9,10],[7,8]]
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 2
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1),(4,5)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 4
([(0,1),(0,2),(0,3),(1,5),(2,4),(3,4),(4,5)],6)
=> [8,2,2]
=> [[1,2,7,8,9,10,11,12],[3,4],[5,6]]
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(3,5),(4,5),(5,1)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 2
([(0,3),(0,4),(0,5),(4,2),(5,1)],6)
=> [7,6,6]
=> [[1,2,3,4,5,6,19],[7,8,9,10,11,12],[13,14,15,16,17,18]]
=> ?
=> ? = 6
([(0,2),(0,3),(0,4),(3,5),(4,1),(4,5)],6)
=> [10,7]
=> [[1,2,3,4,5,6,7,15,16,17],[8,9,10,11,12,13,14]]
=> ?
=> ? = 7
([(0,1),(0,2),(0,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [7,2,2,2,2]
=> [[1,2,11,12,13,14,15],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 2
([(1,2),(1,3),(2,5),(3,5),(5,4)],6)
=> [10,2,2]
=> [[1,2,7,8,9,10,11,12,13,14],[3,4],[5,6]]
=> ?
=> ? = 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 2
([(1,3),(1,4),(3,5),(4,2),(4,5)],6)
=> [10,4,4]
=> [[1,2,3,4,13,14,15,16,17,18],[5,6,7,8],[9,10,11,12]]
=> [[1,5,9],[2,6,10],[3,7,11],[4,8,12],[13],[14],[15],[16],[17],[18]]
=> ? = 4
([(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> [4,4,2,2,2,2]
=> [[1,2,11,12],[3,4,15,16],[5,6],[7,8],[9,10],[13,14]]
=> ?
=> ? = 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(5,1)],6)
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 2
([(0,4),(0,5),(4,3),(5,1),(5,2)],6)
=> [6,4,3,3]
=> [[1,2,3,10,15,16],[4,5,6,14],[7,8,9],[11,12,13]]
=> ?
=> ? = 3
([(0,3),(0,4),(3,5),(4,1),(4,2),(4,5)],6)
=> [6,5,4]
=> [[1,2,3,4,9,15],[5,6,7,8,14],[10,11,12,13]]
=> [[1,5,10],[2,6,11],[3,7,12],[4,8,13],[9,14],[15]]
=> ? = 4
([(0,3),(0,4),(3,2),(3,5),(4,1),(4,5)],6)
=> [8,4,2]
=> [[1,2,5,6,11,12,13,14],[3,4,9,10],[7,8]]
=> ?
=> ? = 2
([(0,2),(0,3),(2,4),(2,5),(3,1),(3,4),(3,5)],6)
=> [5,4,2,2]
=> [[1,2,7,8,13],[3,4,11,12],[5,6],[9,10]]
=> [[1,3,5,9],[2,4,6,10],[7,11],[8,12],[13]]
=> ? = 2
([(0,1),(0,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> [4,2,2,2,2]
=> [[1,2,11,12],[3,4],[5,6],[7,8],[9,10]]
=> ?
=> ? = 2
([(2,3),(3,5),(5,4)],6)
=> [10,10]
=> [[1,2,3,4,5,6,7,8,9,10],[11,12,13,14,15,16,17,18,19,20]]
=> [[1,11],[2,12],[3,13],[4,14],[5,15],[6,16],[7,17],[8,18],[9,19],[10,20]]
=> ? = 10
([(1,4),(4,5),(5,2),(5,3)],6)
=> [10,2,2]
=> [[1,2,7,8,9,10,11,12,13,14],[3,4],[5,6]]
=> ?
