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Your data matches 80 different statistics following compositions of up to 3 maps.
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Matching statistic: St000667
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(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00202: Integer partitions —first row removal⟶ Integer partitions
St000667: 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
St000667: Integer partitions ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [1]
=> 1
([],3)
=> [1,1,1]
=> [1,1]
=> 1
([(1,2)],3)
=> [2,1]
=> [1]
=> 1
([],4)
=> [1,1,1,1]
=> [1,1,1]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [1,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [1,1,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [1,1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [1]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [1,1,1,1,1]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [1,1,1,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [2,1,1]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [1,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [3]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [2,2]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [2,1]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2]
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [1]
=> 1
Description
The greatest common divisor of the parts of the partition.
Matching statistic: St000993
(load all 4 compositions to match this statistic)
(load all 4 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
St000993: Integer partitions ⟶ ℤResult quality: 96% ●values known / values provided: 96%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [2]
=> 1
([],3)
=> [1,1,1]
=> [3]
=> 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 1
([],4)
=> [1,1,1,1]
=> [4]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [6]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [4,2]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,2,1]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [2,2,2]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [3,3]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,2,1]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> ? = 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> ? = 1
Description
The multiplicity of the largest part of an integer partition.
Matching statistic: St000326
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00104: Binary words —reverse⟶ Binary words
St000326: Binary words ⟶ ℤResult quality: 95% ●values known / values provided: 95%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> 110 => 011 => 2 = 1 + 1
([],3)
=> [1,1,1]
=> 1110 => 0111 => 2 = 1 + 1
([(1,2)],3)
=> [2,1]
=> 1010 => 0101 => 2 = 1 + 1
([],4)
=> [1,1,1,1]
=> 11110 => 01111 => 2 = 1 + 1
([(2,3)],4)
=> [2,1,1]
=> 10110 => 01101 => 2 = 1 + 1
([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 01001 => 2 = 1 + 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => 0011 => 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => 01001 => 2 = 1 + 1
([],5)
=> [1,1,1,1,1]
=> 111110 => 011111 => 2 = 1 + 1
([(3,4)],5)
=> [2,1,1,1]
=> 101110 => 011101 => 2 = 1 + 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => 01011 => 2 = 1 + 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => 011001 => 2 = 1 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => 00101 => 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => 010001 => 2 = 1 + 1
([],6)
=> [1,1,1,1,1,1]
=> 1111110 => 0111111 => 2 = 1 + 1
([(4,5)],6)
=> [2,1,1,1,1]
=> 1011110 => 0111101 => 2 = 1 + 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => 0111001 => 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => 0110001 => 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> 110110 => 011011 => 2 = 1 + 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 1000110 => 0110001 => 2 = 1 + 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => 0111001 => 2 = 1 + 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> 100100 => 001001 => 3 = 2 + 1
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => 0110001 => 2 = 1 + 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 1000110 => 0110001 => 2 = 1 + 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> 11000 => 00011 => 4 = 3 + 1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => 0110001 => 2 = 1 + 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> 11100 => 00111 => 3 = 2 + 1
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> 100100 => 001001 => 3 = 2 + 1
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 101010 => 010101 => 2 = 1 + 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> 100100 => 001001 => 3 = 2 + 1
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> 1000010 => 0100001 => 2 = 1 + 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> 10000000000000110 => ? => ? = 1 + 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> 1000000000110 => ? => ? = 1 + 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> 1000000000000010 => ? => ? = 1 + 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> 1000000001110 => ? => ? = 1 + 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 1 + 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> 1000000000011110 => ? => ? = 1 + 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 1 + 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> 10000000000001110 => ? => ? = 1 + 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> 100000000110 => 011000000001 => ? = 1 + 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> 10000000000110 => ? => ? = 1 + 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> 1000000000000110 => ? => ? = 1 + 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> 100000001110 => 011100000001 => ? = 1 + 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> 1000000000001110 => ? => ? = 1 + 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 1 + 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> 100000000000110 => ? => ? = 1 + 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> 100000001110 => 011100000001 => ? = 1 + 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 1 + 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> 100000000011110 => ? => ? = 1 + 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> 100000000000110 => ? => ? = 1 + 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> 100000000000000110 => ? => ? = 1 + 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 1 + 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 1 + 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> 1000000000000110 => ? => ? = 1 + 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> 1000000000110 => ? => ? = 1 + 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
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00044: Integer partitions —conjugate⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
St000297: Binary words ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [2]
=> 100 => 1
([],3)
=> [1,1,1]
=> [3]
=> 1000 => 1
([(1,2)],3)
=> [2,1]
=> [2,1]
=> 1010 => 1
([],4)
=> [1,1,1,1]
=> [4]
=> 10000 => 1
([(2,3)],4)
=> [2,1,1]
=> [3,1]
=> 10010 => 1
([(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [2,2]
=> 1100 => 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [2,1,1]
=> 10110 => 1
([],5)
=> [1,1,1,1,1]
=> [5]
=> 100000 => 1
([(3,4)],5)
=> [2,1,1,1]
=> [4,1]
=> 100010 => 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [3,2]
=> 10100 => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [3,1,1]
=> 100110 => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [2,2,1]
=> 11010 => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [2,1,1,1]
=> 101110 => 1
([],6)
=> [1,1,1,1,1,1]
=> [6]
=> 1000000 => 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [5,1]
=> 1000010 => 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [4,2]
=> 100100 => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,2,1]
=> 101010 => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [4,1,1]
=> 1000110 => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [2,2,2]
=> 11100 => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [3,1,1,1]
=> 1001110 => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [3,3]
=> 11000 => 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [3,2,1]
=> 101010 => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [2,2,1,1]
=> 110110 => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [2,1,1,1,1]
=> 1011110 => 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 10011111111111110 => ? = 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> 1001111111110 => ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [2,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1011111111111110 => ? = 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [4,1,1,1,1,1,1,1,1]
=> 1000111111110 => ? = 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> 100011111111110 => ? = 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1,1]
=> 1000011111111110 => ? = 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> 100011111111110 => ? = 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1,1]
=> 10001111111111110 => ? = 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [3,1,1,1,1,1,1,1,1]
=> 100111111110 => ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1]
=> 10011111111110 => ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1001111111111110 => ? = 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> 100011111110 => ? = 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1,1]
=> 1000111111111110 => ? = 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> 10001111111110 => ? = 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [6,1,1,1,1,1]
=> 100000111110 => ? = 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> 100111111111110 => ? = 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [4,1,1,1,1,1,1,1]
=> 100011111110 => ? = 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> 10001111111110 => ? = 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [5,1,1,1,1,1,1,1,1,1]
=> 100001111111110 => ? = 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1]
=> 100111111111110 => ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1,1,1]
=> 100111111111111110 => ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1,1]
=> 100011111111110 => ? = 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [4,1,1,1,1,1,1,1,1,1]
=> 10001111111110 => ? = 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [3,1,1,1,1,1,1,1,1,1,1,1,1]
=> 1001111111111110 => ? = 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [3,1,1,1,1,1,1,1,1,1]
=> 1001111111110 => ? = 1
Description
The number of leading ones in a binary word.
Matching statistic: St000382
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
([(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> [1,2] => 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [1,1,2] => 1
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [1,3] => 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [1,1,1,2] => 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [1,2,2] => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [1,1,3] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [2,3] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [1,4] => 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [1,1,1,1,2] => 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [1,1,2,2] => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [1,1,1,3] => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [1,1,4] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [1,2,3] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [2,4] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [1,5] => 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3]]
=> ? => ? = 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ? => ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,3,4,5,6,7,8,9,10,11,12,13,14,15],[2]]
=> ? => ? = 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,5,6,7,8,9,10,11,12],[2],[3],[4]]
=> ? => ? = 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? => ? = 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14,15],[2],[3],[4],[5]]
=> ? => ? = 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? => ? = 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3],[4]]
=> ? => ? = 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ? => ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13],[2],[3]]
=> ? => ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ? => ? = 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? = 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15],[2],[3],[4]]
=> ? => ? = 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? => ? = 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ? => ? = 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ? => ? = 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ? => ? = 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? => ? = 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14],[2],[3],[4],[5]]
=> ? => ? = 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ? => ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[2],[3]]
=> ? => ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ? => ? = 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ? => ? = 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ? => ? = 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ? => ? = 1
Description
The first part of an integer composition.