=> ? = 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> [5,2,2,2]
=> [[1,2,9,10,11],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10],[11]]
=> ? = 2
([(0,5),(1,5),(4,2),(5,3),(5,4)],6)
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 2
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 2
([(0,4),(1,2),(1,4),(2,5),(4,5),(5,3)],6)
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> 5
([(0,4),(0,5),(1,4),(1,5),(4,2),(4,3),(5,2),(5,3)],6)
=> [4,2,2,2]
=> [[1,2,9,10],[3,4],[5,6],[7,8]]
=> [[1,3,5,7],[2,4,6,8],[9],[10]]
=> 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [5,2,2]
=> [[1,2,7,8,9],[3,4],[5,6]]
=> [[1,3,5],[2,4,6],[7],[8],[9]]
=> 2
([(0,4),(0,5),(1,4),(1,5),(2,3),(4,2),(5,3)],6)
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> 2
([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> [5,5]
=> [[1,2,3,4,5],[6,7,8,9,10]]
=> [[1,6],[2,7],[3,8],[4,9],[5,10]]
=> 5
([(0,2),(0,4),(1,5),(2,5),(3,1),(4,3)],6)
=> [6,4]
=> [[1,2,3,4,9,10],[5,6,7,8]]
=> [[1,5],[2,6],[3,7],[4,8],[9],[10]]
=> 4
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [6,2]
=> [[1,2,5,6,7,8],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8]]
=> 2
([(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,1)],6)
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> 2
([(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [8,2]
=> [[1,2,5,6,7,8,9,10],[3,4]]
=> [[1,3],[2,4],[5],[6],[7],[8],[9],[10]]
=> 2
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St001491
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00307: Posets —promotion cycle type⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 12%
Mp00095: Integer partitions —to binary word⟶ Binary words
St001491: Binary words ⟶ ℤResult quality: 5% ●values known / values provided: 5%●distinct values known / distinct values provided: 12%
Values
([],2)
=> [2]
=> 100 => 1 = 2 - 1
([],3)
=> [3,3]
=> 11000 => ? = 2 - 1
([(0,1),(0,2)],3)
=> [2]
=> 100 => 1 = 2 - 1
([(0,2),(1,2)],3)
=> [2]
=> 100 => 1 = 2 - 1
([(2,3)],4)
=> [4,4,4]
=> 1110000 => ? = 6 - 1
([(1,2),(1,3)],4)
=> [8]
=> 100000000 => ? = 2 - 1
([(0,1),(0,2),(0,3)],4)
=> [3,3]
=> 11000 => ? = 2 - 1
([(0,1),(0,2),(1,3),(2,3)],4)
=> [2]
=> 100 => 1 = 2 - 1
([(1,2),(2,3)],4)
=> [4]
=> 10000 => ? = 4 - 1
([(0,3),(3,1),(3,2)],4)
=> [2]
=> 100 => 1 = 2 - 1
([(1,3),(2,3)],4)
=> [8]
=> 100000000 => ? = 2 - 1
([(0,3),(1,3),(3,2)],4)
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(1,3),(2,3)],4)
=> [3,3]
=> 11000 => ? = 2 - 1
([(0,3),(1,2)],4)
=> [4,2]
=> 100100 => ? = 3 - 1
([(0,3),(1,2),(1,3)],4)
=> [3,2]
=> 10100 => ? = 3 - 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(1,2),(1,3),(1,4)],5)
=> [15,15]
=> 11000000000000000 => ? = 2 - 1
([(0,2),(0,3),(0,4),(4,1)],5)
=> [4,4,4]
=> 1110000 => ? = 6 - 1
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> [8]
=> 100000000 => ? = 2 - 1
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> [3,3]
=> 11000 => ? = 2 - 1
([(0,3),(0,4),(4,1),(4,2)],5)
=> [8]
=> 100000000 => ? = 2 - 1
([(1,2),(1,3),(2,4),(3,4)],5)
=> [5,5]
=> 1100000 => ? = 2 - 1
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(0,4),(3,2),(4,1)],5)
=> [4,2]
=> 100100 => ? = 3 - 1
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> [3,2]
=> 10100 => ? = 4 - 1
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(2,3),(3,4)],5)
=> [5,5,5,5]
=> 111100000 => ? = 4 - 1
([(1,4),(4,2),(4,3)],5)
=> [5,5]
=> 1100000 => ? = 2 - 1
([(0,4),(4,1),(4,2),(4,3)],5)
=> [3,3]
=> 11000 => ? = 2 - 1
([(1,4),(2,4),(4,3)],5)
=> [5,5]
=> 1100000 => ? = 2 - 1