Matching statistic: St000383
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00095: Integer partitions —to binary word⟶ Binary words
Mp00097: Binary words —delta morphism⟶ Integer compositions
St000383: Integer compositions ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> 110 => [2,1] => 1
([],3)
=> [1,1,1]
=> 1110 => [3,1] => 1
([(1,2)],3)
=> [2,1]
=> 1010 => [1,1,1,1] => 1
([],4)
=> [1,1,1,1]
=> 11110 => [4,1] => 1
([(2,3)],4)
=> [2,1,1]
=> 10110 => [1,1,2,1] => 1
([(1,3),(2,3)],4)
=> [3,1]
=> 10010 => [1,2,1,1] => 1
([(0,3),(1,2)],4)
=> [2,2]
=> 1100 => [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> 10010 => [1,2,1,1] => 1
([],5)
=> [1,1,1,1,1]
=> 111110 => [5,1] => 1
([(3,4)],5)
=> [2,1,1,1]
=> 101110 => [1,1,3,1] => 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => [1,3,1,1] => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> 11010 => [2,1,1,1] => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> 100010 => [1,3,1,1] => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => [1,1,1,2] => 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> 100110 => [1,2,2,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => [1,3,1,1] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> 100010 => [1,3,1,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => [1,3,1,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> 10100 => [1,1,1,2] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> 100010 => [1,3,1,1] => 1
([],6)
=> [1,1,1,1,1,1]
=> 1111110 => [6,1] => 1
([(4,5)],6)
=> [2,1,1,1,1]
=> 1011110 => [1,1,4,1] => 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> 110110 => [2,1,2,1] => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> 1001110 => [1,2,3,1] => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> 100100 => [1,2,1,2] => 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> 11000 => [2,3] => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> 1000110 => [1,3,2,1] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> 11100 => [3,2] => 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> 100100 => [1,2,1,2] => 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> 101010 => [1,1,1,1,1,1] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> 100100 => [1,2,1,2] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> 1000010 => [1,4,1,1] => 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> 10000000000000110 => ? => ? = 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> 1000000000110 => ? => ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> 1000000000000010 => ? => ? = 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> 1000000001110 => ? => ? = 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> 1000000000011110 => ? => ? = 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> 10000000000001110 => ? => ? = 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> 100000000110 => ? => ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> 10000000000110 => ? => ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> 1000000000000110 => ? => ? = 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> 100000001110 => ? => ? = 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> 1000000000001110 => ? => ? = 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> 100000111110 => [1,5,5,1] => ? = 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> 100000000000110 => ? => ? = 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> 100000001110 => ? => ? = 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> 100000000011110 => ? => ? = 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> 100000000000110 => ? => ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> 100000000000000110 => ? => ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> 100000000001110 => ? => ? = 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> 10000000001110 => ? => ? = 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> 1000000000000110 => ? => ? = 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> 1000000000110 => ? => ? = 1
Description
The last part of an integer composition.
Matching statistic: St000733
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000733: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [[1,3],[2]]
=> 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [[1,3,4],[2]]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [[1,4],[2],[3]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [[1,3,4,5],[2]]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [[1,3,5],[2,4]]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [[1,4,5],[2],[3]]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [[1,4],[2,5],[3]]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [[1,5],[2],[3],[4]]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [[1,3,4,5,6],[2]]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [[1,3,5,6],[2,4]]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [[1,4,5,6],[2],[3]]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [[1,5,6],[2],[3],[4]]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [[1,4,6],[2,5],[3]]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [[1,5],[2,6],[3],[4]]
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [[1,6],[2],[3],[4],[5]]
=> 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15],[16]]
=> ?
=> ? = 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ?
=> ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15]]
=> ?
=> ? = 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
=> [[1,10,11,12],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14],[15]]
=> ?
=> ? = 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15],[16]]
=> ?
=> ? = 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> [[1,10,11],[2],[3],[4],[5],[6],[7],[8],[9]]
=> ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13]]
=> ?
=> ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ?
=> ? = 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14],[15]]
=> ?
=> ? = 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ?
=> ? = 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ?
=> ? = 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> [[1,9,10,11],[2],[3],[4],[5],[6],[7],[8]]
=> ? = 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13],[14]]
=> ?
=> ? = 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ?
=> ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[16],[17]]
=> ?
=> ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ?
=> ? = 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ?
=> ? = 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ?
=> ? = 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ?
=> ? = 1
Description
The row containing the largest entry of a standard tableau.
Matching statistic: St000745
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00045: Integer partitions —reading tableau⟶ Standard tableaux
Mp00084: Standard tableaux —conjugate⟶ Standard tableaux
St000745: Standard tableaux ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [[1],[2]]
=> [[1,2]]
=> 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [[1,2,3]]
=> 1
([(1,2)],3)
=> [2,1]
=> [[1,3],[2]]
=> [[1,2],[3]]
=> 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [[1,2,3,4]]
=> 1
([(2,3)],4)
=> [2,1,1]
=> [[1,4],[2],[3]]
=> [[1,2,3],[4]]
=> 1
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [[1,3],[2,4]]
=> 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,3,4],[2]]
=> [[1,2],[3],[4]]
=> 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [[1,2,3,4,5]]
=> 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,5],[2],[3],[4]]
=> [[1,2,3,4],[5]]
=> 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,3],[2,5],[4]]
=> [[1,2,4],[3,5]]
=> 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,4,5],[2],[3]]
=> [[1,2,3],[4],[5]]
=> 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,5],[3,4]]
=> [[1,3],[2,4],[5]]
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,3,4,5],[2]]
=> [[1,2],[3],[4],[5]]
=> 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [[1,2,3,4,5,6]]
=> 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,6],[2],[3],[4],[5]]
=> [[1,2,3,4,5],[6]]
=> 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,4],[2,6],[3],[5]]
=> [[1,2,3,5],[4,6]]
=> 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,5,6],[2],[3],[4]]
=> [[1,2,3,4],[5],[6]]
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [[1,4],[2,5],[3,6]]
=> 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,4,5,6],[2],[3]]
=> [[1,2,3],[4],[5],[6]]
=> 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [[1,3,5],[2,4,6]]
=> 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,3,6],[2,5],[4]]
=> [[1,2,4],[3,5],[6]]
=> 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,5,6],[3,4]]
=> [[1,3],[2,4],[5],[6]]
=> 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [[1,3,4,5,6],[2]]
=> [[1,2],[3],[4],[5],[6]]
=> 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3]]
=> ?
=> ? = 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ?
=> ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,3,4,5,6,7,8,9,10,11,12,13,14,15],[2]]
=> ?
=> ? = 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,5,6,7,8,9,10,11,12],[2],[3],[4]]
=> ?
=> ? = 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ?
=> ? = 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14,15],[2],[3],[4],[5]]
=> ?
=> ? = 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ?
=> ? = 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15,16],[2],[3],[4]]
=> ?
=> ? = 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,4,5,6,7,8,9,10,11],[2],[3]]
=> ?
=> ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13],[2],[3]]
=> ?
=> ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ?
=> ? = 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ?
=> ? = 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14,15],[2],[3],[4]]
=> ?
=> ? = 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ?
=> ? = 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,7,8,9,10,11],[2],[3],[4],[5],[6]]
=> ?
=> ? = 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ?
=> ? = 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,5,6,7,8,9,10,11],[2],[3],[4]]
=> ?
=> ? = 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ?
=> ? = 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,6,7,8,9,10,11,12,13,14],[2],[3],[4],[5]]
=> ?
=> ? = 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14],[2],[3]]
=> ?
=> ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15,16,17],[2],[3]]
=> ?
=> ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13,14],[2],[3],[4]]
=> ?
=> ? = 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,5,6,7,8,9,10,11,12,13],[2],[3],[4]]
=> ?
=> ? = 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,4,5,6,7,8,9,10,11,12,13,14,15],[2],[3]]
=> ?
=> ? = 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,4,5,6,7,8,9,10,11,12],[2],[3]]
=> ?
=> ? = 1
Description
The index of the last row whose first entry is the row number in a standard Young tableau.
Matching statistic: St000996
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00081: Standard tableaux —reading word permutation⟶ Permutations
St000996: Permutations ⟶ ℤResult quality: 94% ●values known / values provided: 94%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [[1],[2]]
=> [2,1] => 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [3,2,1] => 1
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [3,1,2] => 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [4,3,2,1] => 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [4,3,1,2] => 1
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [3,4,1,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [4,1,2,3] => 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [5,4,3,2,1] => 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [5,4,3,1,2] => 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [5,3,4,1,2] => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [5,4,1,2,3] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [4,5,1,2,3] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [5,1,2,3,4] => 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [6,5,4,3,2,1] => 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [6,5,4,3,1,2] => 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [6,5,3,4,1,2] => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [6,5,4,1,2,3] => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [4,5,6,1,2,3] => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [6,5,1,2,3,4] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [5,6,3,4,1,2] => 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [6,4,5,1,2,3] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [5,6,1,2,3,4] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [6,1,2,3,4,5] => 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15],[16]]
=> ? => ? = 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ? => ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15]]
=> ? => ? = 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
=> ? => ? = 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14],[15]]
=> ? => ? = 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15],[16]]
=> ? => ? = 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13]]
=> ? => ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ? => ? = 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14],[15]]
=> ? => ? = 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ? => ? = 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13],[14]]
=> ? => ? = 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ? => ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[16],[17]]
=> ? => ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ? => ? = 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ? => ? = 1
Description
The number of exclusive left-to-right maxima of a permutation.