([(0,4),(1,4),(4,2),(4,3)],5)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(1,4),(2,4),(3,4)],5)
=> [15,15]
=> 11000000000000000 => ? = 2 - 1
([(0,4),(1,4),(2,4),(4,3)],5)
=> [3,3]
=> 11000 => ? = 2 - 1
([(0,4),(1,4),(2,3)],5)
=> [10,10]
=> 110000000000 => ? = 3 - 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> [12,4]
=> 10000000010000 => ? = 2 - 1
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [14]
=> 100000000000000 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> [6,6]
=> 11000000 => ? = 2 - 1
([(0,4),(1,4),(2,3),(4,2)],5)
=> [2]
=> 100 => 1 = 2 - 1
([(0,4),(1,3),(2,3),(3,4)],5)
=> [8]
=> 100000000 => ? = 2 - 1
([(0,4),(1,4),(2,3),(2,4)],5)
=> [10,4,4]
=> 1000000110000 => ? = 3 - 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> [4,4,4]
=> 1110000 => ? = 6 - 1
([(1,4),(2,3)],5)
=> [5,5,5,5,5,5]
=> 11111100000 => ? = 6 - 1
([(1,4),(2,3),(2,4)],5)
=> [15,5,5]
=> 100000000001100000 => ? = 6 - 1
([(0,4),(1,2),(1,4),(2,3)],5)
=> [5,4]
=> 1010000 => ? = 3 - 1
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => ? = 4 - 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [5,5,5,5]
=> 111100000 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> [6]
=> 1000000 => ? = 2 - 1
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,4),(1,2),(1,3)],5)
=> [10,10]
=> 110000000000 => ? = 3 - 1
([(0,4),(1,2),(1,3),(1,4)],5)
=> [10,4,4]
=> 1000000110000 => ? = 3 - 1
([(0,2),(0,4),(3,1),(4,3)],5)
=> [4]
=> 10000 => ? = 4 - 1
([(0,4),(1,2),(1,3),(3,4)],5)
=> [4,4,3]
=> 1101000 => ? = 2 - 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> [8]
=> 100000000 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,4)],5)
=> [12,4]
=> 10000000010000 => ? = 2 - 1
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [14]
=> 100000000000000 => ? = 2 - 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> [6,6]
=> 11000000 => ? = 2 - 1
([(0,3),(1,2),(1,4),(3,4)],5)
=> [5,4]
=> 1010000 => ? = 3 - 1
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> [6]
=> 1000000 => ? = 2 - 1
([(0,3),(3,4),(4,1),(4,2)],5)
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(1,2),(2,4),(3,4)],5)
=> [4,2]
=> 100100 => ? = 3 - 1
([(0,4),(1,2),(2,3),(3,4)],5)
=> [4]
=> 10000 => ? = 4 - 1
([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> [2]
=> 100 => 1 = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> [3,3]
=> 11000 => ? = 2 - 1
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,1)],6)
=> [10,10]
=> 110000000000 => ? = 3 - 1
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,5),(1,5),(3,2),(4,3),(5,4)],6)
=> [2]
=> 100 => 1 = 2 - 1
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],6)
=> [2]
=> 100 => 1 = 2 - 1
([(0,4),(3,5),(4,3),(5,1),(5,2)],6)
=> [2]
=> 100 => 1 = 2 - 1
([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> [2]
=> 100 => 1 = 2 - 1
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,4),(1,6),(2,6),(4,5),(5,1),(5,2),(6,3)],7)
=> [2]
=> 100 => 1 = 2 - 1
([(0,6),(1,6),(3,4),(4,2),(5,3),(6,5)],7)
=> [2]
=> 100 => 1 = 2 - 1
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,2),(0,3),(2,6),(3,6),(4,1),(5,4),(6,5)],7)
=> [2]
=> 100 => 1 = 2 - 1
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> [2,2]
=> 1100 => 1 = 2 - 1
([(0,5),(3,4),(4,6),(5,3),(6,1),(6,2)],7)
=> [2]
=> 100 => 1 = 2 - 1
([(0,5),(2,6),(3,6),(4,1),(5,2),(5,3),(6,4)],7)
=> [2]
=> 100 => 1 = 2 - 1
([(0,4),(1,6),(2,6),(3,5),(4,3),(5,1),(5,2)],7)
=> [2]
=> 100 => 1 = 2 - 1
Description
The number of indecomposable projective-injective modules in the algebra corresponding to a subset.