This is the number of left-to-right maxima that are not right-to-left minima.
Matching statistic: St000657
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Mp00037: Graphs —to partition of connected components⟶ Integer partitions
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Mp00042: Integer partitions —initial tableau⟶ Standard tableaux
Mp00207: Standard tableaux —horizontal strip sizes⟶ Integer compositions
St000657: Integer compositions ⟶ ℤResult quality: 93% ●values known / values provided: 93%●distinct values known / distinct values provided: 100%
Values
([],2)
=> [1,1]
=> [[1],[2]]
=> [1,1] => 1
([],3)
=> [1,1,1]
=> [[1],[2],[3]]
=> [1,1,1] => 1
([(1,2)],3)
=> [2,1]
=> [[1,2],[3]]
=> [2,1] => 1
([],4)
=> [1,1,1,1]
=> [[1],[2],[3],[4]]
=> [1,1,1,1] => 1
([(2,3)],4)
=> [2,1,1]
=> [[1,2],[3],[4]]
=> [2,1,1] => 1
([(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
([(0,3),(1,2)],4)
=> [2,2]
=> [[1,2],[3,4]]
=> [2,2] => 2
([(1,2),(1,3),(2,3)],4)
=> [3,1]
=> [[1,2,3],[4]]
=> [3,1] => 1
([],5)
=> [1,1,1,1,1]
=> [[1],[2],[3],[4],[5]]
=> [1,1,1,1,1] => 1
([(3,4)],5)
=> [2,1,1,1]
=> [[1,2],[3],[4],[5]]
=> [2,1,1,1] => 1
([(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
([(1,4),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
([(1,4),(2,3)],5)
=> [2,2,1]
=> [[1,2],[3,4],[5]]
=> [2,2,1] => 1
([(1,4),(2,3),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
([(0,1),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
([(2,3),(2,4),(3,4)],5)
=> [3,1,1]
=> [[1,2,3],[4],[5]]
=> [3,1,1] => 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> [3,2]
=> [[1,2,3],[4,5]]
=> [3,2] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [4,1]
=> [[1,2,3,4],[5]]
=> [4,1] => 1
([],6)
=> [1,1,1,1,1,1]
=> [[1],[2],[3],[4],[5],[6]]
=> [1,1,1,1,1,1] => 1
([(4,5)],6)
=> [2,1,1,1,1]
=> [[1,2],[3],[4],[5],[6]]
=> [2,1,1,1,1] => 1
([(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
([(2,5),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(2,5),(3,4)],6)
=> [2,2,1,1]
=> [[1,2],[3,4],[5],[6]]
=> [2,2,1,1] => 1
([(2,5),(3,4),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
([(1,2),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
([(3,4),(3,5),(4,5)],6)
=> [3,1,1,1]
=> [[1,2,3],[4],[5],[6]]
=> [3,1,1,1] => 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(0,1),(2,5),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(2,4),(2,5),(3,4),(3,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
([(0,5),(1,5),(2,4),(3,4)],6)
=> [3,3]
=> [[1,2,3],[4,5,6]]
=> [3,3] => 3
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,1,1]
=> [[1,2,3,4],[5],[6]]
=> [4,1,1] => 1
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(0,5),(1,4),(2,3)],6)
=> [2,2,2]
=> [[1,2],[3,4],[5,6]]
=> [2,2,2] => 2
([(1,5),(2,4),(3,4),(3,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(0,1),(2,5),(3,4),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 2
([(1,2),(3,4),(3,5),(4,5)],6)
=> [3,2,1]
=> [[1,2,3],[4,5],[6]]
=> [3,2,1] => 1
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)
=> [4,2]
=> [[1,2,3,4],[5,6]]
=> [4,2] => 2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [5,1]
=> [[1,2,3,4,5],[6]]
=> [5,1] => 1
([(1,2),(1,7),(1,9),(2,6),(2,8),(3,4),(3,6),(3,8),(3,9),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,9),(7,8),(8,9)],10)
=> [9,1]
=> [[1,2,3,4,5,6,7,8,9],[10]]
=> [9,1] => ? = 1
([(2,10),(3,6),(3,10),(3,13),(4,9),(4,11),(4,14),(4,15),(5,12),(5,13),(5,14),(5,15),(6,12),(6,14),(6,15),(7,8),(7,9),(7,12),(7,14),(7,15),(8,11),(8,13),(8,14),(8,15),(9,11),(9,13),(9,15),(10,12),(10,14),(10,15),(11,12),(11,14),(11,15),(12,13),(13,14),(13,15)],16)
=> [14,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15],[16]]
=> ? => ? = 1
([(2,9),(2,10),(2,11),(3,4),(3,5),(3,8),(3,11),(4,5),(4,7),(4,10),(5,6),(5,9),(6,7),(6,8),(6,10),(6,11),(7,8),(7,9),(7,11),(8,9),(8,10),(9,10),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ? => ? = 1
([(1,2),(1,10),(1,12),(1,14),(2,9),(2,11),(2,13),(3,7),(3,8),(3,11),(3,12),(3,13),(3,14),(4,9),(4,10),(4,11),(4,12),(4,13),(4,14),(5,6),(5,8),(5,9),(5,11),(5,12),(5,13),(5,14),(6,7),(6,10),(6,11),(6,12),(6,13),(6,14),(7,8),(7,9),(7,11),(7,13),(7,14),(8,10),(8,12),(8,13),(8,14),(9,10),(9,12),(9,14),(10,11),(10,13),(11,12),(11,14),(12,13),(13,14)],15)
=> [14,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14],[15]]
=> ? => ? = 1
([(4,9),(5,8),(6,7),(7,9),(8,9)],10)
=> [6,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10]]
=> [6,1,1,1,1] => ? = 1
([(3,9),(4,5),(4,11),(5,10),(6,10),(6,11),(7,8),(7,11),(8,9),(8,10),(9,11),(10,11)],12)
=> [9,1,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11],[12]]
=> ? => ? = 1
([(3,11),(4,10),(5,8),(5,13),(6,9),(6,13),(7,12),(7,13),(8,10),(8,12),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 1
([(4,12),(5,11),(6,13),(6,14),(7,9),(7,14),(8,10),(8,14),(9,11),(9,13),(10,12),(10,13),(11,14),(12,14),(13,14)],15)
=> [11,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14],[15]]
=> ? => ? = 1
([(3,6),(3,9),(4,5),(4,9),(5,8),(6,8),(7,8),(7,9),(8,9)],10)
=> [7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [7,1,1,1] => ? = 1
([(3,12),(3,13),(4,5),(4,13),(5,12),(6,9),(6,10),(6,11),(7,8),(7,10),(7,11),(7,12),(8,9),(8,11),(8,13),(9,10),(9,12),(10,13),(11,12),(11,13),(12,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 1
([(3,13),(4,14),(4,15),(5,6),(5,15),(6,14),(7,10),(7,11),(7,12),(7,15),(8,9),(8,11),(8,12),(8,13),(9,10),(9,12),(9,15),(10,11),(10,13),(10,14),(11,14),(11,15),(12,13),(12,14),(13,15),(14,15)],16)
=> [13,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15],[16]]
=> ? => ? = 1
([(2,6),(2,10),(3,7),(3,8),(3,9),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(5,10),(6,7),(6,8),(6,9),(7,10),(8,10),(9,10)],11)
=> [9,1,1]
=> [[1,2,3,4,5,6,7,8,9],[10],[11]]
=> ? => ? = 1
([(2,9),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,6),(5,8),(5,9),(6,10),(6,11),(6,12),(7,8),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(9,10),(9,11),(9,12)],13)
=> [11,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13]]
=> ? => ? = 1
([(2,9),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,9),(7,9),(8,9)],10)
=> [8,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10]]
=> [8,1,1] => ? = 1
([(2,9),(2,10),(2,11),(2,14),(3,6),(3,7),(3,8),(3,13),(4,6),(4,7),(4,8),(4,13),(4,14),(5,9),(5,10),(5,11),(5,13),(5,14),(6,9),(6,10),(6,11),(6,12),(6,14),(7,9),(7,10),(7,11),(7,12),(7,14),(8,9),(8,10),(8,11),(8,12),(8,14),(9,12),(9,13),(10,12),(10,13),(11,12),(11,13),(12,13),(12,14),(13,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ? => ? = 1