Let $A_n=K[x]/(x^n)$.
We associate to a nonempty subset S of an (n-1)-set the module $M_S$, which is the direct sum of $A_n$-modules with indecomposable non-projective direct summands of dimension $i$ when $i$ is in $S$ (note that such modules have vector space dimension at most n-1). Then the corresponding algebra associated to S is the stable endomorphism ring of $M_S$. We decode the subset as a binary word so that for example the subset $S=\{1,3 \} $ of $\{1,2,3 \}$ is decoded as 101.
Matching statistic: St001630
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ? = 2
([],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 2
([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> ? = 2
([(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6
([(1,2),(1,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,1),(0,2),(0,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(1,2),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(0,3),(3,1),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(1,3),(3,2)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> ? = 2
([(0,3),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 3
([(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 2
([(0,2),(0,3),(0,4),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6
([(0,1),(0,2),(0,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(4,1),(4,2)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 2
([(0,2),(0,3),(2,4),(3,4),(4,1)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
([(0,3),(0,4),(3,2),(4,1)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 3
([(0,2),(0,3),(2,4),(3,1),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(0,1),(0,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(2,3),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,28),(1,29),(1,30),(2,9),(2,13),(2,18),(2,19),(2,30),(3,8),(3,12),(3,16),(3,17),(3,30),(4,11),(4,15),(4,17),(4,19),(4,29),(5,10),(5,14),(5,16),(5,18),(5,29),(6,12),(6,13),(6,14),(6,15),(6,28),(7,8),(7,9),(7,10),(7,11),(7,28),(8,20),(8,21),(8,32),(9,22),(9,23),(9,32),(10,20),(10,22),(10,33),(11,21),(11,23),(11,33),(12,24),(12,25),(12,32),(13,26),(13,27),(13,32),(14,24),(14,26),(14,33),(15,25),(15,27),(15,33),(16,20),(16,24),(16,31),(17,21),(17,25),(17,31),(18,22),(18,26),(18,31),(19,23),(19,27),(19,31),(20,34),(21,34),(22,34),(23,34),(24,34),(25,34),(26,34),(27,34),(28,32),(28,33),(29,31),(29,33),(30,31),(30,32),(31,34),(32,34),(33,34)],35)
=> ? = 4
([(1,4),(4,2),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 2
([(0,4),(4,1),(4,2),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(1,4),(2,4),(4,3)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 2
([(0,4),(1,4),(4,2),(4,3)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,16),(1,21),(1,23),(2,8),(2,16),(2,20),(2,22),(3,10),(3,15),(3,20),(3,23),(4,11),(4,15),(4,21),(4,22),(5,13),(5,14),(5,22),(5,23),(6,12),(6,14),(6,20),(6,21),(7,8),(7,9),(7,10),(7,11),(7,12),(7,13),(8,17),(8,24),(8,26),(9,17),(9,25),(9,27),(10,18),(10,24),(10,27),(11,18),(11,25),(11,26),(12,19),(12,24),(12,25),(13,19),(13,26),(13,27),(14,19),(14,28),(15,18),(15,28),(16,17),(16,28),(17,29),(18,29),(19,29),(20,24),(20,28),(21,25),(21,28),(22,26),(22,28),(23,27),(23,28),(24,29),(25,29),(26,29),(27,29),(28,29)],30)
=> ? = 2
([(0,4),(1,4),(2,4),(4,3)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 3
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 2
([(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,4),(1,4),(2,3),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
([(0,4),(1,3),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,4),(1,4),(2,3),(2,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,10),(1,11),(2,8),(2,9),(2,11),(3,6),(3,7),(3,11),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,12),(7,12),(8,12),(9,12),(10,12),(11,12)],13)
=> ? = 6
([(1,4),(2,3)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,16),(1,20),(1,24),(1,30),(1,32),(2,15),(2,19),(2,23),(2,30),(2,31),(3,17),(3,21),(3,23),(3,29),(3,32),(4,18),(4,22),(4,24),(4,29),(4,31),(5,10),(5,13),(5,14),(5,17),(5,18),(5,30),(6,10),(6,11),(6,12),(6,15),(6,16),(6,29),(7,9),(7,11),(7,13),(7,19),(7,22),(7,32),(8,9),(8,12),(8,14),(8,20),(8,21),(8,31),(9,35),(9,36),(9,41),(10,33),(10,34),(10,41),(11,25),(11,38),(11,41),(12,26),(12,37),(12,41),(13,28),(13,39),(13,41),(14,27),(14,40),(14,41),(15,25),(15,33),(15,37),(16,26),(16,33),(16,38),(17,27),(17,34),(17,39),(18,28),(18,34),(18,40),(19,25),(19,35),(19,39),(20,26),(20,36),(20,40),(21,27),(21,36),(21,37),(22,28),(22,35),(22,38),(23,37),(23,39),(24,38),(24,40),(25,42),(26,42),(27,42),(28,42),(29,34),(29,37),(29,38),(30,33),(30,39),(30,40),(31,35),(31,37),(31,40),(32,36),(32,38),(32,39),(33,42),(34,42),(35,42),(36,42),(37,42),(38,42),(39,42),(40,42),(41,42)],43)
=> ? = 6
([(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,10),(1,11),(1,24),(1,26),(2,8),(2,9),(2,24),(2,25),(3,16),(3,17),(3,18),(3,19),(3,24),(4,9),(4,14),(4,15),(4,17),(4,26),(5,8),(5,12),(5,13),(5,16),(5,26),(6,11),(6,13),(6,15),(6,19),(6,25),(7,10),(7,12),(7,14),(7,18),(7,25),(8,27),(8,31),(9,28),(9,31),(10,29),(10,32),(11,30),(11,32),(12,20),(12,27),(12,29),(13,21),(13,27),(13,30),(14,22),(14,28),(14,29),(15,23),(15,28),(15,30),(16,20),(16,21),(16,31),(17,22),(17,23),(17,31),(18,20),(18,22),(18,32),(19,21),(19,23),(19,32),(20,33),(21,33),(22,33),(23,33),(24,31),(24,32),(25,27),(25,28),(25,32),(26,29),(26,30),(26,31),(27,33),(28,33),(29,33),(30,33),(31,33),(32,33)],34)
=> ? = 6
([(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 3
([(0,3),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,13),(1,14),(1,15),(1,17),(2,10),(2,11),(2,12),(2,17),(3,7),(3,8),(3,9),(3,17),(4,9),(4,12),(4,15),(4,16),(5,8),(5,11),(5,14),(5,16),(6,7),(6,10),(6,13),(6,16),(7,18),(7,21),(8,19),(8,21),(9,20),(9,21),(10,18),(10,22),(11,19),(11,22),(12,20),(12,22),(13,18),(13,23),(14,19),(14,23),(15,20),(15,23),(16,21),(16,22),(16,23),(17,18),(17,19),(17,20),(18,24),(19,24),(20,24),(21,24),(22,24),(23,24)],25)