([(3,10),(4,9),(5,8),(5,9),(6,7),(6,10),(7,8),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 1
([(3,11),(4,9),(4,14),(5,6),(5,11),(5,13),(6,12),(6,14),(7,12),(7,13),(7,14),(8,10),(8,13),(8,14),(9,10),(9,13),(10,12),(10,14),(11,12),(11,14),(12,13),(13,14)],15)
=> [12,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14],[15]]
=> ? => ? = 1
([(3,8),(3,12),(4,7),(4,11),(5,9),(5,11),(5,12),(6,10),(6,11),(6,12),(7,9),(7,12),(8,10),(8,11),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 1
([(5,10),(6,9),(7,8),(8,10),(9,10)],11)
=> [6,1,1,1,1,1]
=> [[1,2,3,4,5,6],[7],[8],[9],[10],[11]]
=> ? => ? = 1
([(2,9),(2,13),(3,10),(3,11),(3,12),(4,10),(4,11),(4,12),(5,7),(5,8),(5,9),(5,13),(6,7),(6,10),(6,11),(6,12),(6,13),(7,10),(7,11),(7,12),(8,10),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(10,13),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ? => ? = 1
([(3,8),(4,10),(5,9),(6,7),(6,10),(7,9),(8,10),(9,10)],11)
=> [8,1,1,1]
=> [[1,2,3,4,5,6,7,8],[9],[10],[11]]
=> ? => ? = 1
([(3,12),(4,11),(5,7),(6,8),(7,11),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 1
([(4,11),(5,10),(6,12),(7,13),(8,9),(8,12),(9,13),(10,12),(11,13),(12,13)],14)
=> [10,1,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13],[14]]
=> ? => ? = 1
([(2,4),(2,13),(3,11),(3,12),(3,13),(4,11),(4,12),(5,8),(5,9),(5,10),(5,13),(6,8),(6,9),(6,10),(6,13),(7,8),(7,9),(7,10),(7,12),(7,13),(8,11),(8,12),(9,11),(9,12),(10,11),(10,12),(11,13),(12,13)],14)
=> [12,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12],[13],[14]]
=> ? => ? = 1
([(2,4),(2,15),(3,9),(3,15),(3,16),(4,9),(4,16),(5,11),(5,12),(5,13),(5,16),(6,11),(6,12),(6,13),(6,16),(7,10),(7,14),(7,15),(7,16),(8,11),(8,12),(8,13),(8,14),(8,16),(9,10),(9,14),(9,15),(10,11),(10,12),(10,13),(10,16),(11,14),(11,15),(12,14),(12,15),(13,14),(13,15),(14,16),(15,16)],17)
=> [15,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13,14,15],[16],[17]]
=> ? => ? = 1
([(3,8),(4,6),(4,9),(5,7),(5,9),(6,7),(6,8),(7,9),(8,9)],10)
=> [7,1,1,1]
=> [[1,2,3,4,5,6,7],[8],[9],[10]]
=> [7,1,1,1] => ? = 1
([(3,11),(3,12),(3,13),(4,6),(4,8),(4,10),(5,9),(5,11),(5,12),(5,13),(6,7),(6,8),(6,9),(7,10),(7,11),(7,12),(7,13),(8,11),(8,12),(8,13),(9,10),(9,11),(9,12),(9,13),(10,11),(10,12),(10,13)],14)
=> [11,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11],[12],[13],[14]]
=> ? => ? = 1
([(3,4),(3,12),(4,11),(5,11),(5,12),(6,9),(6,10),(7,8),(7,10),(7,11),(8,9),(8,12),(9,10),(9,11),(10,12),(11,12)],13)
=> [10,1,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12],[13]]
=> ? => ? = 1
([(2,5),(2,12),(2,13),(2,14),(3,4),(3,9),(3,10),(3,11),(4,12),(4,13),(4,14),(5,9),(5,10),(5,11),(6,9),(6,10),(6,11),(6,12),(6,13),(6,14),(7,9),(7,10),(7,11),(7,12),(7,13),(7,14),(8,9),(8,10),(8,11),(8,12),(8,13),(8,14),(9,12),(9,13),(9,14),(10,12),(10,13),(10,14),(11,12),(11,13),(11,14)],15)
=> [13,1,1]
=> [[1,2,3,4,5,6,7,8,9,10,11,12,13],[14],[15]]
=> ? => ? = 1
([(2,11),(3,7),(3,11),(4,8),(4,9),(4,10),(5,8),(5,9),(5,10),(6,8),(6,9),(6,10),(7,8),(7,9),(7,10),(8,11),(9,11),(10,11)],12)
=> [10,1,1]
=> [[1,2,3,4,5,6,7,8,9,10],[11],[12]]
=> ? => ? = 1
Description
The smallest part of an integer composition.