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(4,2)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(3,2),(4,2)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,2),(1,3)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,9),(1,10),(1,28),(1,29),(2,12),(2,16),(2,20),(2,22),(2,29),(3,11),(3,15),(3,20),(3,21),(3,28),(4,13),(4,17),(4,19),(4,21),(4,29),(5,14),(5,18),(5,19),(5,22),(5,28),(6,8),(6,10),(6,15),(6,16),(6,17),(6,18),(7,8),(7,9),(7,11),(7,12),(7,13),(7,14),(8,27),(8,34),(8,35),(9,27),(9,30),(9,31),(10,27),(10,32),(10,33),(11,23),(11,30),(11,34),(12,24),(12,31),(12,34),(13,23),(13,31),(13,35),(14,24),(14,30),(14,35),(15,25),(15,32),(15,34),(16,26),(16,33),(16,34),(17,25),(17,33),(17,35),(18,26),(18,32),(18,35),(19,35),(19,36),(20,34),(20,36),(21,23),(21,25),(21,36),(22,24),(22,26),(22,36),(23,37),(24,37),(25,37),(26,37),(27,37),(28,30),(28,32),(28,36),(29,31),(29,33),(29,36),(30,37),(31,37),(32,37),(33,37),(34,37),(35,37),(36,37)],38)
=> ? = 3
([(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,7),(1,20),(1,21),(2,9),(2,14),(2,15),(2,21),(3,8),(3,12),(3,13),(3,21),(4,11),(4,13),(4,15),(4,20),(5,10),(5,12),(5,14),(5,20),(6,7),(6,8),(6,9),(6,10),(6,11),(7,22),(7,23),(8,16),(8,17),(8,22),(9,18),(9,19),(9,22),(10,16),(10,18),(10,23),(11,17),(11,19),(11,23),(12,16),(12,24),(13,17),(13,24),(14,18),(14,24),(15,19),(15,24),(16,25),(17,25),(18,25),(19,25),(20,23),(20,24),(21,22),(21,24),(22,25),(23,25),(24,25)],26)
=> ? = 3
([(0,2),(0,4),(3,1),(4,3)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 4
([(0,4),(1,2),(1,3),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(0,4),(1,2),(1,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(1,16),(1,17),(1,18),(1,25),(2,13),(2,14),(2,15),(2,25),(3,10),(3,11),(3,12),(3,25),(4,8),(4,9),(4,12),(4,15),(4,18),(5,7),(5,9),(5,11),(5,14),(5,17),(6,7),(6,8),(6,10),(6,13),(6,16),(7,21),(7,24),(7,29),(8,19),(8,22),(8,29),(9,20),(9,23),(9,29),(10,26),(10,29),(11,27),(11,29),(12,28),(12,29),(13,19),(13,21),(13,26),(14,20),(14,21),(14,27),(15,19),(15,20),(15,28),(16,22),(16,24),(16,26),(17,23),(17,24),(17,27),(18,22),(18,23),(18,28),(19,30),(20,30),(21,30),(22,30),(23,30),(24,30),(25,26),(25,27),(25,28),(26,30),(27,30),(28,30),(29,30)],31)
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,12),(1,16),(2,8),(2,11),(2,16),(3,7),(3,10),(3,16),(4,6),(4,10),(4,11),(4,12),(5,6),(5,7),(5,8),(5,9),(6,13),(6,14),(6,15),(7,13),(7,17),(8,14),(8,17),(9,15),(9,17),(10,13),(10,18),(11,14),(11,18),(12,15),(12,18),(13,19),(14,19),(15,19),(16,17),(16,18),(17,19),(18,19)],20)
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,7),(1,8),(2,6),(2,8),(3,5),(3,8),(4,5),(4,6),(4,7),(5,9),(6,9),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(1,2),(1,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,9),(1,10),(1,11),(1,15),(2,7),(2,8),(2,11),(2,14),(3,6),(3,8),(3,10),(3,13),(4,6),(4,7),(4,9),(4,12),(5,12),(5,13),(5,14),(5,15),(6,18),(6,22),(7,16),(7,22),(8,17),(8,22),(9,19),(9,22),(10,20),(10,22),(11,21),(11,22),(12,16),(12,18),(12,19),(13,17),(13,18),(13,20),(14,16),(14,17),(14,21),(15,19),(15,20),(15,21),(16,23),(17,23),(18,23),(19,23),(20,23),(21,23),(22,23)],24)
=> ? = 3
([(0,3),(0,4),(1,2),(2,3),(2,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ? = 2
([(0,3),(3,4),(4,1),(4,2)],5)
=> ([(3,4)],5)
=> ([(0,1)],2)
=> ? = 2
([(0,3),(1,2),(2,4),(3,4)],5)
=> ([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,8),(1,9),(1,10),(2,6),(2,7),(2,10),(3,5),(3,7),(3,9),(4,5),(4,6),(4,8),(5,11),(6,11),(7,11),(8,11),(9,11),(10,11)],12)
=> ? = 3
([(0,2),(0,3),(0,4),(2,5),(3,5),(4,5),(5,1)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(3,5),(4,5),(5,1),(5,2)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(1,4),(1,5),(2,4),(2,5),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(4,5),(5,1),(5,2),(5,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,5),(1,5),(2,5),(3,4),(5,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,5),(1,5),(4,2),(4,3),(5,4)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,5),(2,5),(3,5),(4,1),(4,2),(4,3)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,5),(3,5),(4,2),(4,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,4),(0,5),(1,4),(1,5),(3,2),(4,3),(5,3)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2)],6)
=> ([(2,5),(3,4)],6)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6),(6,1)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,6),(2,6),(3,5),(5,4),(6,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,6),(2,6),(3,6),(4,5),(5,1),(5,2),(5,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,6),(1,6),(2,5),(3,5),(4,2),(4,3),(6,4)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(0,4),(1,5),(2,5),(3,6),(4,6),(6,1),(6,2)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,6),(4,5),(5,2),(5,3),(6,4)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,6),(1,6),(2,5),(3,5),(5,4),(6,2),(6,3)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(0,6),(1,5),(1,6),(2,3),(4,2),(5,4),(6,4)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(0,4),(2,6),(3,6),(4,6),(5,1),(6,5)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(3,6),(4,6),(5,1),(5,2),(6,5)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,1),(3,2),(4,6),(5,6)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,6),(2,6),(5,1),(5,2),(6,3),(6,4)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
([(0,5),(1,6),(2,6),(3,6),(5,1),(5,2),(5,3),(6,4)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,5),(4,6),(5,4),(6,1),(6,2),(6,3)],7)
=> ([(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(1,5),(1,6),(2,5),(2,6),(3,4),(4,1),(4,2)],7)
=> ([(3,6),(4,5)],7)
=> ([(0,1),(0,2),(1,3),(2,3)],4)
=> 2
Description
The global dimension of the incidence algebra of the lattice over the rational numbers.
The following 12 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St000735The last entry on the main diagonal of a standard tableau. St001184Number of indecomposable injective modules with grade at least 1 in the corresponding Nakayama algebra. St001185The number of indecomposable injective modules of grade at least 2 in the corresponding Nakayama algebra. St001192The maximal dimension of $Ext_A^2(S,A)$ for a simple module $S$ over the corresponding Nakayama algebra $A$. St001215Let X be the direct sum of all simple modules of the corresponding Nakayama algebra. St001481The minimal height of a peak of a Dyck path. St001239The largest vector space dimension of the double dual of a simple module in the corresponding Nakayama algebra. St001473The absolute value of the sum of all entries of the Coxeter matrix of the corresponding LNakayama algebra. St001872The number of indecomposable injective modules with even projective dimension in the corresponding Nakayama algebra. St001803The maximal overlap of the cylindrical tableau associated with a tableau.
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