The following 70 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001803The maximal overlap of the cylindrical tableau associated with a tableau. St001038The minimal height of a column in the parallelogram polyomino associated with the Dyck path. St001075The minimal size of a block of a set partition. St001107The number of times one can erase the first up and the last down step in a Dyck path and still remain a Dyck path. St001829The common independence number of a graph. St001119The length of a shortest maximal path in a graph. St001322The size of a minimal independent dominating set in a graph. St001232The number of indecomposable modules with projective dimension 2 for Nakayama algebras with global dimension at most 2. St001316The domatic number of a graph. St000990The first ascent of a permutation. St001264The smallest index i such that the i-th simple module has projective dimension equal to the global dimension of the corresponding Nakayama algebra. 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. St001481The minimal height of a peak of a Dyck path. St001571The Cartan determinant of the integer partition. St001392The largest nonnegative integer which is not a part and is smaller than the largest part of the partition. St000310The minimal degree of a vertex of a graph. St000264The girth of a graph, which is not a tree. St000456The monochromatic index of a connected graph. St001592The maximal number of simple paths between any two different vertices of a graph. St000379The number of Hamiltonian cycles in a graph. St000455The second largest eigenvalue of a graph if it is integral. St000654The first descent of a permutation. St000090The variation of a composition. St000284The Plancherel distribution on integer partitions. St000314The number of left-to-right-maxima of a permutation. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000706The product of the factorials of the multiplicities of an integer partition. St000781The number of proper colouring schemes of a Ferrers diagram. St000813The number of zero-one matrices with weakly decreasing column sums and row sums given by the partition. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St001568The smallest positive integer that does not appear twice in the partition. St001603The number of colourings of a polygon such that the multiplicities of a colour are given by a partition. St001604The multiplicity of the irreducible representation corresponding to a partition in the relabelling action on polygons. St001605The number of colourings of a cycle such that the multiplicities of colours are given by a partition. St001765The number of connected components of the friends and strangers graph. St000567The sum of the products of all pairs of parts. St000699The toughness times the least common multiple of 1,. St000929The constant term of the character polynomial of an integer partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St001097The coefficient of the monomial symmetric function indexed by the partition in the formal group law for linear orders. St001098The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for vertex labelled trees. St001099The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled binary trees. St001100The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for leaf labelled trees. St001101The coefficient times the product of the factorials of the parts of the monomial symmetric function indexed by the partition in the formal group law for increasing trees. St001123The multiplicity of the dual of the standard representation in the Kronecker square corresponding to a partition. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition. St001586The number of odd parts smaller than the largest even part in an integer partition. St001657The number of twos in an integer partition. St001118The acyclic chromatic index of a graph. St001281The normalized isoperimetric number of a graph. St000464The Schultz index of a connected graph. St001545The second Elser number of a connected graph. St001570The minimal number of edges to add to make a graph Hamiltonian. St001704The size of the largest multi-subset-intersection of the deck of a graph with the deck of another graph. St001613The binary logarithm of the size of the center of a lattice. St001719The number of shortest chains of small intervals from the bottom to the top in a lattice. St001881The number of factors of a lattice as a Cartesian product of lattices. St001845The number of join irreducibles minus the rank of a lattice. St001681The number of inclusion-wise minimal subsets of a lattice, whose meet is the bottom element. St001677The number of non-degenerate subsets of a lattice whose meet is the bottom element. St001621The number of atoms of a lattice. St001877Number of indecomposable injective modules with projective dimension 2. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St000261The edge connectivity of a graph. St001060The distinguishing index of a graph.
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